# Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

# (3+1)-Dimensional Nonlinear Equations and Couplings of Fifth-Order Equations in the Solitary Waves Theory: Multiple Soliton Solutions

• Abdul-Majid Wazwaz
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_5-7

## Keywords

Multiple Soliton Solutions Hirota Bilinear Method Wave Front Solutions Positon Solutions Peakons
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Glossary

### Solitons

Solitons appear as a result of a balance between a weakly nonlinear convection and a linear dispersion. The soliton is a localized highly stable wave that retains its identity – shape and speed – upon interaction and resembles particle-like behavior. It is a localized solitary wave so that it decays or approaches a constant at infinity. In the case of a collision, solitons undergo a phase shift. The stability of solitons stems from the delicate equilibrium between the two effects of nonlinearity and dispersion.

### Types of Traveling Waves

Traveling waves appear in many scientific and engineering applications in solitary wave theory. Solitons, kinks, peakons, cuspons, compactons, negatons, positons, complexitons, and others are examples of solitary waves. Solitons are localized wave packets which are asymptotically zero at large distances with exponential wings or tails. Kink waves are solitons that rise or descend from one asymptotic state to another, and hence another type of traveling waves as in the case of the Burgers hierarchy. Peakons, another type of traveling waves, are peaked solitary wave solutions. For peakons, such as in the case of Camassa-Holm equation, the traveling wave solutions are smooth except for a peak at a corner of its crest. Peakons are the points at which spatial derivative changes sign so that peakons have a finite jump in first derivative of the solution. Cuspons are other forms of solitons where solution exhibits cusps at their crests. Unlike peakons where the derivatives at the peak differ only by a sign, the derivatives at the jump of a cuspon diverges. The compactons are solitons with compact spatial support such that each compacton is a soliton confined to a finite core or a soliton without exponential tails or wings (Wazwaz 2002).

A new classification of solutions of soliton equations, both without and with self-consistent sources, is now used. This classification depends mainly on the property of the associated spectral parameters λ of the Lax pair of each equation (Liu et al. 2004; Ma and Fuchssteiner 1996; Ma et al. 2011). Negaton solution is related to the negative spectral parameter, i.e., for λ < 0. Negaton solutions, like solitons, contain exponential functions of the space variable but not trigonometric functions. Positon solution is related to the positive spectral parameter, i.e., for λ > 0. Positon solution is slowly decreasing and oscillating. Unlike soliton and negaton, positon solution contains trigonometric functions of the space variable but not exponential functions. However, the complexiton solution is related to the complex spectral parameter. Complexiton solution is usually expressed by the combinations of trigonometric functions and hyperbolic functions of the space variable. We recall that if the spectral parameter is zero, i.e., for λ = 0, the resulting solution is a soliton expressed in terms of exponential functions (Geng 2003; Shen and Tu 2011; Zhang and Tam 2010; Zhaqilao 2012).

It is well known that the celebrated modified KdV (mKdV) equation
$${u}_t+\frac{3}{2}{u}^2{u}_x+\frac{1}{4}{u}_{xxx}=0,$$
(1)
is completely integrable and gives multiple soliton solutions. However, in Zhaqilao (2012) the following negaton solution
$$u\left(x,t\right)=-\frac{3{e}^{\frac{11\sqrt{5}}{432}t}+3{e}^{\frac{\sqrt{5}}{3}x}-14{e^{\frac{\sqrt{5}}{864}}}^{\left(144x+11t\right)}}{18{e}^{\frac{11\sqrt{5}}{432}}+18{e}^{\frac{\sqrt{5}}{3}x}-24{e}^{\frac{\sqrt{5}}{864}\left(144x+11t\right)}},$$
(2)
was obtained. Figure 1 below shows the one-negaton solution u(x, t) for the modified KdV (mKdV) Eq. 1.
We can also show that the celebrated modified KdV (mKdV) Eq. 1 gives also a one-positon solution in the form
$$u\left(x,t\right)=\frac{7-2 \cos \left(\frac{\sqrt{15}}{4}\left(2x-\frac{9}{8}t\right)\right)}{8+2 \cos \left(\frac{\sqrt{15}}{4}\left(2x-\frac{9}{8}t\right)\right)}.$$
(3)
Figure 2 below shows the one-positon solution for the modified KdV (mKdV) Eq. 1.
Moreover, we can also show that the modified KdV (mKdV) Eq. 1 gives a one-complexiton solution in the form
$$\begin{array}{cc}\hfill u\left(x,t\right)=\frac{\left(2+i\right){e}^{\left(2x+\frac{1}{2}t\right)}}{\left({e}^t+{e}^{4x}\right) cos\left(x-\frac{11}{4}t\right)-i\left({e}^t-{e}^{4x}\right)\ sin\left(x-\frac{11}{4}t\right)},\hfill & \hfill i=\sqrt{-1}.\hfill \end{array}$$
(4)
Figure 3 below shows the one-complexiton solution u(x, t) for the modified KdV (mKdV) equation.

### The Hirota Bilinear Method

Hirota (1971, 1972; Hirota and Ito 1983) established a method for the determination of exact solutions of nonlinear PDEs. The method is called the Hirota direct method or the Hirota bilinear formalism. A necessary condition for the direct method to be applicable is that the PDE can be brought into a bilinear form. Hirota proposed a bilinear form where it was shown that soliton solutions are just polynomials of exponentials. Finding bilinear forms for nonlinear PDEs, if they exist at all, is highly nontrivial. Hirota constructed the N-soliton solutions of the integrable evolution equations by reducing it to the bilinear form. The completely integrable PDEs are the equations that have infinitely many conservation laws and admit N-soliton solutions of any order (Hereman and Nuseir 1997; Hietarinta 1987; Kadomtsev and Petviashvili 1970; Lenells 2005). The bilinear formalism is a very helpful tool in the study of the nonlinear equations, and it was the most suitable for computer algebra. In what follows we highlight the main steps of this method.

The customary definition of the Hirota bilinear operators are given by
$${D}_t^n{D}_x^ma.b={\left(\frac{\partial }{\partial t}-\frac{\partial }{\partial {t}^{\prime }}\right)}^n{\left(\frac{\partial }{\partial x}-\frac{\partial }{\partial {x}^{\prime }}\right)}^ma\left(x,t\right)b\left({x}^{\prime },{t}^{\prime}\right)\Big|{x}^{\prime }=x,{t}^{\prime }=t.$$
(5)

### The Simplified Hirota Method

Hereman and Nuseir (1997) developed the simplified version of the Hirota method where it was shown that soliton solutions are just polynomials of exponentials. By using the simplified Hirota’s method, there is no need to construct bilinear forms suggested by Hirota method. Instead, we can approach the PDE in a straightforward manner. It was confirmed by many that the simplified Hirota’s method is reliable and efficient for solving nonlinear PDEs (Wazwaz 2007a, b, c, d, 2008a, b, c, d, e, f, g, h, 2009a, 2011a, b, c, d, 2012).

## Definitions of the Subject and Its Importance

The study of dynamical behavior of integrable nonlinear partial differential equations has been a major inspiration for mathematicians as well as physicists for the last few decades. Mathematicians have been doing a lot of interesting works to employ new methods of solving the integrable equations, and physicists usually look for the dynamical behavior of the physical systems.

Nonlinear phenomena play a significant role in many branches of applied sciences such as applied mathematics, physics, biology, chemistry, astronomy, and plasma and fluid dynamics. Nonlinear dispersive equations that govern these phenomena have the genuine soliton property. Solitons are pulses that propagate without any change of their identity, i.e., shape and speed, during their travel through a nonlinear dispersive medium (Ablowitz and Clarkson 1991; Drazin and Johnson 1996; Whitham 1999). Solitons resemble properties of a particle, hence the suffix on is used (Wadati 1972, 2001). Solitons are important solutions for science, engineering, and technology.

Solitons exist in many scientific branches, such as optical fiber photonics, fiber lasers, plasmas, protein molecular systems, laser pulses propagating in solids, liquid crystals, nonlinear optics, cosmology, and condensed-matter physics. Based on its importance in many fields, a huge size of research work has been invested during the last few decades to develop more progress and insights through the soliton phenomenon (Shen and Tu 2011; Veksler and Zarmi 2005; Wadati 1972, 2001).

It is well known that completely integrable equations give multiple soliton solutions and enjoy infinitely many conservation laws (Zabusky and Kruskal 1965). The study of multisoliton solutions for completely integrable equations is important for information technology.

## Introduction

In the context of completely integrable equations, studies are flourishing because these equations are able to describe the real features in a variety of science, technology, and engineering areas. Toward this goal, a variety of powerful methods to construct multiple soliton solutions has been established in the fields of mathematical physics and engineering. Examples of the methods that have been used are the Hirota bilinear method (Hirota 1971, 1972; Hirota and Ito 1983), the simplified Hirota method developed by Hereman and Nuseir (Hereman and Nuseir 1997), the Bäcklund transformation method, Darboux transformation, Pfaffian technique, the inverse scattering method, the Painlevé analysis, the generalized symmetry method, the subsidiary ordinary differential equation method, the coupled amplitude-phase formulation, the sine-cosine method, the sech-tanh method (Malfliet 1992; Malfliet and Hereman 1996a, b), the mapping and deformation approach, and many other methods. The inverse scattering method (Ablowitz et al. 1991) of integrable problems is more general than the Hirota bilinear method which yields special solutions. Moreover, the inverse scattering method is more complex and requires cumbersome work, whereas the Hirota bilinear method is mainly algebraic. The Hirota bilinear method and the simplified Hirota method developed are rather heuristic and significant to handle equations with constant coefficients. These two methods possess powerful features that make practical the determination of single-soliton and multiple-soliton solutions for a wide class of nonlinear evolution equations. The simplified Hirota method does not depend on the construction of the bilinear forms; instead it assumes the multisoliton solutions can be expressed as polynomials of exponential functions. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations.

The aim of this work is to apply the simplified Hereman-Nuseir method to study a variety of (3 + 1)-dimensional nonlinear evolution equations and couplings of fifth-order nonlinear equations. A huge amount of research work was invested on (1 + 1)-dimensional and (2 + 1)-dimensional problems. We therefore will focus our attention on (3 + 1)-dimensional equations for further studies and to make further progress in this area.

## The Hirota Bilinear Method

Hirota (1971, 1972; Hirota and Ito 1983) established a method for the determination of exact solutions of nonlinear PDEs. The method is called the Hirota direct method or the Hirota bilinear formalism. A necessary condition for the direct method to be applicable is that the PDE can be brought into a bilinear form. Hirota proposed a bilinear form where it was shown that soliton solutions are just polynomials of exponentials. Finding bilinear forms for nonlinear PDEs, if they exist at all, is highly nontrivial.

The Hirota bilinear method is widely used especially to handle the multisoliton solutions of many evolution equations. Hirota introduced the customary definition of the Hirota bilinear operators by
$${D}_t^n{D}_x^m\left(a\cdot b\right)={\left(\frac{\partial }{\partial t}-\frac{\partial }{\partial {t}^{\prime }}\right)}^n{\left(\frac{\partial }{\partial x}-\frac{\partial }{\partial {x}^{\prime }}\right)}^ma\left(x,t\right)b\left({x}^{\prime },{t}^{\prime}\right)\Big|{x}^{\prime }=x,{t}^{\prime }=t.$$
(6)
In what follows, we express some of the bilinear differential operators:
$$\begin{array}{l}\hfill {D}_x\left(a\cdot b\right) = {a}_xb-a{b}_x,\hfill \\ {}{D}_x^2\left(a\cdot b\right) = {a}_{2x}b-2{a}_x{b}_x+a{b}_{2x},\hfill \\ {}{D}_x{D}_t\left(a\cdot b\right) = {D}_x\left({a}_tb-a{b}_t\right)={a}_{xt}b-{a}_t{b}_x-{a}_x{b}_t+a{b}_{xt},\hfill \\ {}{D}_x{D}_t\left(a\cdot a\right) = 2\left(a{a}_{xt}-{a}_x{a}_t\right),\hfill \\ {}{D}_x^4\left(a\cdot b\right) = {a}_{4x}b-4{a}_{3x}{b}_x+6{a}_{2x}{b}_{2x}-4{a}_x{b}_{3x}+a{b}_{4x},\hfill \\ {}{D}^n\left(a\cdot a\right) = 0,\mathrm{f}\mathrm{o}\mathrm{r} n \mathrm{is}0 \mathrm{o}\mathrm{dd}.\hfill \end{array}$$
(7)
Moreover, more of the properties of the D operators are as follows:
$$\begin{array}{ll}\frac{D_t^2f\cdot f}{f^2}\hfill & ={\displaystyle \int {\displaystyle \int {u}_{tt}} dx dx,}\hfill \\ {}\frac{D_t{D}_x^3 f\cdot f}{f^2}\hfill & ={u}_{xt}+3u{\displaystyle \int x {u}_t} d{x}^{\prime },\hfill \\ {}\frac{D_x^2 f\cdot f}{f^2}\hfill & =u,\hfill \\ {}\frac{D_x^4 f\cdot f}{f^2}\hfill & ={u}_{2x}+3{u}^2,\hfill \\ {}\frac{D_t{D}_x f\cdot f}{f^2}\hfill & = \ln {\left({f}^2\right)}_{xt},\hfill \\ {}\frac{D_x^6 f\cdot f}{f^2}\hfill & ={u}_{4x}+15u{u}_{2x}+15{u}^3,\hfill \\ {}\frac{D_t^2f\cdot f}{f^2}\hfill & ={\displaystyle \int {\displaystyle \int {u}_{tt}} dx dx,}\hfill \\ {}\frac{D_t{D}_x^3 f\cdot f}{f^2}\hfill & ={u}_{xt}+3u{\displaystyle \int {u}_t} d{x}^{\prime },\hfill \end{array}$$
(8)
where
$$u\left(x,t\right)=2{\left( \ln\ f\left(x,t\right)\right)}_{xx},$$
(9)
For example, the KdV equation
$${u}_t+6u{u}_x+{u}_{xxx}=0,$$
(10)
can be transformed to the bilinear form
$$\left({D}_x{D}_t+{D}_x^4\right)f\cdot f=0,f=2{\left( \ln\ f\right)}_{xx}.$$
(11)
$${\left({u}_t+6u{u}_x+{u}_{xxx}\right)}_x+3{u}_{yy}=0,$$
(12)
can be transformed to the bilinear form
$$\left({D}_x{D}_t+{D}_x^4+3{D}_y^2\right)f\cdot f=0,f=2{\left( \ln\ f\right)}_{xx}.$$
(13)
$${u}_t+15{u}^3+15u{u}_{xx}{u}_{xxxxx}=0,$$
(14)
can be transformed to the bilinear form
$${D}_x\left({D}_t+{D}_x^5\right)f\cdot f=0,f=2{\left( \ln\ f\right)}_{xx}.$$
(15)
In a like manner, the Boussinesq equation
$${u}_{tt}-{u}_{xx}-3{\left({u}^2\right)}_{xx}-{u}_{xxxx}=0,$$
(16)
can be transformed to the bilinear form
$$\left({D}_t^2-{D}_x^2-{D}_x^4\right)f\cdot f=0,f=2{\left( \ln\ f\right)}_{xx}.$$
(17)
For other nonlinear equations, the reader is advised to see Hereman and Nuseir (1997), Hietarinta (1987), Hirota (1971), Hirota (1972), and Hirota and Ito (1983).

## The Simplified Hirota Method

Hereman and Nuseir (1997) developed the simplified version of the Hirota method where it was shown that soliton solutions are just polynomials of exponentials. By using the simplified Hirota method, there is no need to construct bilinear forms suggested by Hirota method. Instead, we can approach the PDE in a straightforward manner. In what follows we list the few steps of the simplified method.

We first substitute
$$u\left(x,y,z,t\right)={e}^{kx+ry+sz-\omega t},$$
(18)
into the linear terms of the equation under discussion to determine the dispersion relation between k, r, s, and ω. We then substitute the single-soliton solution
$$u\left(x,y,z,t\right)=R\frac{\partial \ln\ f\left(x,y,z,t\right)}{\partial x}=R\frac{f_x\left(x,y,z,t\right)}{f\left(x,y,z,t\right)},$$
(19)
into the equation under discussion, where the auxiliary function f (x, y, z, t) is given by
$$f\left(x,y,z,t\right)=1+{f}_1\left(x,y,z,t\right)=1+{e}^{\theta_1},$$
(20)
and the auxiliary variable
$${\theta}_i={k}_ix+{r}_iy+{s}_iz-{\omega}_it,i=1,2,3.$$
(21)
We then solve the resulting equation to determine the numerical value for R. Notice that the N-soliton solutions can be obtained for the given equation by using the following steps:
1. (i)
For dispersion relation, we use
$$u\left(x,y,z,t\right)={e}^{\theta_i},{\theta}_i={k}_ix+{r}_iy+{s}_iz-{\omega}_it.$$
(22)

2. (ii)
For single soliton, we use
$$f\left(x,y,z,t\right)=1+{e}^{\theta_1}.$$
(23)

3. (iii)
For two-soliton solutions, we use
$$f\left(x,y,z,t\right)=1+{e}^{\theta_1}+{e}^{\theta_2}+{a}_{12}{e}^{\theta_1+{\theta}_2}.$$
(24)

4. (iv)
For three-soliton solutions, we use
$$f\left(x,y,z,t\right)=1+{e}^{\theta_1}+{e}^{\theta_2}+{e}^{\theta_3}+{a}_{12}{e}^{\theta_1+{\theta}_2}+{a}_{23}{e}^{\theta_2+{\theta}_3}+{a}_{13}{e}^{\theta_1+{\theta}_3}+{b}_{123}{e}^{\theta_1+{\theta}_2+{\theta}_3}.$$
(25)

Notice that we use Eq. 22 to determine the dispersion relation, Eq. 24 to determine the phase shift a 12 to be generalized for the other factors a ij , and finally we use Eq. 25 to determine b 123, which is given by b 123 = a 12 a 23 a 13 for completely integrable equations. The determination of three-soliton solutions confirms the fact that N-soliton solutions exist for any order.

It is to be noted that the existence of three multiple-soliton solutions, and hence the multiple-soliton solutions of any equation, often indicates the integrability of that equation. However, this is not sufficient, and other methods such as Lax pairs should be used to justify the integrability concept.

However, for the multiple–singular soliton solutions (Hereman and Nuseir 1997; Malfliet and Hereman 1996a, b; Shen and Tu 2011; Veksler and Zarmi 2005; Wadati 1972, 2001; Wazwaz 2007a, b, c, d, 2008a, b, c), we use the following steps:
1. (i)
For dispersion relation, we use
$$u\left(x,y,z,t\right)={e}^{\theta_i},{\theta}_i={k}_ix+{r}_iy+{s}_iz-{\omega}_it.$$
(26)

2. (ii)
For single singular soliton, we use
$$f\left(x,y,z,t\right)=1-{e}^{\theta_1}.$$
(27)

3. (iii)
For two–singular soliton solutions, we use
$$f\left(x,y,z,t\right)=1-{e}^{\theta_1}-{e}^{\theta_2}+{a}_{12}{e}^{\theta_1+{\theta}_2}.$$
(28)

4. (iv)
For three–singular soliton solutions, we use
$$f\left(x,y,z,t\right)=1-{e}^{\theta_1}-{e}^{\theta_2}-{e}^{\theta_3}+{a}_{12}{e}^{\theta_1+{\theta}_2}+{a}_{23}{e}^{\theta_2+{\theta}_3}+{a}_{13}{e}^{\theta_1+{\theta}_3}-{b}_{123}{e}^{\theta_1+{\theta}_2+{\theta}_3}.$$
(29)

As indicated before, we will apply the simplified Hirota method to study three nonlinear equations, each of (3 + 1) dimensions. In addition couplings of the fifth-order Sawada-Kotera and Lax equations will be examined as well by the same method. We aim to formally derive multiple-soliton solutions for these models and to examine the constraints that may generate more than one set of multiple-soliton solutions.

## Three Extended (3 + 1)-Dimensional Nonlinear Equations with Multiple-Soliton Solutions

In this section, three extended (3 + 1)-dimensional nonlinear evolution equations will be investigated. The simplified form of the Hirota bilinear method is applied to determine the necessary conditions for the complete integrability of these equations.

There is sufficient work in the literature that examines the (1 + 1)-dimensional equations and the (2 + 1)-dimensional equations. We will concern ourselves in this section on a variety of (3 + 1)-dimensional nonlinear evolution equations. As a first example, the (3 + 1)-dimensional nonlinear evolution equation (Geng 2003; Wazwaz 2011a)
$$3{w}_{xz}-{\left(2{w}_t+{w}_{xxx}- aw{w}_x-bw{w}_y\right)}_y+c{\left({w}_x{\partial}_x^{-1}{w}_y\right)}_x=0,$$
(30)
was examined, and multiple-soliton solutions were derived using a variety of distinct approaches. The inverse operator $${\partial}_x^{-1}$$ is defined by
$$\left({\partial}_x^{-1}f\right)\;(x)={\displaystyle {\mathit{\int}}_{ -\infty}^xf(t)}dt,$$
(31)
under the decaying condition at infinity. Note that $${\partial}_x{\partial}_x^{-1}={\partial}_x^{-1}{\partial}_x=1$$. The necessary condition for the integrability of Eq. 30 has been investigated in Geng (2003).

We aim here to determine multiple-soliton solutions and multiple–singular soliton solutions for three distinct extensions of the (3 + 1)-dimensional nonlinear evolution Eq. 30.

To achieve our goal, we introduce the following three extensions of Eq. 30 in the forms
$$3{w}_{xz}-{\left(2{w}_t+{w}_{xxx}-2w{w}_x-2w{w}_y\right)}_y+{\left({w}_x{\partial}_x^{-1}{w}_y\right)}_x+{\left(w{\partial}_x^{-1}{w}_{yy}\right)}_y=0,$$
(32)
$$3{w}_{xz}-{\left(2{w}_t+{w}_{xxx}-2w{w}_x-2w{w}_y\right)}_y+{\left({\partial}_x^{-1}{w}_z{\partial}_x^{-1}{w}_{yy}\right)}_y+{\left({\partial}_x^{-1}{w}_y{\partial}_x^{-1}{w}_{zz}\right)}_z=0,$$
(33)
and
$$3{w}_{xz}-{\left(2{w}_t+{w}_{xxx}-2w{w}_x-2w{w}_y\right)}_y+{\left(w{\partial}_x^{-1}{w}_{zz}\right)}_z+{\left({w}_x{\partial}_x^{-1}{w}_z\right)}_x=0,$$
(34)
that will be named first, second, and third model respectively. In each model, we replaced the last term of Eq. 30 by two distinct nonlinear terms, and we selected a = 2, b = 2, c = 1. This means that the difference between these three models stems from the last two nonlinear terms of each model. In addition to the determination of multiple-soliton solutions and multiple–singular soliton solutions for each model, we will investigate the effect of these terms on the structure of the dispersion relations and hence on the dispersion variables.

In what follows we will apply the simplified form of the Hirota method to the aforementioned three models (Eqs. 32, 33, and 34).

### The First Model

In this section we apply the simplified Hirota method to the first model that reads
$$3{w}_{xz}-{\left(2{w}_t+{w}_{xxx}-2w{w}_x-2w{w}_y\right)}_y+{\left({w}_x{\partial}_x^{-1}{w}_y\right)}_x+{\left(w{\partial}_x^{-1}{w}_{yy}\right)}_y=0.$$
(35)
Multiple-soliton solutions
We first remove the integral term in Eq. 35 by introducing the potential
$$w\left(x,y,z,t\right)={u}_x\left(x,y,z,t\right),$$
(36)
to carry Eq. 35 to the equation
$$3{u}_{xxz}-{\left(2{u}_{xt}+{u}_{xxxx}-2{u}_x{u}_{xx}-2{u}_x{u}_{xy}\right)}_y+{\left({u}_{xx}{u}_y\right)}_x+{\left({u}_{yy}{u}_x\right)}_y=0.$$
(37)
Substituting
$$u\left(x,y,z,t\right)={e}^{\theta_i},{\theta}_i={k}_ix+{r}_iy+{s}_iz-{\omega}_it,$$
(38)
into the linear terms of Eq. 37 and solving the resulting equation for ω i we obtain the dispersion relation
$${\omega}_i=\frac{k_i^3{r}_i-3{k}_i{s}_i}{2{r}_i},i=1,2,3,$$
(39)
and hence the dispersion variables θ i become
$${\theta}_i={k}_ix+{r}_iy+{s}_iz-\left(\frac{k_i^3{r}_i-3{k}_i{s}_i}{2{r}_i}\right)t.$$
(40)
To determine R, we substitute
$$u\left(x,y,z,t\right)=R\frac{\partial \ln\ f\left(x,y,z,t\right)}{\partial x}=R\frac{f_x\left(x,y,z,t\right)}{f\left(x,y,z,t\right)},$$
(41)
where the auxiliary function is given by
$$f\left(x,y,z,t\right)=1+{e}^{k_1x+{r}_1y+{s}_1z-\left(\frac{k_1^3{r}_1-3{k}_1{s}_1}{2{r}_1}\right) t},$$
(42)
into Eq. 37 and solve to find that
$$R=-2,$$
(43)
and the complete integrability is satisfied only if
$${r}_i={k}_i-3{k}_i,$$
(44)
and the parameter s i is left free.
The last result reduces the dispersion relation (Eq. 39) to two distinct values
$${\omega}_i=\frac{k_i^3-3{s}_i}{2},i=1,2,3,$$
(45)
or
$${\omega}_i=\frac{k_i^3-{s}_i}{2},i=1,2,3.$$
(46)
Consequently, the auxiliary function (Eq. 42) takes the two distinct forms
$$f\left(x,y,z,t\right)=1+{e}^{k_1x+{k}_1y+{s}_1z-\frac{k_1^3-3{s}_1}{2}t},$$
(47)
and
$$f\left(x,y,z,t\right)=1+{e}^{k_1x-3{k}_1y+{s}_1z-\frac{k_1^3-{s}_1}{2}t}.$$
(48)
Substituting Eqs. 45 and 47 into Eq. 41 gives
$$u\left(x,y,z,t\right)=-\frac{2{k}_1{e}^{k_1x+{k}_1y+{s}_1z-\left(\frac{k_1^3-3{s}_1}{2}\right)t}}{\left(1+{e}^{k_1x+{k}_1y+{s}_1z-\left(\frac{k_1^3-3{s}_1}{2}\right)t}\right)},$$
(49)
and by substituting Eqs. 46 and 48 into Eq. 41 we find
$$\left(x,y,z,t\right)=-\frac{2{k}_1{e}^{k_1x+{k}_1y+{s}_1z-\left(\frac{k_1^3+{s}_1}{2}\right)t}}{\left(1+{e}^{k_1x+{k}_1y+{s}_1z-\left(\frac{k_1^3+{s}_1}{2}\right)t}\right)},$$
(50)
respectively. Recall that the single-soliton solutions are obtained by using w = u x .
Figure 4 below shows the soliton solution w(x, y, z, t) for this case.
For the two-soliton solution, we substitute
$$u\left(x,y,z,t\right)=-2\frac{\partial \ln\ f\left(x,y,z,t\right)}{\partial x},$$
(51)
where the auxiliary function for the two-soliton solution is given by
$$f\left(x,y,z,t\right)=1+{e}^{\theta_1}+{e}^{\theta_2}+{a}_{12}{e}^{\theta_1+{\theta}_2},$$
(52)
into Eq. 37, where θ 1 and θ 2 are given in Eq. 40, to obtain the phase shift a 12 by
$${a}_{12}=\frac{{\left({k}_1-{k}_2\right)}^2}{{\left({k}_1+{k}_2\right)}^2},$$
(53)
and hence generalized to
$${a}_{ij}=\frac{{\left({k}_i-{k}_j\right)}^2}{{\left({k}_i+{k}_j\right)}^2},1\le i<j\le 3.$$
(54)
We point out that the (3 + 1)-dimensional Eq. 37 does not show any resonant phenomenon because the phase shift term a 12 in Eq. 53 cannot be 0 or ∞ for |k 1| ≠ |k 2|.
This in turn gives
$$f\left(x,y,z,t\right)=1+{e}^{\theta_1}+{e}^{\theta_2}+\frac{{\left({k}_1-{k}_2\right)}^2}{{\left({k}_1+{k}_2\right)}^2}{e}^{\theta_1+{\theta}_2}.$$
(55)
To determine the two-soliton solutions explicitly, we substitute Eq. 55 into the formula (Eq. 51), and then we use the potential w = u x as defined in Eq. 36. Note that two distinct sets of two-soliton solutions are obtained depending on the dispersion relations that we use. Figure 5 below shows the two-soliton solution w(x, y, z, t) for this case.
To determine the three-soliton solutions, we substitute the auxiliary function
$$f\left(x,y,z,t\right)=1+{e}^{\theta_1}+{e}^{\theta_2}+{e}^{\theta_3}+{a}_{12}{e}^{\theta_1+{\theta}_2}+{a}_{13}{e}^{\theta_1+{\theta}_3}+{a}_{23}{e}^{\theta_2+{\theta}_3}+{b}_{123}{e}^{\theta_1+{\theta}_2+{\theta}_3},$$
(56)
and proceed as before to find that
$${b}_{123}={a}_{12}{a}_{13}{a}_{23}.$$
(57)
To determine the three-soliton solutions explicitly, we substitute the last result for f(x, y, z, t) in the formula w(x, y, z, t) = −2(ln f (x, y, z, t)) xx . The higher-level soliton solutions for N ≥ 4 can be obtained in a parallel manner. This shows that the three-soliton solutions and hence the multiple-soliton solutions exist for finite N, where N ≥ 1. Note that two distinct sets of three-soliton solutions are derived depending on the dispersion relation that we use. The existence of multiple-soliton solutions often indicates the integrability of the equation, but further conditions are needed such as the Lax pairs or the Painlevé analysis.

In conclusion, it is obvious that two distinct dispersion relations are formally derived that provide two sets of distinct multiple-soliton solutions. Unlike other evolution equations, the multiple-soliton solutions for this nonlinear evolution equation is not unique.

Multiple–singular soliton solutions

To determine the singular-soliton solutions for the first model, we normally use the same approach presented above with one main difference. The auxiliary functions that should be used are given by
$$f\left(x,y,z,t\right)=1-{e}^{k_1x+{k}_1y+{s}_1z-\frac{k_1^3-3{s}_1}{2}t},$$
(58)
and
$$f\left(x,y,z,t\right)=1-{e}^{k_1x-3{k}_1y+{s}_1z-\frac{k_1^3+{s}_1}{2}t}.$$
(59)
In this case the set of distinct singular solutions reads
$$u\left(x,y,z,t\right)=\frac{2{k}_1{e}^{k_1x+{k}_1y+{s}_1z-\left(\frac{k_1^3-3{s}_1}{2}\right)t}}{\left(1-{e}^{k_1x+{k}_1y+{s}_1z-\left(\frac{k_1^3-3{s}_1}{2}\right)t}\right)},$$
(60)
and
$$u\left(x,y,z,t\right)=\frac{2{k}_1{e}^{k_1x+{k}_1y+{s}_1z-\left(\frac{k_1^3+{s}_1}{2}\right)t}}{\left(1-{e}^{k_1x+{k}_1y+{s}_1z-\left(\frac{k_1^3+{s}_1}{2}\right)t}\right)},$$
(61)
respectively. Consequently, the single–singular soliton solutions are obtained by using w = u x .
To determine the two–singular soliton and three–singular soliton solutions we use the auxiliary functions by
$$f\left(x,y,z,t\right)=1-{e}^{\theta_1}-{e}^{\theta_2}+{a}_{12}{e}^{\theta_1+{\theta}_2},$$
(62)
and
$$f\left(x,y,z,t\right)=1-{e}^{\theta_1}-{e}^{\theta_2}-{e}^{\theta_3}+{a}_{12}{e}^{\theta_1+{\theta}_2}+{a}_{13}{e}^{\theta_1+{\theta}_3}+{a}_{23}{e}^{\theta_2+{\theta}_3}-{a}_{12}{a}_{13}{a}_{23}{e}^{\theta_1+{\theta}_2+{\theta}_3},$$
(63)
respectively, where the phase shifts a ij and the dispersion variables θ i are as defined earlier.

### The Second Model

In this part we examine the second model
$$3{w}_{xz}-{\left(2{w}_t+{w}_{xxx}-2w{w}_x-2w{w}_y\right)}_y+{\left({\partial}_x^{-1}{w}_z{\partial}_x^{-1}{w}_{yy}\right)}_y+{\left({\partial}_x^{-1}{w}_y{\partial}_x^{-1}{w}_{zz}\right)}_z=0.$$
(64)
Multiple-soliton solutions
To remove the integral term in Eq. 64 we use the potential
$$w\left(x,y,z,t\right)={u}_x\left(x,y,z,t\right),$$
(65)
to carry Eq. 64 to the equation
$$3{u}_{xxz}-{\left(2{u}_{xt}+{u}_{xxxx}-2{u}_x{u}_{xx}-2{u}_x{u}_{xy}\right)}_y+{\left({u}_{yy}{u}_z\right)}_y+{\left({u}_{zz}{u}_y\right)}_z=0.$$
(66)
Substituting
$$u\left(x,y,z,t\right)={e}^{\theta_i},{\theta}_i={k}_ix+{r}_iy+{s}_iz-{\omega}_it,$$
(67)
into the linear terms of Eq. 66, and solving the resulting equation for ω i we obtain the dispersion relation
$${\omega}_i=\frac{k_i^3{r}_i-3{k}_i{s}_i}{2{r}_i},i=1,2,3,$$
(68)
and hence θ i becomes
$${\theta}_i={k}_ix+{r}_iy+{s}_iz-\left(\frac{k_i^3{r}_i-3{k}_i{s}_i}{2{r}_i}\right) t.$$
(69)
To determine R, we substitute
$$u\left(x,y,z,t\right)=R\frac{\partial \ln\ f\left(x,y,z,t\right)}{\partial x},$$
(70)
where
$$f\left(x,y,z,t\right)=1+{e}^{k_1x+{r}_1y+{s}_1z-\left(\frac{k_1^3{r}_1-3{k}_1{s}_1}{2{r}_1}\right)t},$$
(71)
into Eq. 66 and solve to find that
$$R=-2,$$
(72)
and the complete integrability is satisfied only if
$${r}_1=\frac{-2{k}_1^2\pm \sqrt{k_1^4+4{s}_1{k}_1^3-{s}_1^4}}{2{s}_1},$$
(73)
and the parameter s 1 in this case should satisfy
$$0<{s}_1\le {k}_1,$$
(74)
that can be generalized to
$$0<{s}_i\le {k}_i,i=1,2,3.$$
(75)
For simplicity reasons we select the following sets for the parameters given by
$${s}_i={k}_i,{r}_i={k}_i,$$
(76)
and
$${s}_i={k}_i,{r}_i=-3{k}_i.$$
(77)
The last results reduce the dispersion relation (Eq. 68) to two distinct values
$${\omega}_i=\frac{k_i^3-3{k}_i}{2},i=1,2,3,$$
(78)
or
$${\omega}_i=\frac{k_i^3+{k}_i}{2},i=1,2,3.$$
(79)
Consequently, the auxiliary function (Eq. 71) takes the two forms
$$f\left(x,y,z,t\right)=1+{e}^{k_1x+{k}_1y+{k}_1z-\frac{k_1^3-3{k}_1}{2}t},$$
(80)
and
$$f\left(x,y,z,t\right)=1+{e}^{k_1x-3{k}_1y+{k}_1z-\frac{k_1^3+{k}_1}{2}t},$$
(81)
Substituting Eqs. 78 and 80, then Eqs. 79 and 81 into Eq. 70 gives two distinct solutions
$$u\left(x,y,z,t\right)=-\frac{2{k}_1{e}^{k_1x+{k}_1y+{k}_1z-\left(\frac{k_1^3-3{k}_1}{2}\right)t}}{\left(1+{e}^{k_1x+{k}_1y+{k}_1z-\left(\frac{k_1^3-3{k}_1}{2}\right)t}\right)},$$
(82)
and
$$u\left(x,y,z,t\right)=-\frac{2{k}_1{e}^{k_1x-3{k}_1y+{k}_1z-\left(\frac{k_1^3+{k}_1}{2}\right)t}}{\left(1+{e}^{k_1x-3{k}_1y+{k}_1z-\left(\frac{k_1^3+{k}_1}{2}\right)t}\right)},$$
(83)
respectively. Recall that the single-soliton solutions are obtained by using w = u x .
For the two-soliton solutions, we substitute
$$u\left(x,y,z,t\right)=-2\frac{\partial \ln\ f\left(x,y,z,t\right)}{\partial x},$$
(84)
where
$$f\left(x,y,z,t\right)=1+{e}^{\theta_1}+{e}^{\theta_2}+{a}_{12}{e}^{\theta_1+{\theta}_2},$$
(85)
into Eq. 66, where θ 1 and θ 2 are given in Eq. 69, to obtain the phase shift a 12 by
$${a}_{12}=\frac{{\left({k}_1-{k}_2\right)}^2}{{\left({k}_1+{k}_2\right)}^2},$$
(86)
and hence
$${a}_{ij}=\frac{{\left({k}_i-{k}_j\right)}^2}{{\left({k}_i+{k}_j\right)}^2},1\le i<j\le 2.$$
(87)
We point out that the (3 + 1)-dimensional Eq. 66 does not show any resonant phenomenon because the phase shift term a 12 in Eq. 86 cannot be 0 or ∞ for |k 1| ≠ |k 2|.
This in turn gives
$$f\left(x,y,z,t\right)=1+{e}^{\theta_1}+{e}^{\theta_2}+\frac{{\left({k}_1-{k}_2\right)}^2}{{\left({k}_1+{k}_2\right)}^2}{e}^{\theta_1+{\theta}_2}.$$
(88)
To determine the two-soliton solutions explicitly, we substitute Eq. 88 into the formula (Eq. 84), and then we use the potential w = u x as defined in Eq. 65. Note that two distinct sets of two-soliton solutions are obtained depending on the dispersion relation that we use.
To determine the three-soliton solutions, we substitute the auxiliary function
$$f\left(x,y,z,t\right)=1+{e}^{\theta_1}+{e}^{\theta_2}+{e}^{\theta_3}+{a}_{12}{e}^{\theta_1+{\theta}_2}+{a}_{13}{e}^{\theta_1+{\theta}_3}+{a}_{23}{e}^{\theta_2+{\theta}_3}+{b}_{123}{e}^{\theta_1+{\theta}_2+{\theta}_3},$$
(89)
and proceed as before to find that
$${b}_{123}={a}_{12}{a}_{13}{a}_{23}.$$
(90)
To determine the three-soliton solutions explicitly, we substitute the last result for f(x, y, z, t) in the formula w(x, y, z, t) = −2(ln f(x, y, z, t)) xx . The higher-level soliton solutions for N ≥ 4 can be obtained in a parallel manner. This shows that the three-soliton solutions and hence the multiple-soliton solutions exist for finite N, where N ≥ 1. Note that two distinct sets of three-soliton solutions are derived depending on the dispersion relation that we use. The existence of multiple-soliton solutions often indicates the integrability of the equation, but further conditions are needed such as the Lax pairs or the Painlevé analysis.

In conclusion, it is obvious that two distinct dispersion relations are formally derived that provide two sets of distinct multiple-soliton solutions. However, by selecting other values of s i , we can determine other sets of multiple-soliton solutions. Unlike other evolution equations, the multiple-soliton solution for this nonlinear evolution equation is not unique.

Multiple–singular soliton solutions

To determine the singular-soliton solutions for the second model, we normally use the approach presented above with one main difference. The auxiliary functions that should be used are given by
$$f\left(x,y,z,t\right)=1-{e}^{k_1x+{k}_1y+{k}_1z-\frac{k_1^3-3{k}_1}{2}t},$$
(91)
and
$$f\left(x,y,z,t\right)=1-{e}^{k_1x-3{k}_1y+{k}_1z-\frac{k_1^3+{k}_1}{2}t}.$$
(92)
In this case the set of distinct singular solutions reads
$$u\left(x,y,z,t\right)=\frac{2{k}_1{e}^{k_1x+{k}_1y+{k}_1z-\left(\frac{k_1^3-3{k}_1}{2}\right)t}}{\left(1-{e}^{k_1x+{k}_1y+{k}_1z-\left(\frac{k_1^3-3{k}_1}{2}\right)t}\right)},$$
(93)
and
$$u\left(x,y,z,t\right)=\frac{2{k}_1{e}^{k_1x-3{k}_1y+{k}_1z-\left(\frac{k_1^3+{k}_1}{2}\right)t}}{\left(1-{e}^{k_1x-3{k}_1y+{k}_1z-\left(\frac{k_1^3+{k}_1}{2}\right)t}\right)},$$
(94)
respectively. Recall that the single-soliton solutions are obtained by using w = u x .
To determine the two–singular soliton and three–singular soliton solutions we use the auxiliary functions by
$$f\left(x,y,z,t\right)=1-{e}^{\theta_1}-{e}^{\theta_2}+{a}_{12}{e}^{\theta_1+{\theta}_2},$$
(95)
and
$$f\left(x,y,z,t\right)=1-{e}^{\theta_1}-{e}^{\theta_2}-{e}^{\theta_3}+{a}_{12}{e}^{\theta_1+{\theta}_2}+{a}_{13}{e}^{\theta_1+{\theta}_3}+{a}_{23}{e}^{\theta_2+{\theta}_3}-{a}_{12}{a}_{13}{a}_{23}{e}^{\theta_1+{\theta}_2+{\theta}_3},$$
(96)
respectively, where the phase shifts a ij and the dispersion variables θ i are as defined earlier.

### The Third Model

In this section we proceed as before and examine the third model
$$3{w}_{xz}-{\left(2{w}_t+{w}_{xxx}-2w{w}_x-2w{w}_y\right)}_y+{\left(w{\partial}_x^{-1}{w}_{zz}\right)}_z+{\left({w}_x{\partial}_x^{-1}{w}_z\right)}_x=0.$$
(97)
Using the potential
$$w\left(x,y,z,t\right)={u}_x\left(x,y,z,t\right),$$
(98)
carries Eq. 97 to the equation
$$3{u}_{xxz}-{\left(2{u}_{xt}+{u}_{xxxx}-2{u}_x{u}_{xx}-2{u}_x{u}_{xy}\right)}_y+{\left({u}_{zz}{u}_x\right)}_z+{\left({u}_{xx}{u}_z\right)}_x=0.$$
(99)
Proceeding as before, we find the dispersion relation and the dispersion variable as
$${\omega}_i=\frac{k_i^3{r}_i-3{k}_i{s}_i}{2{r}_i},i=1,2,3,$$
(100)
and hence θ i becomes
$${\theta}_i={k}_ix+{r}_iy+{s}_iz-\left(\frac{k_i^3{r}_i-3{k}_i{s}_i}{2{r}_i}\right)t.$$
(101)
We next use the transformation
$$u\left(x,y,z,t\right)=-2\frac{\partial \ln\ f\left(x,y,z,t\right)}{\partial x},$$
(102)
where the auxiliary function is given by
$$f\left(x,y,z,t\right)=1+{e}^{k_1x+{r}_1y+{s}_1z-\left(\frac{k_1^3{r}_1-3{k}_1{s}_1}{2{r}_1}\right)t}.$$
(103)
Following the discussion presented before, we find
$${r}_1=\frac{4{k}_1^2\pm 2\sqrt{k_1^4-2{s}_1{k}_1^3-2{k}_1{s}_1^3}}{4{k}_1},$$
(104)
and the parameter s 1 in this case should satisfy
$${s}_1\le {k}_1,$$
(105)
that can be generalized to
$${s}_i\le {k}_i,i=1,2,3.$$
(106)
Based on this result we select the following three sets for the parameters given by
$${s}_i={k}_i,{r}_i={k}_i,$$
(107)
$${s}_i=-3{k}_i,{r}_i=-3{k}_i,$$
(108)
and
$${s}_i=-3{k}_i,{r}_i=5{k}_i,$$
(109)
where other values can be selected as well.
The last results reduce the dispersion relation (Eq. 100) to three distinct values
$${\omega}_i=\frac{k_i^3-3{k}_i}{2},i=1,2,3,$$
(110)
$${\omega}_i=\frac{k_i^3-3{k}_i}{2},i=1,2,3,$$
(111)
and
$${\omega}_i=\frac{5{k}_i^3+9{k}_i}{10},i=1,2,3,$$
(112)
Consequently, the auxiliary function (Eq. 103) takes the three forms
$$f\left(x,y,z,t\right)=1+{e}^{k_1x+{k}_1y+{k}_1z-\frac{k_1^3-3{k}_1}{2}t},$$
(113)
$$f\left(x,y,z,t\right)=1+{e}^{k_1x-3{k}_1y-3{k}_1z-\frac{k_i^3-3{k}_i}{2}t},$$
(114)
and
$$f\left(x,y,z,t\right)=1+{e}^{k_1x+5{k}_1y-3{k}_1z-\frac{5{k}_i^3+9{k}_i}{10}t},$$
(115)
Substituting Eqs. 113, 114, 115 into Eq. 102 gives three distinct solutions
$$u\left(x,y,z,t\right)=-\frac{2{k}_1{e}^{k_1x+{k}_1y+{k}_1z-\left(\frac{k_1^3-3{k}_1}{2}\right)t}}{\left(1+{e}^{k_1x+{k}_1y+{k}_1z-\left(\frac{k_1^3-3{k}_1}{2}\right)t}\right)},$$
(116)
$$u\left(x,y,z,t\right)=-\frac{2{k}_1{e}^{k_1x-3{k}_1y-3{k}_1z-\left(\frac{k_1^3-{k}_1}{2}\right)t}}{\left(1+{e}^{k_1x-3{k}_1y-3{k}_1z-\left(\frac{k_1^3-{k}_1}{2}\right)t}\right)},$$
(117)
and
$$u\left(x,y,z,t\right)=-\frac{2{k}_1{e}^{k_1x+5{k}_1y-3{k}_1z-\left(\frac{5{k}_i^3+9{k}_i}{10}\right)t}}{\left(1+{e}^{k_1x+5{k}_1y-3{k}_1z-\left(\frac{5{k}_i^3+9{k}_i}{10}\right)t}\right)},$$
(118)
respectively. Recall that the single-soliton solutions are obtained by using w = u x .

For the two-soliton solutions, and the three-soliton solutions we follow the same analysis used before. In conclusion, it is obvious that at least three distinct dispersion relations are formally derived that provide at least three sets of distinct multiple-soliton solutions. However, by selecting other values of s i we can determine other sets of multiple-soliton solutions. Unlike other evolution equations, the multiple-soliton solution for this nonlinear evolution equation is not unique.

Multiple–singular soliton solutions

To determine the singular-soliton solutions for the third model, we normally use the approach presented above with one main difference. The auxiliary functions should be given as
$$f\left(x,y,z,t\right)=1-{e}^{k_1x+{k}_1y+{k}_1z-\frac{k_1^3-3{k}_1}{2}t},$$
(119)
$$f\left(x,y,z,t\right)=1-{e}^{k_1x-3{k}_1y-3{k}_1z-\frac{k_i^3-3{k}_i}{2}t},$$
(120)
and
$$f\left(x,y,z,t\right)=1-{e}^{k_1x+5{k}_1y-3{k}_1z-\frac{5{k}_i^3+9{k}_i}{10}t}.$$
(121)
This in turn will give the following single-soliton solutions
$$u\left(x,y,z,t\right)=\frac{2{k}_1{e}^{k_1x+{k}_1y+{k}_1z-\left(\frac{k_1^3-3{k}_1}{2}\right)t}}{\left(1-{e}^{k_1x+{k}_1y+{k}_1z-\left(\frac{k_1^3-3{k}_1}{2}\right)t}\right)},$$
(122)
$$u\left(x,y,z,t\right)=\frac{2{k}_1{e}^{k_1x-3{k}_1y-3{k}_1z-\left(\frac{k_1^3-{k}_1}{2}\right)t}}{\left(1-{e}^{k_1x-3{k}_1y-3{k}_1z-\left(\frac{k_1^3-{k}_1}{2}\right)t}\right)},$$
(123)
and
$$u\left(x,y,z,t\right)=\frac{2{k}_1{e}^{k_1x+5{k}_1y-3{k}_1z-\left(\frac{5{k}_i^3+9{k}_i}{10}\right)t}}{\left(1-{e}^{k_1x+5{k}_1y-3{k}_1z-\left(\frac{5{k}_i^3+9{k}_i}{10}\right)t}\right)},$$
(124)
respectively.

For the two–singular soliton solutions and three–singular soliton solutions we follow the approach applied before in the other sections.

### Discussion

In this work we examined three distinct (3 + 1)-dimensional nonlinear models. Multiple-soliton solutions and multiple–singular soliton solutions were formally derived. For each model we obtained more than one dispersion relation, and this in turn provided more than one set of multiple-soliton solutions. Unlike many nonlinear integrable equations where the result of multiple-soliton solutions is unique, more than one result was derived for each model. This of course is due to the existence of more than one dispersion relation for each model. The three models do not show resonance phenomenon. A summary of distinct dispersion relations is shown in Table 1. Table 1 below summarizes the parameters and the dispersion relations.
Table 1

Parameters and dispersion relations of the three models

Model

The coefficients k i , r i , s i

Dispersion relations

First model

k i , r i = k i, and s i is free

$$\frac{k_i^3-3{s}_i}{2}$$

k i , r i = −3k i, and s i is free

$$\frac{k_i^3-3{s}_i}{2}$$

Second model

$${k}_i,{r}_i=\frac{-2{k}_i^2\pm \sqrt{k_i^4+4{s}_i{k}_i^3-{s}_i^4}}{2{s}_i},{s}_i$$ is free $$0<{s}_i\le {k}_i,i=1,2,3$$

select k i , r i = k i and s i = k i

$$\frac{k_i^3-3{k}_i}{2}$$

select k i , r i = −3k i and s i = k i

$$\frac{k_i^3+{k}_i}{2}$$

Third model

$${k}_i,{r}_i=\frac{4{k}_i^2\pm 2\sqrt{k_i^4-2{s}_i{k}_i^3-2{k}_i{s}_i^3}}{4{k}_i},{s}_i$$ is free $${s}_i\le {k}_i,i=1,2,3$$

select k i , r i = k i and s i = k i

$$\frac{k_i^3-3{k}_i}{2}$$

select k i , r i = −3k i and s i = −3k i

$$\frac{k_i^3-3{k}_i}{2}$$

select k i , r i = 5k i and s i = −3k i

$$\frac{5{k}_i^3+9{k}_i}{10}$$

## An Extended (3 + 1)-Dimensional Modified Kadomtsev-Petviashvili Equation with Multiple-Front Waves

We present here an extended (3 + 1)-dimensional modified Kadomtsev-Petviashvili (mKP) equation. We will also apply the simplified form of the Hirota bilinear method to conduct a reliable study. Multiple-front wave solutions are formally derived for this extended equation. We also show that the extension terms do not kill the integrability of the mKP equation.

The modified Kadomtsev-Petviashvili (mKP) equation (Ma et al. 2011; Wazwaz 2011a) reads
$$4{v}_t+{v}_{xxx}-6{v}^2{v}_x+6{v}_x{\partial}_x^{-1}{v}_y+3{\partial}_x^{-1}{v}_{yy}=0.$$
(125)
The mKP equation (Eq. 125) describes the propagation of ion-acoustic waves in a plasma with non isothermal electrons. This equation can describe the evolution of various solitary waves in the multitemperature electron plasmas, in which there exists a collision-less multicomponent plasma conceiving cold ions and two-temperature electrons having different Maxwellian distributions rendered in the form of two Boltzmann relations.
An extended (2 + 1)-dimensional form of Eq.125 was presented in Wazwaz (2011a), given by
$$4{v}_t+{v}_{xxx}-6{v}^2{v}_x+6{v}_x{\partial}_x^{-1}{v}_y+3{\partial}_x^{-1}{v}_{yy}+4\alpha {v}_y+4\beta {v}_x=0,$$
(126)
where α and β are arbitrary constants. This extended form of the mKP equation is obtained by adding the terms 4αv x and 4βv y to the mKP equation (Eq. 125), where v x and v y are the potentials in the x and the y directions respectively.
In this work we aim to develop a new extended form of Eq. 126 where the effect of the potential v z should be studied, and not to focus only on the two potentials v x and v y. For this reason, we introduce the new extended (3 + 1)-dimensional mKP equation
$$4{v}_t+{v}_{xxx}-6{v}^2{v}_x+6{v}_x{\partial}_x^{-1}{v}_y+3{\partial}_x^{-1}{v}_{yy}+3{\partial}_x^{-1}{v}_{zz}+4\alpha {v}_y+4\beta {v}_x+4\gamma {v}_z=0,$$
(127)
where the $$3{\partial}_x^{-1}{v}_{zz}$$ term and the potential v z in the z direction are added to Eq. 126.

We aim in this work to derive multiple-front wave solutions and multiple–singular front wave solutions of the extended model (Eq. 127). Our next goal is to show that the additional terms $$3{\partial}_x^{-1}{v}_{zz}$$ and 4γv z do not kill the integrability of the mKP equation (Eq. 125), but will change the dispersion relation of the mKP equation.

As stated before, we will study the (3 + 1)-dimensional extended form of the modified Kadomtsev-Petviashvili (mKP) equation that reads
$$4{v}_t+{v}_{xxx}-6{v}^2{v}_x+6{v}_x{\partial}_x^{-1}{v}_y+3{\partial}_x^{-1}{v}_{yy}+3{\partial}_x^{-1}{v}_{zz}+4\alpha {v}_y+4\beta {v}_x+4\gamma {v}_z=0.$$
(128)
To remove the integral operator we use the potential
$$v={u}_x,$$
(129)
that carries Eq. 128) to
$$4{u}_{xt}+{u}_{xxxx}-6\left({u}_x^2-{u}_y\right){u}_{xx}+3{u}_{yy}+3{u}_{zz}+4\alpha {u}_{xy}+4\beta {u}_{xx}+4\gamma {u}_{xz}=0.$$
(130)

### Multiple-Front Wave Solutions

Substituting
$$u\left(x,y,z,t\right)={e}^{\theta_i},{\theta}_i={k}_ix+{r}_iy+{s}_iz-{\omega}_it,$$
(131)
into the linear terms of Eq. 130 gives the dispersion relation by
$${\omega}_i=\frac{k_i^4+3\left({r}_i^2+{s}_i^2\right)+4{k}_i\left(\alpha {r}_i+\beta {k}_i+\gamma {s}_i\right)}{4{k}_i},$$
(132)
and as a result the dispersion variable becomes
$${\theta}_i={k}_ix+{r}_iy+{s}_iz=\frac{k_i^4+3\left({r}_i^2+{s}_i^2\right)+4{k}_i\left(\alpha {r}_i+\beta {k}_i+\gamma {s}_i\right)}{4{k}_i}t.$$
(133)
To determine the multifront wave solutions of Eq. 128, the Cole-Hopf transformation method admits the use of
$$u\left(x,y,z,t\right)=R{\left( \ln\ f\right)}_x=R\frac{f_x}{f},$$
(134)
and this in turn gives
$$u\left(x,y,z,t\right)=R \ln\ f\left(x,y,z,t\right),$$
(135)
where the auxiliary function f (x, y, z, t) for the single-front wave solution is given by
$$f\left(x,y,z,t\right)=1+{e}^{\theta_1}=1+{e}^{k_1x+{r}_1y+{s}_1z-\frac{k_1^4+3\left({r}_1^2+{s}_1^2\right)+4{k}_i\left(\alpha {r}_1+\beta {k}_1+\gamma {s}_1\right)}{4{k}_1}t}.$$
(136)
Substituting Eq. 135 into Eq. 130 and solving for R and r 1, s 1 we find
$$\begin{array}{c}\hfill R=1,\hfill \\ {}\hfill {r}_1={k}_1^2,\hfill \\ {}\hfill {s}_1={k}_1.\hfill \end{array}$$
(137)
This immediately can be generalized to
$$\begin{array}{c}\hfill R=1,\hfill \\ {}\hfill {r}_i={k}_i^2,\hfill \\ {}\hfill {s}_i={k}_i,\hfill \\ {}\hfill {\omega}_i=\frac{4{k}_i^3+4\alpha {k}_i^2+\left(3+4\beta +4\gamma \right){k}_i}{4},\hfill \\ {}\hfill {\theta}_i\left(x,y,z,t\right)={k}_ix+{k}_i^2y+{k}_iz-\left(\frac{4{k}_i^3+4\alpha {k}_i^2+\left(3+4\beta +4\gamma \right){k}_i}{4}\right) t,i=1,2,3.\hfill \end{array}$$
(138)
This means that for the single-front wave solution we set
$$f\left(x,y,z,t\right)=1+{e}^{k_1x+{k}_1^2y+{k}_1z-\left(\frac{4{k}_i^3+4\alpha {k}_i^2+\left(3+4\beta +4\gamma \right){k}_i}{4}\right)t}.$$
(139)
Substituting Eq. 139 into Eq. 135 gives
$$u\left(x,y,z,t\right)= \ln \left(1+{e}^{k_1x+{k}_1^2y+{k}_1z-\left(\frac{4{k}_1^3+4\alpha {k}_1^2+\left(3+4\beta +4\gamma \right){k}_1}{4}\right)t}\right),$$
(140)
and as a result the single-front wave solution
$$u\left(x,y,z,t\right)=\frac{k_1{e}^{k_1x+{k}_1^2y+{k}_1z-\left(\frac{4{k}_1^3+4\alpha {k}_1^2+\left(3+4\beta +4\gamma \right){k}_1}{4}\right)t}}{1+{e}^{k_1x+{k}_1^2y+{k}_1z-\left(\frac{4{k}_1^3+4\alpha {k}_1^2+\left(3+4\beta +4\gamma \right){k}_1}{4}\right)t}},$$
(141)
follows immediately upon using Eq. 129. Fig. 6 below shows the obtained kink solution v(x, y, z, t).
For the two-front wave solutions we use
$$f=1+{e}^{k_1x+{k}_1^2y+{k}_1z-\left(\frac{4{k}_i^3+4\alpha {k}_i^2+\left(3+4\beta +4\gamma \right){k}_i}{4}\right)t}+{e}^{k_2x+{k}_2^2y+{k}_2z-\left(\frac{4{k}_2^3+4\alpha {k}_2^2+\left(3+4\beta +4\gamma \right){k}_2}{4}\right)t},$$
(142)
where for the two-front wave solutions, the phase shift does not exist.
Using Eq. 142 gives the two-front wave solutions
$$v\left(x,y,z,t\right)=\frac{k_1{e}^{\theta_1}+{k}_2{e}^{\theta_2}}{1+{e}^{\theta_1}+{e}^{\theta_2}},$$
(143)
where θ i , i = 1, 2 are given in Eq. 138. For the three-front wave solutions, we set
$$f\left(x,y,z,t\right)=1+{e}^{\theta_1}+{e}^{\theta_2}+{e}^{\theta_3},$$
(144)
and proceeding as before we find the three-front wave solutions
$$v\left(x,y,z,t\right)=\frac{k_1{e}^{\theta_1}+{k}_2{e}^{\theta_2}+{k}_3{e}^{\theta_3}}{1+{e}^{\theta_1}+{e}^{\theta_2}+{e}^{\theta_3}},$$
(145)
obtained upon using Eq. 129.

This shows that the (3 + 1)-dimensional extended form of the modified KP equation gives multiple-wavefront solutions. The additional extension terms did not kill the integrability of the modified KP equation. However, the additional terms affected the dispersion relation.

Based on the obtained results, the general-front wave solutions can be set in the form
$$v\left(x,y,z,t\right)=\frac{{\displaystyle {\sum}_{i=1}^N{k}_i{e}^{\theta_i}}}{1+{\displaystyle {\sum}_{i=1}^N{e}^{\theta_i}}}.$$
(146)

### Multiple–Singular Front Wave Solutions

To determine the singular-front wave solutions, we use the auxiliary function
$$f\left(x,y,z,t\right)=1-{e}^{k_1x+{k}_1^2y+{k}_1z-\left(\frac{4{k}_i^3+4\alpha {k}_i^2+\left(3+4\beta +4\gamma \right){k}_i}{4}\right)t}.$$
(147)
This in turn gives the single–singular front wave solution
$$u\left(x,y,z,t\right)=-\frac{k_1{e}^{k_1x+{k}_1^2y+{k}_1z-\left(\frac{4{k}_1^3+4\alpha {k}_1^2+\left(3+4\beta +4\gamma \right){k}_1}{4}\right)t}}{1-{e}^{k_1x+{k}_1^2y+{k}_1z-\left(\frac{4{k}_1^3+4\alpha {k}_1^2+\left(3+4\beta +4\gamma \right){k}_1}{4}\right)t}},$$
(148)
Fig. 7 below shows the obtained singular-kink solution v(x, y, z, t).
For the two–singular front wave solutions we use the auxiliary function
$$f\left(x,y,z,t\right)=1-{e}^{\theta_1}-{e}^{\theta_2},$$
(149)
and proceed as before to obtain the two–singular front wave solutions
$$v\left(x,y,z,t\right)=-\frac{k_1{e}^{\theta_1}+{k}_2{e}^{\theta_2}}{1-{e}^{\theta_1}-{e}^{\theta_2}},$$
(150)
where θ i , i = 1, 2 are given in Eq. 138.
For the three-front wave solutions, we obtain
$$f\left(x,y,z,t\right)=1-{e}^{\theta_1}-{e}^{\theta_2}-{e}^{\theta_3}.$$
(151)
Proceeding as before leads to the three–singular wavefront solutions
$$u\left(x,y,z,t\right)= \ln \left(1-{e}^{k_1x+{k}_1^2y-{k}_1^3t}-{e}^{k_2x+{k}_2^2y-{k}_2^3t}-{e}^{k_3x+{k}_3^2y-{k}_3^3t}\right),$$
(152)
and as a result, the three–singular front wave solutions are given by
$$u\left(x,y,z,t\right)=-\frac{k_1{e}^{\theta_1}+{k}_2{e}^{\theta_2}+{k}_3{e}^{\theta_3}}{1-{e}^{\theta_1}-{e}^{\theta_2}-{e}^{\theta_3}},$$
(153)
The general–singular front wave solutions can be set in the form
$$v\left(x,y,z,t\right)=-\frac{{\displaystyle {\sum}_{i=1}^N{k}_i{e}^{\theta_i}}}{1-{\displaystyle {\sum}_{i=1}^N{e}^{\theta_i}}}.$$
(154)

### Discussion

The (3 + 1)-dimensional extended form of the modified KP equation is examined. Multiple-front wave solutions and multiple–singular front wave solutions are formally derived. The analysis confirms the integrability of this equation, and it also shows that the extension terms did not kill the integrability of the modified KP equation.

## An Extended (3 + 1)-Dimensional B-Type Kadomtsev-Petviashvili Equation with a Variety of Distinct Multiple-Soliton Solutions

In this work, we examine an extended (3 + 1)-dimensional B-type Kadomtsev-Petviashvili (BKP) equation. We plan to derive a variety of distinct multiple-soliton solutions. We will apply the simplified form of the Hirota direct method to derive sets of distinct kinds of multiple-soliton solutions under specific conditions. The coefficients of the spatial variables play the role of showing that the multiple-soliton solution of this problem is not a unique set.

It is well known that the Kadomtsev-Petviashvili (KP) equation (Kadomtsev and Petviashvili 1970) describes weakly dispersive and small-amplitude waves propagating in a quasi–two-dimensional media. The KP equation is integrable and can be expressed in the Lax form. The KP hierarchy of B type (BKP) possesses many integrable structures such as Lax formulation, tau function, and fermion representation. The BKP equation was given this name because it is a B-type KP equation (Ma et al. 2011; Wazwaz 2011b).

A (3 + 1)-dimensional nonlinear generalized B-type KP equation (Wazwaz 2011e), given by
$${u}_{yt}-{u}_{xxxy}-3{\left({u}_x{u}_y\right)}_x+3{u}_{xx}+3{u}_{zz}=0,$$
(155)
was investigated, and multiple-soliton solutions were formally derived.
To make further progress, we introduce an extension to Eq. 155 in the form
$${u}_{yt}-{u}_{xxxy}-3{\left({u}_x{u}_y\right)}_x+3{u}_{xx}+3{u}_{yy}+3{u}_{zz}=0,$$
(156)
where we added the term 3u yy which is the second derivative with respect to the spatial variable y. The aim of this section is twofold. First we will show that a variety of distinct multiple-soliton solutions exists under specific conditions of the coefficients of the spatial variables. We next aim to show that this problem gives more than one set of multiple-soliton solutions. We will achieve our goal by using the simplified form of the Hirota direct method. The steps of the simplified form were introduced earlier; hence we focus on implementing this method.

### Multiple-Soliton Solutions

In this section we will study the extended (3 + 1)-dimensional B-type Kadomtsev-Petviashvili (BKP) equation
$${u}_{yt}-{u}_{xxxy}-3{\left({u}_x{u}_y\right)}_x+3{u}_{xx}+3{u}_{yy}+3{u}_{zz}=0.$$
(157)
Substituting
$$u\left(x,y,z,t\right)={e}^{\theta_i},{\theta}_i={k}_ix+{r}_iy+{s}_iz-{\omega}_it,$$
(158)
into the linear terms of Eq. 157, and solving the resulting equation for ω i , the dispersion relation
$${\omega}_i=\frac{3\left({k}_i^2+{r}_i^2+{s}_i^2\right)-{r}_i{k}_i^3}{r_i},i=1,2,3,$$
(159)
follows immediately. As a result, the dispersion variable θ i becomes
$${\theta}_i={k}_ix+{r}_iy+{s}_iz-\left(\frac{3\left({k}_i^2+{r}_i^2+{s}_i^2\right)-{r}_i{k}_i^3}{r_i}\right) t.$$
(160)
We next use Cole-Hopf transformation of the solution
$$u\left(x,y,z,t\right)=R\left( \ln\ f\right)x,$$
(161)
to determine R, where the auxiliary function f(x, y, z, t) is given by
$$f\left(x,y,z,t\right)=1+{e}^{k_ix+{r}_iy+{s}_iz-\left(\frac{3\left({k}_i^2+{r}_i^2+{s}_i^2\right)-{r}_i{k}_i^3}{r_i}\right)t},$$
(162)
into Eq. 157 and solve to find that
$$R=2.$$
(163)
Moreover, to obtain multiple-soliton solutions, we were able to find five distinct sets of specific constraints on the coefficients k i , r i , and s i of the spatial variables x, y, and z respectively. The specific constraints that we will examine in the forthcoming sections are given by
$$\begin{array}{c}\hfill {r}_i={k}_i^n,n\ge 0\hfill \\ {}{s}_i={k}_i,\hfill \end{array}$$
(164)
$$\begin{array}{c}{r}_i={k}_i\hfill \\ {}\hfill {s}_i={k}_i^n,n\ge 0,\hfill \end{array}$$
(165)
$$\begin{array}{c}\hfill {r}_i={k}_i^n,n\ge 0\hfill \\ {}{s}_i={k}_i^n,\hfill \end{array}$$
(166)
$$\begin{array}{c}\hfill {r}_i={k}_i^{\frac{1}{n}},n\ge 0\hfill \\ {}{s}_i={k}_i,\hfill \end{array}$$
(167)
and
$$\begin{array}{c}\hfill {r}_i={k}_i^{\frac{1}{n}},n\ge 0,\hfill \\ {}{s}_i={k}_i^{\frac{1}{n}}.\hfill \end{array}$$
(168)
where k i is left as a free parameter.

In what follows we will investigate each case independently. The dispersion relation and the set of multiple-soliton solutions will be derived for each case.

### First Case

$${r}_i={k}_i^n$$, s i = k i. For the first case, the dispersion relation and the dispersion variable read
$${\omega}_i=\frac{3\left(2{k}_i^2+{k}_i^{2n}\right)-{k}_i^{n+3}}{k_i^n},i=1,2,3,$$
(169)
and
$${\theta}_i={k}_ix+{k}_i^ny+{k}_iz-\left(\frac{3\left(2{k}_i^2+{k}_i^{2n}\right)-{k}_i^{n+3}}{k_i^n}\right) t.$$
(170)
respectively.
Substituting $${r}_1={k}_1^n$$, s 1 = k 1 in Eq. 161 the single-soliton solution for the extended (3 + 1)-dimensional nonlinear evolution equation (Eq. 157) is given by
$$u\left(x,y,z,t\right)=\frac{2{k}_1{e}^{k_1x+{k}_2^ny+{k}_1z-\left(\frac{3\left(2{k}_1^2+{k}_1^{2n}\right)-{k}_1^{n+3}}{k_1^n}\right)t}}{1+{e}^{k_1x+{k}_1^ny+{k}_1z-\left(\frac{3\left(2{k}_1^2+{k}_1^{2n}\right)-{k}_1^{n+3}}{k_1^n}\right)t}}.$$
(171)
Figure 8 below shows the soliton solution u(x, y, z, t) for this case for n = 3 and n = 7 respectively.
It is obvious that the single-soliton solution depends on n among the other factors. For the two-soliton solutions, we use the auxiliary function
$$f\left(x,y,z,t\right)=1+{e}^{\theta_1}+{e}^{\theta_2}+{a}_{12}{e}^{\theta_1+{\theta}_2},$$
(172)
into Eq. 157, where θ 1 and θ 2 are given in Eq. 170, to obtain the phase shift a 12 by
$${a}_{12}=\frac{{\left({k}_1{k}_2\right)}^{n+1}\left({k}_1^n-{k}_2^n\right)\;\left({k}_1-{k}_2\right)+2{\left({k}_1^n{k}_2-{k}_2^n{k}_1\right)}^2}{{\left({k}_1{k}_2\right)}^{n+1}\left({k}_1^n+{k}_2^n\right)\;\left({k}_1+{k}_2\right)+2{\left({k}_1^n{k}_2-{k}_2^n{k}_1\right)}^2},$$
(173)
and this can be generalized to
$${a}_{ij}=\frac{{\left({k}_i{k}_j\right)}^{n+1}\left({k}_i^n-{k}_j^n\right)\;\left({k}_i-{k}_j\right)+2{\left({k}_i^n{k}_j-{k}_j^n{k}_i\right)}^2}{{\left({k}_i{k}_j\right)}^{n+1}\left({k}_i^n+{k}_j^n\right)\;\left({k}_i+{k}_j\right)+2{\left({k}_i^n{k}_j-{k}_j^n{k}_i\right)}^2},1\le i<j\le 3.$$
(174)
Substituting the last results into Eq. 161 gives the two-soliton solutions.
For the three-soliton solutions, we use the auxiliary function
$$f\left(x,y,z,t\right)=1+{e}^{\theta_1}+{e}^{\theta_2}+{e}^{\theta_3}+{a}_{12}{e}^{\theta_1+{\theta}_2}+{a}_{13}{e}^{\theta_1+{\theta}_3}+{a}_{23}{e}^{\theta_2+{\theta}_3}+{b}_{123}{e}^{\theta_1+{\theta}_2+{\theta}_3},$$
(175)
into Eq. 157 and solve to find that
$${b}_{123}={a}_{12}{a}_{13}{a}_{23}.$$
(176)
This result indicates that three-soliton solutions and hence multiple-soliton solutions exist for finite N.

### Second Case

r i = k i , $${s}_i={k}_i^n.$$ For the second case, the dispersion relation and the dispersion variable read
$${\omega}_i=\frac{3\left(2{k}_i^2+{k}_i^{2n}\right)-{k}_i^4}{k_i},i=1,2,3,$$
(177)
and
$${\theta}_i={k}_ix+{k}_iy+{k}_i^nz-\left(\frac{3\left(2{k}_i^2+{k}_i^{2n}\right)-{k}_i^4}{k_i}\right)\;t.$$
(178)
respectively.
Substituting r 1 = k 1, $${s}_1={k}_1^n$$ in Eq. 161 the single-soliton solution for the extended (3 + 1)-dimensional nonlinear evolution equation (Eq. 157) is given by
$$u\left(x,y,z,t\right)=\frac{2{k}_1{e}^{k_1x+{k}_1y+{k}_1^nz-\left(\frac{3\left(2{k}_1^2+{k}_1^{2n}\right)-{k}_1^4}{k_1}\right)t}}{1+{e}^{k_1x+{k}_1y+{k}_1^nz-\left(\frac{3\left(2{k}_1^2+{k}_1^{2n}\right)-{k}_1^4}{k_1}\right)t}}.$$
(179)
For the two-soliton solutions, we use the auxiliary function
$$f\left(x,y,z,t\right)=1+{e}^{\theta_1}+{e}^{\theta_2}+{a}_{12}{e}^{\theta_1+{\theta}_2},$$
(180)
into Eq. 157, where θ 1 and θ 2 are given in Eq. 186, to obtain the phase shift a 12 by
$${a}_{12}=\frac{{\left({k}_1{k}_2\right)}^2{\left({k}_1-{k}_2\right)}^2+{\left({k}_1^n{k}_2-{k}_2^n{k}_1\right)}^2}{{\left({k}_1{k}_2\right)}^2{\left({k}_1+{k}_2\right)}^2+{\left({k}_1^n{k}_2-{k}_2^n{k}_1\right)}^2},$$
(181)
and this can be generalized to
$${a}_{ij}=\frac{{\left({k}_i{k}_j\right)}^2{\left({k}_i-{k}_j\right)}^2+{\left({k}_i^n{k}_j-{k}_j^n{k}_i\right)}^2}{{\left({k}_i{k}_j\right)}^2{\left({k}_i+{k}_j\right)}^2+{\left({k}_i^n{k}_j-{k}_j^n{k}_i\right)}^2},1\le i<j\le 3.$$
(182)
Substituting the last results into Eq. 161 gives the two-soliton solutions.
For the three-soliton solutions, we use the auxiliary function
$$f\left(x,y,z,t\right)=1+{e}^{\theta_1}+{e}^{\theta_2}+{e}^{\theta_3}+{a}_{12}{e}^{\theta_1+{\theta}_2}+{a}_{13}{e}^{\theta_1+{\theta}_3}+{a}_{23}{e}^{\theta_2+{\theta}_2}+{b}_{123}{e}^{\theta_1+{\theta}_2+{\theta}_3},$$
(183)
into Eq. 157, and solve to find that
$${b}_{123}={a}_{12}{a}_{13}{a}_{23}.$$
(184)
This result indicates that three-soliton solutions and hence multiple-soliton solutions exist for finite N.

### Third Case

$${r}_i={k}_i^n$$, $${s}_i={k}_i^n.$$ For the fourth case, the dispersion relation and the dispersion variable read
$${\omega}_i=\frac{3\left({k}_i^2+2{k}_i^{2n}\right)-{k}_i^{n+3}}{k_i^n},i=1,2,3,$$
(185)
and
$${\theta}_i={k}_ix+{k}_i^ny+{k}_i^nz-\left(\frac{3\left({k}_i^2+2{k}_i^{2n}\right)-{k}_i^{n+3}}{k_i^n}\right) t.$$
(186)
respectively.
Substituting $${r}_1={k}_1^n$$, $${s}_1={k}_1^n$$ in Eq. 161 the single-soliton solution for the extended (3 + 1)-dimensional nonlinear evolution equation (Eq. 157) is given by
$$u\left(x,y,z,t\right)=\frac{2{k}_1{e}^{k_1x+{k}_1^ny+{k}_1^nz-\left(\frac{3\left({k}_1^2+2{k}_1^{2n}\right)-{k}_1^{n+3}}{k_1^n}\right)t}}{1+{e}^{k_1x+{k}_1^ny+{k}_1^nz-\left(\frac{3\left({k}_1^2+2{k}_1^{2n}\right)-{k}_1^{n+3}}{k_1^n}\right)t}}.$$
(187)
For the two-soliton solutions, we use the auxiliary function
$$f\left(x,y,z,t\right)=1+{e}^{\theta_1}+{e}^{\theta_2}+{a}_{12}{e}^{\theta_1+{\theta}_2},$$
(188)
into Eq. 157, where θ 1 and θ 2 are given in Eq. 186, to obtain the phase shift a 12 by
$${a}_{12}=\frac{{\left({k}_1{k}_2\right)}^{n+1}\left({k}_1^n-{k}_2^n\right)\;\left({k}_1-{k}_2\right)+{\left({k}_1^n{k}_2-{k}_2^n{k}_1\right)}^2}{{\left({k}_1{k}_2\right)}^{n+1}\left({k}_1^n+{k}_2^n\right)\;\left({k}_1+{k}_2\right)+{\left({k}_1^n{k}_2+{k}_2^n{k}_1\right)}^2},$$
(189)
and this can be generalized to
$${a}_{ij}=\frac{{\left({k}_i{k}_j\right)}^{n+1}\left({k}_i^n-{k}_j^n\right)\;\left({k}_i-{k}_j\right)+{\left({k}_i^n{k}_j-{k}_j^n{k}_i\right)}^2}{{\left({k}_i{k}_j\right)}^{n+1}\left({k}_i^n+{k}_j^n\right)\;\left({k}_i+{k}_j\right)+{\left({k}_i^n{k}_j+{k}_j^n{k}_i\right)}^2},1\le i<j\le 3.$$
(190)
Substituting the last results into Eq. 161 gives the two-soliton solutions.
For the three-soliton solutions, we use the auxiliary function
$$f\left(x,y,z,t\right)=1+{e}^{\theta_1}+{e}^{\theta_2}+{e}^{\theta_3}+{a}_{12}{e}^{\theta_1+{\theta}_2}+{a}_{13}{e}^{\theta_1+{\theta}_3}+{a}_{23}{e}^{\theta_2+{\theta}_2}+{b}_{123}{e}^{\theta_1+{\theta}_2+{\theta}_3},$$
(191)
into Eq. 157 and solve to find that
$${b}_{123}={a}_{12}{a}_{13}{a}_{23}.$$
(192)
This result indicates that three-soliton solutions and hence multiple-soliton solutions exist for finite N.

### Fourth Case

$${r}_i=\frac{1}{k_i^n}$$, s i = k i. For the fourth case, the dispersion relation and the dispersion variable read
$${\omega}_i=3\left(2{k}_i^{n+2}+{k}_i^{-n}\right)-{k}_i^3,i=1,2,3,$$
(193)
and
$${\theta}_i={k}_ix+{k}_i^{-n}y+{k}_iz-\left(3\left(2{k}_i^{n+2}+{k}_i^{-n}\right)-{k}_i^3\right) t.$$
(194)
respectively.
Substituting $${r}_1=\frac{1}{k_1^n}$$, s 1 = k 1 (Eq. 161) the single-soliton solution for the extended (3 + 1)-dimensional nonlinear evolution equation (Eq. 157) is given by
$$u\left(x,y,z,t\right)=\frac{2{k}_1{e}^{k_1x+{k}_1^{-n}y+{k}_1z-\left(3\left(2{k}_1^{n+2}+{k}_1^{-n}\right)-{k}_1^3\right)t}}{1+{e}^{k_1x+{k}_1^{-n}y+{k}_1z-\left(3\left(2{k}_1^{n+2}+{k}_1^{-n}\right)-{k}_1^3\right)t}}.$$
(195)
For the two-soliton solutions, we use the auxiliary function
$$f\left(x,y,z,t\right)=1+{e}^{\theta_1}+{e}^{\theta_2}+{a}_{12}{e}^{\theta_1+{\theta}_2},$$
(196)
into Eq. 157, where θ 1 and θ 2 are defined earlier, to obtain the phase shift a 12 by
$${a}_{12}=\frac{{\left({k}_1{k}_2\right)}^{1-2n}\left({k}_2^n-{k}_1^n\right)\;\left({k}_1-{k}_2\right)+2{\left({k}_1{k}_2\right)}^{-2n}{\left({k}_2^{n+1}-{k}_1^{n+1}\right)}^2}{{\left({k}_1{k}_2\right)}^{1-2n}\left({k}_2^n+{k}_1^n\right)\;\left({k}_1+{k}_2\right)+2{\left({k}_1{k}_2\right)}^{-2n}{\left({k}_2^{n+1}-{k}_1^{n+1}\right)}^2},$$
(197)
and this can be generalized to
$${a}_{ij}=\frac{{\left({k}_i{k}_j\right)}^{1-2n}\left({k}_j^n-{k}_i^n\right)\;\left({k}_i-{k}_j\right)+2{\left({k}_i{k}_j\right)}^{-2n}{\left({k}_j^{n+1}-{k}_i^{n+1}\right)}^2}{{\left({k}_i{k}_j\right)}^{1-2n}\left({k}_j^n+{k}_i^n\right)\;\left({k}_i+{k}_j\right)+2{\left({k}_i{k}_j\right)}^{-2n}{\left({k}_j^{n+1}-{k}_i^{n+1}\right)}^2}.$$
(198)
Substituting the last results into Eq. 161 gives the two-soliton solutions.
For the three-soliton solutions, we proceed as before to find that
$${b}_{123}={a}_{12}{a}_{13}{a}_{23}.$$
(199)
This result indicates that three-soliton solutions and hence multiple-soliton solutions exist for finite N.

### Fifth Case

$${r}_i=\frac{1}{k_i^n}$$, $${s}_i=\frac{1}{k_i^n}.$$ For the last case, the dispersion relation and the dispersion variable read
$${\omega}_i=3\left({k}_i^{n+2}+2{k}_i^{-n}\right)-{k}_i^3,i=1,2,3,$$
(200)
and
$${\theta}_i={k}_ix+{k}_i^{-n}y+{k}_i^{-n}z-\left(3\left({k}_i^{n+2}+2{k}_i^{-n}\right)-{k}_i^3\right) t.$$
(201)
respectively.
Substituting $${r}_1=\frac{1}{k_1^n}$$, $${s}_1=\frac{1}{k_1^n}$$ (Eq. 161) the single-soliton solution for the extended (3 + 1)-dimensional nonlinear evolution equation (Eq. 157) is given by
$$u\left(x,y,z,t\right)=\frac{2{k}_1{e}^{k_1x+{k}_1^{-n}y+{k}_1^{-n}z-\left(3\left({k}_1^{n+2}+2{k}_1^{-n}\right)-{k}_1^3\right)t}}{1+{e}^{k_1x+{k}_1^{-n}y+{k}_1^{-n}z-\left(3\left({k}_1^{n+2}+2{k}_1^{-n}\right)-{k}_1^3\right)t}}.$$
(202)
For the two-soliton solutions, we use the auxiliary function
$$f\left(x,y,z,t\right)=1+{e}^{\theta_1}+{e}^{\theta_2}+{a}_{12}{e}^{\theta_1+{\theta}_2},$$
(203)
into Eq. 157, where θ 1 and θ 2 are given before, to obtain the phase shift a 12 by
$${a}_{12}=\frac{{\left({k}_1{k}_2\right)}^{1-2n}\left({k}_2^n-{k}_1^n\right)\;\left({k}_1-{k}_2\right)+{\left({k}_1{k}_2\right)}^{-2n}{\left({k}_2^{n+1}-{k}_1^{n+1}\right)}^2}{{\left({k}_1{k}_2\right)}^{1-2n}\left({k}_2^n+{k}_1^n\right)\;\left({k}_1+{k}_2\right)+{\left({k}_1{k}_2\right)}^{-2n}{\left({k}_2^{n+1}-{k}_1^{n+1}\right)}^2},$$
(204)
and this can be generalized to
$${a}_{ij}=\frac{{\left({k}_i{k}_j\right)}^{1-2n}\left({k}_j^n-{k}_i^n\right)\;\left({k}_i-{k}_j\right)+{\left({k}_i{k}_j\right)}^{-2n}{\left({k}_j^{n+1}-{k}_i^{n+1}\right)}^2}{{\left({k}_i{k}_j\right)}^{1-2n}\left({k}_j^n+{k}_i^n\right)\;\left({k}_i+{k}_j\right)+{\left({k}_i{k}_j\right)}^{-2n}{\left({k}_i^{n+1}-{k}_j^{n+1}\right)}^2}.$$
(205)
Substituting the last results into Eq. 161 gives the two-soliton solutions.
For the three-soliton solutions, we proceed as before to find that
$${b}_{123}={a}_{12}{a}_{13}{a}_{23}.$$
(206)
This result indicates that three-soliton solutions and hence multiple-soliton solutions exist for finite N.

### Discussion

In this work we conducted an analysis on an extended (3 + 1)-dimensional B-type KP equation. We obtained five sets of specific values of the coefficients of the spatial variables. This result led to a variety of different kinds of multiple-soliton solutions for the same (3 + 1)-dimensional nonlinear equation, where each set of specific multiple-soliton solutions has significant features and distinct physical structures. Table 2 below summarizes the parameters and the dispersion relations.
Table 2

Parameters and dispersion relations of the five case models

Case

The coefficients k i , r i , s i

Dispersion relations

First case

$${k}_i,{r}_i={k}_i^n \mathrm{and} {s}_i={k}_i$$

$${\omega}_i=\frac{3\left({2}_i^2+{k}_i^{2n}\right)-{k}_i^{n+3}}{k_i^n}$$

Second case

$${k}_i,\;{r}_i={k}_i, {s}_i={k}_i^n$$

$${\omega}_i=\frac{3\left(2{k}_i^2+{k}_i^{2n}\right)-{k}_i^4}{k_i}$$

Third case

$${k}_i,\;{r}_i={k}_i^n, {s}_i={k}_i^n$$

$${\omega}_i=\frac{3\left(2{k}_i^2+2{k}_i^{2n}\right)-{k}_i^{n+3}}{k_i^n}$$

Fourth case

$${k}_i,\;{r}_i=\frac{1}{k_i^n}, {s}_i={k}_i$$

$${\omega}_i=3\left(2{k}_i^{n+2}+{k}_i^{-n}\right)-{k}_i^3$$

Fifth case

$${k}_i,\;{r}_i=\frac{1}{k_i^n}, {s}_i=\frac{1}{k_i^n}$$

$${\omega}_i=3\left({k}_i^{n+2}+2{k}_i^{-n}\right)-{k}_i^3$$

Moreover, we also obtained distinct phase shifts a ij , 1 ≤ i < j < 3. Table 3 below shows the distinct phase shifts for all five cases.
Table 3

The phase shifts a ij , 1 ≤ i < j < 3, of the five case models

Case

The phase shifts a ij , 1 ≤ i < j < 3

First case

$${a}_{ij}=\frac{{\left({k}_i{k}_j\right)}^{n+1}\left({k}_i^n-{k}_j^n\right)\left({k}_i-{k}_j\right)+2{\left({k}_i^n{k}_j-{k}_j^n{k}_i\right)}^2}{{\left({k}_i{k}_j\right)}^{n+1}\left({k}_i^n+{k}_j^n\right)\left({k}_i+{k}_j\right)+2{\left({k}_i^n{k}_j-{k}_j^n{k}_i\right)}^2}$$

Second case

$${a}_{ij}=\frac{{\left({k}_i{k}_j\right)}^2{\left({k}_i-{k}_j\right)}^2+{\left({k}_i^n{k}_j-{k}_j^n{k}_i\right)}^2}{{\left({k}_i{k}_j\right)}^2{\left({k}_i+{k}_j\right)}^2+{\left({k}_i^n{k}_j-{k}_j^n{k}_i\right)}^2}$$

Third case

$${a}_{ij}=\frac{{\left({k}_i{k}_j\right)}^{n+1}\left({k}_i^n-{k}_j^n\right)\left({k}_i-{k}_j\right)+{\left({k}_i^n{k}_j-{k}_j^n{k}_i\right)}^2}{{\left({k}_i{k}_j\right)}^{n+1}\left({k}_i^n+{k}_j^n\right)\left({k}_i+{k}_j\right)+2{\left({k}_i^n{k}_j-{k}_j^n{k}_i\right)}^2}$$

Fourth case

$${a}_{ij}=\frac{{\left({k}_i{k}_j\right)}^{1-2n}\left({k}_j^n-{k}_i^n\right)\left({k}_i-{k}_j\right)+2{\left({k}_i{k}_j\right)}^{-2n}{\left({k}_i^{n+1}-{k}_j^{n+1}\right)}^2}{{\left({k}_i{k}_j\right)}^{1-2n}\left({k}_j^n+{k}_i^n\right)\left({k}_i+{k}_j\right)+2\Big({k}_i{k}_j-2n{\left({k}_j^{n+1}{k}_i^{n+1}\right)}^2}$$

Fifth case

$${a}_{ij}=\frac{{\left({k}_i{k}_j\right)}^{1-2n}\left({k}_j^n-{k}_i^n\right)\left({k}_i-{k}_j\right)+2{\left({k}_i{k}_j\right)}^{-2n}{\left({k}_i^{n+1}-{k}_j^{n+1}\right)}^2}{{\left({k}_i{k}_j\right)}^{1-2n}\left({k}_j^n+{k}_i^n\right)\left({k}_i+{k}_j\right)+{\left({k}_i{k}_j\right)}^{-2n}{\left({k}_j^{n+1}-{k}_i^{n+1}\right)}^2}$$

To our understanding, the presence of a variety of multiple-soliton solutions for the same nonlinear equation has not been examined before. Does this variety of multiple-soliton solutions exist for other models? This question will be addressed in further works.

## Couplings of the Fifth-Order Integrable Sawada-Kotera and Lax Equations

We construct nonlinear integrable couplings of the fifth-order nonlinear integrable Sawada- Kotera (SK) equation and Lax (Lax) equation. We use the algebra of coupled scalars to construct the two classes of couplings. We study the constructed couplings by using the simplified Hirota method. We show that these classes of couplings possess the same multiple-soliton solutions as the multiple-soliton solutions of the SK and the Lax equations, with one change in the sign of the transformation used. This change of signs exhibits soliton solution, then antisoliton solution consecutively.

The theory of nonlinear integrable couplings of ordinary soliton systems attracted researchers for more works on this topic. The couplings of nonlinear equations was presented in Zhang and Tam (2010) and further studied by Blaszak et al. (2012) and by some of the references therein. Integrable couplings can be defined as coupled systems of integrable equations. Many powerful methods for constructing integrable couplings have been developed, such as the perturbation method used by Ma et al. in Ma and Fuchssteiner (1996), the enlarged Lie algebra method used by Zhang et al. (Zhang and Tam 2010), the non-semisimple Lie algebras, the algebra of coupled scalars (Blaszak et al. 2012), and other methods as well. It is known that for any integrable couplings, it must include the given integrable equation as a subsystem.

The well-known fifth-order KdV (fKdV) equation in its standard form reads (Kadomtsev and Petviashvili 1970)
$${u}_t+\alpha {u}^2{u}_x+\beta {u}_x{u}_{xx}+{\gamma uu}_{3x}+{u}_{5x}0,$$
(207)
where α, β, and γ are arbitrary nonzero and real parameters and u = u(x, t) is a sufficiently smooth function. The fifth-order KdV equations (Eq. 207) involve two dispersive terms u3x and u5x. Because the parameters α, β, and γ are arbitrary and take different values, this will drastically change the characteristics of the fKdV equation (Eq. 207). A variety of the fKdV equations can be developed by changing the real values of the parameters α, β, and γ.
However, five well-known forms of the fKdV are of particular interest given by Kadomtsev and Petviashvili (1970):
1. (i)
The Sawada-Kotera (SK) equation is given by
$${u}_t+5{u}^2{u}_x+5{u}_x{u}_{xx}+5u{u}_{3x}+{u}_{5x}=0,$$
(208)
characterized by
$$\beta =\gamma, \alpha =\frac{1}{5}{\gamma}^2,$$
(209)
where γ = 5 is selected.

2. (ii)
The Caudrey-Dodd-Gibbon equation (CDG) is given by
$${u}_t+180{u}^2ux+30{u}_x{u}_{xx}+30u{u}_{xxx}+{u}_{xxxxx}=0,$$
(210)
characterized by
$$\beta =\gamma, \alpha =\frac{1}{5}{\gamma}^2,$$
(211)
where γ = 30 is selected.

3. (iii)
$$ut+30{u}^2{u}_x+20{u}_x{u}_{xx}+10u{u}_{3x}+{u}_{5x}=0,$$
(212)
characterized by
$$\beta =2\gamma, \alpha =\frac{3}{10}{\gamma}^2,$$
(213)
where γ = 10 is selected.

4. (iv)
$${u}_t+20{u}^2{u}_x+25{u}_x{u}_{xx}+10u{u}_{3x}+{u}_{5x}=0,$$
(214)
characterized by
$$\beta =\frac{5}{2}\gamma, \alpha =\frac{1}{5}{\gamma}^2,$$
(215)
where γ = 10 is selected.

5. (v)
The Ito equation is given as
$${u}_t+2{u}^2ux+6{u}_x{u}_{xx}+3u{u}_{3x}+{u}_{5x}=0,$$
(216)
characterized by
$$\beta =2\gamma, \alpha =\frac{2}{9}{\gamma}^2,$$
(217)
where γ = 3 is selected.

The first four aforementioned equations SK, CDG, Lax, and KK are completely integrable equations that have infinite sets of conserved quantities and give multiple-soliton solutions. However, the Ito equation is not completely integrable but has a limited number of conserved quantities.

Recently, two other fifth-order integrable equations were established by Wazwaz (Lenells 2005; Liu et al. 2004), given by
$${u}_{ttt}-{u}_{txxxx}-4{\left({u}_x{u}_t\right)}_{xx}-4{\left({u}_x{u}_{xt}\right)}_x=0,$$
(218)
and
$${u}_{ttt}-{u}_{txxxx}-\alpha \left({u}_x{u}_t\right)xx-\beta {\left({u}_x{u}_{xt}\right)}_x=0,$$
(219)
that give multiple-kink solutions. The third-order derivative with respect to the temporal variable t and the mixed fifth-order derivative compared to the first-order derivative with respect to t and the fifth-order derivative with respect to the spatial variable x of the aforementioned fifth-order equations SK, CDG, Lax, KK, and Ito equations are particularly noteworthy.

Our aim from this work is twofold. The first goal is to employ the newly developed algebra of coupled scalars to construct nonlinear integrable couplings for the fifth-order SK equation and the Lax equation, hence we will use first the generalized form (Eq. 207). We aim secondly to study the developed classes of the couplings of the SK equation (Eq. 208) and the couplings of the Lax equation (Eq. 212) respectively. We aim to show that each coupling possesses the same features as the fifth-order standard equation but differs only in the signs of the transformations used. This difference exhibits soliton solutions for some equations and antisoliton solutions for others.

### Constructing Nonlinear Integrable Couplings

Blaszak et al. (2012) introduced a practical method developed to construct couplings of the integrable equations. To summarize the approach, it was assumed that if
$$\mathrm{a}={\displaystyle \sum_{i=1}^n{a}_i{\mathrm{e}}_i},$$
(220)
where ei are the basis vectors, then
$$\mathrm{a}\cdot \mathrm{b}=\left(\begin{array}{r}\hfill {a}_1\\ {}\hfill {a}_2\\ {}\hfill \vdots \\ {}\hfill {a}_n\end{array}\right)\cdot \left(\begin{array}{r}\hfill {b}_1\\ {}\hfill {b}_2\\ {}\hfill \vdots \\ {}\hfill {b}_n\end{array}\right)=\left(\begin{array}{r}\hfill {c}_1\\ {}\hfill {c}_2\\ {}\hfill \vdots \\ {}\hfill {c}_n\end{array}\right)$$
(221)
where
$${c}_i={a}_i{b}_i+{a}_i\left({\displaystyle \sum_{k=1}^{i-1}{b}_k}\right)+\left({\displaystyle \sum_{k=1}^{i-1}{a}_k}\right){b}_i.$$
(222)
This means that the value of the coefficient c i is given by a i b i plus terms depending on lower-order elements a k , b k with k < i. This method was called the algebra of coupled scalars and was found to be unital, commutative, and associative. For n = 3, we find
$$\left(\begin{array}{c}\hfill {a}_1\hfill \\ {}\hfill {a}_2\hfill \\ {}\hfill {a}_3\hfill \end{array}\right)\cdot \left(\begin{array}{c}\hfill {b}_1\hfill \\ {}\hfill {b}_2\hfill \\ {}\hfill {b}_3\hfill \end{array}\right)=\left(\begin{array}{c}\hfill {a}_1{b}_1\hfill \\ {}\hfill {a}_2{b}_2+{a}_2{b}_1+{a}_1{b}_2\hfill \\ {}\hfill {a}_3{b}_3+{a}_3\left({b}_1+{b}_2\right)+\left({a}_1+{a}_2\right){b}_3\hfill \end{array}\right).$$
(223)
For more details about the algebra of coupled scalars and its properties, read Ref. Hirota and Ito 1983.
Using the algebra of coupled scalars developed in Hirota and Ito (1983), we introduce a one-field soliton system
$${u}_t=K\left[u\right]\equiv K\left[u,{u}_x,{u}_{xx},{u}_{xxx},\cdots \right],$$
(224)
that can be extended to the system of coupled PDEs of the form
$${\mathrm{u}}_t=K\left[\mathrm{u}\right] \equiv K\left[\mathrm{u},{\mathrm{u}}_x,{\mathrm{u}}_{xx},\cdots \right],$$
(225)
where
$$u=\left(\begin{array}{r}\hfill {u}_1\\ {}\hfill {u}_2\\ {}\hfill \vdots \\ {}\hfill {u}_n\end{array}\right).$$
(226)
Accordingly, the system (Eq. 225) takes the form (Hirota and Ito 1983)
$$\begin{array}{l}{\left({u}_1\right)}_t=K\left[{u}_1\right],\hfill \\ {}{\left({u}_k\right)}_t=K\left[{\displaystyle \sum_{i=1}^k{u}_i}\right]-K\left[{\displaystyle \sum_{i=1}^{k-1}{u}_i}\right],k=2,3,\cdots n.\hfill \end{array}$$
(227)
Using Eq. 207, we can set
$${\mathbf{u}}_t=-{\mathbf{u}}_{xxxxx}-\alpha {\mathbf{u}}^2{\mathbf{u}}_x-\beta {\mathbf{u}}_x{\mathbf{u}}_{xx}-\gamma \mathbf{u}{\mathbf{u}}_{\mathrm{xxx}}.$$
(228)
Inserting Eq. 228 into Eq. 227, we develop the n-coupled fifth-order equation, given by
$$\begin{array}{c}\hfill {\left({u}_1\right)}_t=-{\left({u}_1\right)}_{xxxxx}-\alpha {u}_1^2{\left({u}_1\right)}_x-\beta {\left({u}_1\right)}_x{\left({u}_1\right)}_{xx}-\gamma \left({u}_1\right)\;{\left({u}_1\right)}_{xxx},\hfill \\ {}\hfill {\left({u}_2\right)}_t=-{\left({u}_2\right)}_{xxxxx}-\alpha {u}_1^2{\left({u}_2\right)}_x-\alpha \left(2{u}_1{u}_2+{u}_2^2\right){\left({u}_1+{u}_2\right)}_x,\hfill \\ {}\hfill -\beta {\left({u}_1\right)}_x{\left({u}_2\right)}_{xx}-\beta {\left({u}_2\right)}_x{\left({u}_1+{u}_2\right)}_{xx}-\gamma \left({u}_1\right)\;{\left({u}_2\right)}_{xxx}-\gamma {u}_2{\left({u}_1+{u}_2\right)}_{xxx},\hfill \\ {}\hfill {\left({u}_3\right)}_t=-{\left({u}_3\right)}_{xxxxx}-\alpha {\left({u}_1+{u}_2\right)}^2{\left({u}_3\right)}_x-\alpha \left[2\left({u}_1+{u}_2\right){u}_3+{u}_3^2\right]\;\left[{\left({u}_1+{u}_2\right)}_x+{\left({u}_3\right)}_x\right]\hfill \\ {}\hfill -\beta {\left({u}_1+{u}_2\right)}_x{\left({u}_3\right)}_{xx}-\beta {\left({u}_3\right)}_x{\left({u}_1+{u}_2+{u}_3\right)}_{xx}\hfill \\ {}\hfill -\gamma \left({u}_1+{u}_2\right){\left({u}_3\right)}_{xxx}-\gamma {u}_3{\left({u}_1+{u}_2+{u}_3\right)}_{xxx},\hfill \\ {}\hfill {\left({u}_4\right)}_t=-{\left({u}_4\right)}_{xxxxx}-\alpha {\left({u}_1+{u}_2+{u}_3\right)}^2-{\left({u}_4\right)}_x\hfill \\ {}\hfill -\alpha \left[2\left({u}_1+{u}_2+{u}_3\right){u}_4+{u}_4^2\right] \left[{\left({u}_1+{u}_2+{u}_3\right)}_x+{\left({u}_4\right)}_x\right]\hfill \\ {}\hfill -\beta {\left({u}_1+{u}_2+{u}_3\right)}_x{\left({u}_4\right)}_{xx}-\beta {\left({u}_4\right)}_x{\left({u}_1+{u}_2+{u}_3+{u}_4\right)}_{xx}\hfill \\ {}\hfill -\gamma \left({u}_1+{u}_2+{u}_3\right){\left({u}_4\right)}_{xxx}-\gamma {u}_4{\left({u}_1+{u}_2+{u}_3+{u}_4\right)}_{xxx},\hfill \\ {}\hfill \vdots ,\hfill \\ {}\hfill {\left({u}_n\right)}_t=-{\left({u}_n\right)}_{xxxxx}-\alpha {\left({\displaystyle \sum_{k=1}^{n-1}{u}_k}\right)}^2{\left({u}_n\right)}_x-\alpha \left[2\left({\displaystyle \sum_{k=1}^{n-1}{u}_k}\right) \left({u}_n\right)+{\left({u}_n\right)}^2\right] {\left({\displaystyle \sum_{k=1}^n{u}_k}\right)}_x\hfill \\ {}\hfill -\beta \left[\left({\displaystyle \sum_{k=1}^{n-1}{u}_k}\right) {\left({u}_n\right)}_{xx}\right]-\beta {\left({u}_n\right)}_x{\left({\displaystyle \sum_{k=1}^n{u}_k}\right)}_{xx}\hfill \\ {}\hfill -\gamma \left[\left({\displaystyle \sum_{k=1}^{n-1}{u}_k}\right) {\left({u}_n\right)}_{xxx}\right]-\gamma \left({u}_n\right){\left({\displaystyle \sum_{k=1}^n{u}_k}\right)}_{xxx},n\ge 2.\hfill \end{array}$$
(229)
We first substitute the transformation
$$u\left(x,t\right)=R\frac{\partial^2 \ln\ f\left(x,t\right)}{\partial {x}^2}=R\frac{f {f}_{2x}-{\left({f}_x\right)}^2}{f^2},$$
(230)
into Eq. 207, where the auxiliary function as assumed by the simplified Hirota method reads
$$f=1+{e}^{\theta },$$
(231)
where the wave variable is given by
$$\theta =kx-ct,$$
(232)
and solving the outcome we get
$$\begin{array}{l}\alpha =\frac{\gamma^2+\gamma \beta }{10},\hfill \\ {}R=\frac{60}{\gamma +\beta },\hfill \end{array}$$
(233)
that works for the aforementioned fifth-order equations.

### Couplings of the Fifth-Order Sawada-Kotera Equation

In this section we will examine the couplings of the Sawada-Kotera equation given by
$$\begin{array}{ll}\left({u}_1\right)t\hfill & =-{\left({u}_1\right)}_{xxxxx}-\frac{\gamma^2}{5}{u}_1^2{\left({u}_1\right)}_x-\gamma {\left({u}_1\right)}_x{\left({u}_1\right)}_{xx}-\gamma \left({u}_1\right){\left({u}_1\right)}_{xxx},\hfill \\ {}\left({u}_2\right)t\hfill & =-{\left({u}_2\right)}_{xxxxx}-\frac{\gamma^2}{5}{u}_1^2{\left({u}_2\right)}_x-\frac{\gamma^2}{5}\left(2{u}_1{u}_2+{u}_2^2\right){\left({u}_1+{u}_2\right)}_x\hfill \\ {}\hfill & -\gamma {\left({u}_1\right)}_x{\left({u}_2\right)}_{xx}-\gamma {\left({u}_2\right)}_x{\left({u}_1+{u}_2\right)}_{xx}-\gamma \left({u}_1\right){\left({u}_2\right)}_{xxx}-\gamma {u}_2{\left({u}_1+{u}_2\right)}_{xxx},\hfill \\ {}\left({u}_3\right)t\hfill & =-{\left({u}_3\right)}_{xxxxx}-\frac{\gamma^2}{5}{\left({u}_1+{u}_2\right)}^2{\left({u}_3\right)}_x-\frac{\gamma^2}{5}\left[2\left({u}_1+{u}_2\right){u}_3+{u}_3^2\right]\left[{\left({u}_1+{u}_2\right)}_x+{\left({u}_3\right)}_x\right]\hfill \\ {}\hfill & -\gamma {\left({u}_1+{u}_2\right)}_x{\left({u}_3\right)}_{xx}-\gamma {\left({u}_3\right)}_x{\left({u}_1+{u}_2+{u}_3\right)}_{xx}\hfill \\ {}\hfill & -\gamma \left({u}_1+{u}_2\right){\left({u}_3\right)}_{xxx}-\gamma {u}_3{\left({u}_1+{u}_2+{u}_3\right)}_{xxx},\hfill \\ {}\hfill & \vdots ,\hfill \\ {}\left({u}_n\right)t\hfill & ={\left({u}_n\right)}_{xxxxx}-\frac{\gamma^2}{5}{\left({\displaystyle \sum_{k=1}^{n-1}{u}_k}\right)}^2{\left({u}_n\right)}_x-\frac{\gamma^2}{5}\left[2\left({\displaystyle \sum_{k=1}^{n-1}{u}_k}\right)\left({u}_n\right)+{\left({u}_n\right)}^2\right]{\left({\displaystyle \sum_{k=1}^n{u}_k}\right)}_x\hfill \\ {}\hfill & -\gamma \left[{\left({\displaystyle \sum_{k=1}^{n-1}{u}_k}\right)}_x{\left({u}_n\right)}_{xx}\right]-\gamma {\left({u}_n\right)}_x{\left({\displaystyle \sum_{k=1}^n{u}_k}\right)}_{xx}\hfill \\ {}\hfill & -\gamma \left[\left({\displaystyle \sum_{k=1}^{n-1}{u}_k}\right){\left({u}_n\right)}_{xxx}\right]-\gamma \left({u}_n\right){\left({\displaystyle \sum_{k=1}^n{u}_k}\right)}_{xxx}, n\ge 2,\hfill \end{array}$$
(234)
where we replaced $$\alpha =\frac{\gamma^2}{5}$$ and $$\beta =\gamma$$ as given by the characterization of the Sawada-Kotera equation given earlier. Using the transformation
$${u}_i\left(x,t\right)={R}_i\Big( \ln f{\left(x,t\right)}_{xx}, 1\le i\le n,$$
(235)
into the members of the couplings (Eq. 234) and solving we find that
$${R}_i={\left(-1\right)}^{i+1}\frac{30}{\gamma }, 1\le i\le n.$$
(236)
We next substitute the transformation
$${u}_i\left(x,t\right)={\left(-1\right)}^{i+1}\frac{30}{\gamma }{\left( \ln f\left(x,t\right)\right)}_{xx},$$
(237)
into the linear terms of each member of the couplings (Eq. 234), where the auxiliary function as assumed by the simplified Hirota method reads
$$f=1+{e}^{\theta },$$
(238)
and the wave variable is given by
$$\theta =kx-ct,$$
(239)
and solving the outcome we get the dispersion relation by
$$c={k}^5,$$
(240)
and in view of this result we obtain
$${\theta}_i={k}_ix-{k}_i^5t.$$
(241)
Consequently, for the one-soliton solution, we set
$$f\left(x,t\right)=1+{e}^{k_1x-{k}_1^5t}.$$
(242)
The one-soliton solution is therefore given by
$${u}_i\left(x,t\right)={\left(-i\right)}^{i+1}\frac{30{k}_1^2{e}^{k_1x-{k}_1^5t}}{\gamma {\left(1+{e}^{k_1x-{k}_1^5t}\right)}^2}.$$
(243)
This clearly shows that equations of the couplings give soliton solution if i is odd and antisoliton solution if i is even but with the same amplitude.
To determine the two-soliton solution, we set the auxiliary function by
$$f\left(x,t\right)=1+{e}^{k_1x-{k}_1^5t}+{e}^{k_2x-{k}_2^5t}+{a}_{12}{e}^{\left({k}_1+{k}_2\right)x-\left({k}_1^5+{k}_2^5\right)t},$$
(244)
into Eq. 234 and proceed as before to obtain the phase factor a 12 by
$${a}_{12}=\frac{{\left({k}_1-{k}_2\right)}^2\left({k}_1^2-{k}_1{k}_2+{k}_2^2\right)}{{\left({k}_1+{k}_2\right)}^2\left({k}_1^2+{k}_1{k}_2+{k}_2^2\right)},$$
(245)
and hence can be generalized to
$${a}_{ij}=\frac{{\left({k}_i-{k}_j\right)}^2\left({k}_i^2-{k}_i{k}_j+{k}_j^2\right)}{{\left({k}_i+{k}_j\right)}^2\left({k}_i^2-{k}_i{k}_j+{k}_j^2\right)}, 1\le i<j\le 3.$$
(246)
This in turn gives
$$f\left(x,t\right)=1+{e}^{k_1x-{k}_1^5t}+{e}^{k_2x-{k}_2^5t}+\frac{{\left({k}_1-{k}_2\right)}^2\left({k}_1^2-{k}_1{k}_2+{k}_2^2\right)}{{\left({k}_1+{k}_2\right)}^2\left({k}_1^2+{k}_1{k}_2+{k}_2^2\right)}{e}^{\left({k}_1+{k}_2\right)x-\left({k}_1^5+{k}_2^5\right)t}.$$
(247)
The two-soliton solutions and the two-antisoliton solutions can be obtained by using Eq. 235 for the function f in Eq. 247.
To determine the three soliton solutions we proceed as before and set
$$\begin{array}{l}f\left(x,t\right)=1+{e}^{k_1x-{k}_1^5t}+{e}^{k_2x-{k}_2^5t}+{e}^{k_3x-{k}_3^5t}\hfill \\ {} +{a}_{12}{e}^{\left({k}_1+{k}_2\right)x-\left({k}_1^5+{k}_2^5\right)t}+{a}_{13}{e}^{\left({k}_1+{k}_3\right)x-\left({k}_1^5+{k}_3^5t\right)}+{a}_{23}{e}^{\left({k}_2+{k}_3\right)x-\left({k}_2^5+{k}_3^5t\right)}\\ {} +{b}_{123}{e}^{\left({k}_1+{k}_2+{k}_3\right)x-\left({k}_1^5+{k}_2^5+{k}_3^5\right)t},\end{array}$$
(248)
and proceeding as before to find that
$${b}_{123}={a}_{12}{a}_{13}{a}_{23}.$$
(249)
This shows that the coupling (Eq. 234) gives three-soliton solutions, and hence multiple-soliton solutions. To determine the three-soliton solution explicitly, we proceed as before.

### Couplings of the Fifth-Order Lax Equation

In this section we will examine the couplings of the Lax equation given by
$$\begin{array}{ll}\left({u}_1\right)t\hfill & =-{\left({u}_1\right)}_{xxxxx}-\frac{3}{10}{\gamma}^2{u}_1^2{\left({u}_1\right)}_x-2\gamma {\left({u}_1\right)}_x{\left({u}_1\right)}_{xx}-\gamma \left({u}_1\right){\left({u}_1\right)}_{xxx},\hfill \\ {}\left({u}_2\right)t\hfill & =-{\left({u}_2\right)}_{xxxxx}-\frac{3}{10}{\gamma}^2{u}_1^2{\left({u}_2\right)}_x-\frac{3}{10}{\gamma}^2\left(2{u}_1{u}_2+{u}_2^2\right){\left({u}_1+{u}_2\right)}_x,\hfill \\ {}\hfill & -2\gamma {\left({u}_1\right)}_x{\left({u}_2\right)}_{xx}-2\gamma {\left({u}_2\right)}_x{\left({u}_1+{u}_2\right)}_{xx}-\gamma \left({u}_1\right){\left({u}_2\right)}_{xxx}-\gamma {u}_2{\left({u}_1+{u}_2\right)}_{xxx},\hfill \\ {}\left({u}_3\right)t\hfill & =-{\left({u}_3\right)}_{xxxxx}-\frac{3}{10}{\gamma}^2{\left({u}_1+{u}_2\right)}^2{\left({u}_3\right)}_x-\frac{3}{10}{\gamma}^2\left[2\left({u}_1+{u}_2\right){u}_3+{u}_3^2\right]\left[{\left({u}_1+{u}_2\right)}_x+{\left({u}_3\right)}_x\right]\hfill \\ {}\hfill & -2\gamma {\left({u}_1+{u}_2\right)}_x{\left({u}_3\right)}_{xx}-2\gamma {\left({u}_3\right)}_x{\left({u}_1+{u}_2+{u}_3\right)}_{xx}\hfill \\ {}\hfill & -\gamma \left({u}_1+{u}_2\right){\left({u}_3\right)}_{xxx}-\gamma {u}_3{\left({u}_1+{u}_2+{u}_3\right)}_{xxx},\hfill \\ {}\left({u}_4\right)t\hfill & =-{\left({u}_4\right)}_{xxxxx}-\frac{3}{10}{\gamma}^2{\left({u}_1+{u}_2+{u}_3\right)}^2{\left({u}_4\right)}_x\hfill \\ {}\hfill & -\frac{3}{10}{\gamma}^2\left[2\left({u}_1+{u}_2+{u}_3\right){u}_4+{u}_4^2\right]\left[{\left({u}_1+{u}_2+{u}_3\right)}_x+{\left({u}_4\right)}_x\right]\hfill \\ {}\hfill & -2\gamma {\left({u}_1+{u}_2+{u}_3\right)}_x{\left({u}_4\right)}_{xx}-2\gamma {\left({u}_4\right)}_x{\left({u}_1+{u}_2+{u}_3+{u}_4\right)}_{xx}\hfill \\ {}\hfill & -\gamma \left({u}_1+{u}_2+{u}_3\right){\left({u}_4\right)}_{xxx}-\gamma {u}_4{\left({u}_1+{u}_2+{u}_3+{u}_4\right)}_{xxx},\hfill \\ {}\hfill & \vdots ,\hfill \\ {}\left({u}_n\right)t\hfill & ={\left({u}_n\right)}_{xxxxx}-\frac{3}{10}{\gamma}^2{\left({\displaystyle \sum_{k=1}^{n-1}{u}_k}\right)}^2{\left({u}_n\right)}_x-\frac{3}{10}{\gamma}^2\left[2{\left({\displaystyle \sum_{k=1}^{n-1}{u}_k}\right)}^2\left({u}_n\right)+{\left({u}_n\right)}^2\right]{\left({\displaystyle \sum_{k=1}^n{u}_k}\right)}_x\hfill \\ {}\hfill & -2\gamma \left[{\left({\displaystyle \sum_{k=1}^{n-1}{u}_k}\right)}_x{\left({u}_n\right)}_{xx}\right]-2\gamma {\left({u}_n\right)}_x{\left({\displaystyle \sum_{k=1}^n{u}_k}\right)}_{xx}\hfill \\ {}\hfill & -\gamma \left[\left({\displaystyle \sum_{k=1}^{n-1}{u}_k}\right){\left({u}_n\right)}_{xxx}\right]-\gamma \left({u}_n\right){\left({\displaystyle \sum_{k=1}^n{u}_k}\right)}_{xxx}, n\ge 2.\hfill \end{array}$$
(250)
where we replaced $$\alpha =\frac{3}{10}{\gamma}^2$$, $$\beta =2\gamma$$ as given by the characterization of the Lax equation given earlier. Following the simplified Hirota method we use the transformation
$${u}_i\left(x,t\right)={R}_i{\left( \ln f\left(x,t\right)\right)}_{xx}, 1\le i\le n,$$
(251)
into the members of the couplings (Eq. 250) and solving we find that
$${R}_i={\left(-1\right)}^{i+1}\frac{20}{\gamma }, 1\le i\le n.$$
(252)
We next substitute the transformation
$${u}_i\left(x,t\right)={\left(-1\right)}^{i+1}\frac{20}{\gamma }{\left( \ln f\left(x,t\right)\right)}_{xx},$$
(253)
into the linear terms of each member of the couplings (Eq. 250), where the auxiliary function as assumed by the simplified Hirota method reads
$$f=1+{e}^{\theta },$$
(254)
where the wave variable is given by
$$\theta =kx-ct,$$
(255)
and by solving the outcome we get the dispersion relation by
$$c={k}^5,$$
(256)
and in view of this result we obtain
$${\theta}_i={k}_ix-{k}_i^5t.$$
(257)
Consequently, for the one-soliton solution, we set
$$f\left(x,t\right)=1+{e}^{k_1x-{k}_1^5t}.$$
(258)
The one-soliton solution and the one-antisoliton solution are therefore given by
$${u}_i\left(x,t\right)={\left(-i\right)}^{i+1}\frac{20{k}_1^2{e}^{k_1x-{k}_1^5t}}{\gamma {\left(1+{e}^{k_1x-{k}_1^5t}\right)}^2}.$$
(259)
This clearly shows that equations of the couplings give soliton solution if i is odd and antisoliton solution if i is even but with the same amplitude.
To determine the two-soliton solution, we set the auxiliary function by
$$f\left(x,t\right)=1+{e}^{k_1x-{k}_1^5t}+{e}^{k_2x-{k}_2^5t}+{a}_{12}{e}^{\left({k}_1+{k}_2\right)x-\left({k}_1^5+{k}_2^5\right)t},$$
(260)
into Eq. 252 and proceed as before to obtain the phase factor a 12 by
$${a}_{12}=\frac{{\left({k}_1-{k}_2\right)}^2}{{\left({k}_1+{k}_2\right)}^2},$$
(261)
and hence
$${a}_{ij}=\frac{{\left({k}_i-{k}_j\right)}^2}{{\left({k}_i+{k}_j\right)}^2}, 1\le i<j\le 3.$$
(262)
This in turn gives
$$f\left(x,t\right)=1+{e}^{k_1x-{k}_1^5t}+{e}^{k_2x-{k}_2^5t}+\frac{{\left({k}_1-{k}_2\right)}^2}{{\left({k}_1+{k}_2\right)}^2}{e}^{\left({k}_1+{k}_2\right)x-\left({k}_1^5+{k}_2^5\right)t}.$$
(263)
The two-soliton solutions and the two-antisoliton solutions can be obtained by using Eq. 251 for the function f in Eq. 263.
To determine the three-soliton solutions we proceed as before and set
$$\begin{array}{l}f\left(x,t\right)=1+{e}^{k_1x-{k}_1^5t}+{e}^{k_2x-{k}_2^5t}+{e}^{k_3x-{k}_3^5t}\hfill \\ {} +{a}_{12}{e}^{\left({k}_1+{k}_2\right)x-\left({k}_1^5+{k}_2^5\right)t}+{a}_{13}{e}^{\left({k}_1+{k}_3\right)x-\left({k}_1^5+{k}_3^5t\right)}+{a}_{23}{e}^{\left({k}_2+{k}_3\right)x-\left({k}_2^5+{k}_3^5t\right)}\\ {} +{b}_{123}{e}^{\left({k}_1+{k}_2+{k}_3\right)x-\left({k}_1^5+{k}_2^5+{k}_3^5\right)t},\end{array}$$
(264)
and proceeding as before find that
$${b}_{123}={a}_{12}{a}_{13}{a}_{23}.$$
(265)
This shows that the coupling (Eq. 250) gives three-soliton solutions, and hence multiple-soliton solutions. To determine the three-soliton solution explicitly, we proceed as in the previous section.

### Discussion

In this section we constructed couplings of the fifth-order Sawada-Kotera equation and the fifth-order Lax equation. We used the algebra of coupled scalars for constructing the two classes of couplings. We derived multiple-soliton and multiple-antisoliton solutions for the couplings of the fifth-order Sawada-Kotera equation and the Lax equation. We showed that the derived couplings of the Sawada-Kotera equation possess the same properties as the fifth-order Sawada-Kotera equation: the same phase variable, the same phase shift, and the same amplitude. However, the only difference is that some equations give soliton solutions for i is odd, whereas others give antisoliton solutions for i even. The same conclusion holds for the couplings of the Lax equation. The algebra of coupled scalars is reliable and can be used for constructing other couplings of other integrable equations.

## Future Directions

The most significant advantage of the simplified form of the Hirota method is that it attacks any problem without any need for bilinear forms or any restrictive assumptions that may change the physical behavior of the solution. The field of dynamical integrable systems gave many useful developments. These developments can be attributed to the fruitful relation of mathematics and physics.

As stated before, the simplified form of the Hirota method gives multiple-soliton solutions without using prescribed conditions. The existing techniques require tedious work to evaluate the multiple-soliton solutions. We have shown in the first and the last example that a variety of multiple-soliton solutions can be constructed and not just a unique set. This newly established result needs to be investigated further to examine if it is applicable to other integrable models. Other existing methods may be used to examine this new result.

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