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

**DOI:**https://doi.org/10.1007/978-3-642-27737-5_5-7

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## Keywords

Multiple Soliton Solutions Hirota Bilinear Method Wave Front Solutions Positon Solutions Peakons## 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).

*u*(

*x*,

*t*) for the modified KdV (mKdV) Eq. 1.

*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 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.

*D*operators are as follows:

## 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.

*k*,

*r*,

*s,*and

*ω*. We then substitute the single-soliton solution

*f*(

*x*,

*y*,

*z*,

*t*) is given by

*R*. Notice that the N-soliton solutions can be obtained for the given equation by using the following steps:

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

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

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.

*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

**Multiple-soliton solutions**

*ω*

_{ i }we obtain the dispersion relation

*θ*

_{ i }become

*R*, we substitute

*s*

_{ i }is left free.

*w*=

*u*

_{ x }.

*θ*

_{1}and

*θ*

_{2}are given in Eq. 40, to obtain the phase shift

*a*

_{12}by

*a*

_{12}in Eq. 53 cannot be 0 or ∞ for |

*k*

_{1}| ≠ |

*k*

_{2}|.

*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.

*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**

*w*=

*u*

_{ x }.

*a*

_{ ij }and the dispersion variables

*θ*

_{ i }are as defined earlier.

### The Second Model

**Multiple-soliton solutions**

*ω*

_{ i }we obtain the dispersion relation

*θ*

_{ i }becomes

*R*, we substitute

*s*

_{1}in this case should satisfy

*w*=

*u*

_{ x }.

*θ*

_{1}and

*θ*

_{2}are given in Eq. 69, to obtain the phase shift

*a*

_{12}by

*a*

_{12}in Eq. 86 cannot be 0 or ∞ for |

*k*

_{1}| ≠ |

*k*

_{2}|.

*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.

*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**

*w*=

*u*

_{ x }.

*a*

_{ ij }and the dispersion variables

*θ*

_{ i }are as defined earlier.

### The Third Model

*θ*

_{ i }becomes

*s*

_{1}in this case should satisfy

*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**

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

### Discussion

Parameters and dispersion relations of the three models

Model | The coefficients | Dispersion relations |
---|---|---|

First model | | \( \frac{k_i^3-3{s}_i}{2} \) |

| \( \frac{k_i^3-3{s}_i}{2} \) | |

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 | \( \frac{k_i^3-3{k}_i}{2} \) | |

select | \( \frac{k_i^3+{k}_i}{2} \) | |

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 | \( \frac{k_i^3-3{k}_i}{2} \) | |

select | \( \frac{k_i^3-3{k}_i}{2} \) | |

select | \( \frac{5{k}_i^3+9{k}_i}{10} \) |

## 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.

*α*and

*β*are arbitrary constants. This extended form of the mKP equation is obtained by adding the terms 4

*αv*

_{ x }and 4

*βv*

_{ y }to the mKP equation (Eq. 125), where

*v*

_{ x }and

*v*

_{ y }are the potentials in the

*x*and the

*y*directions respectively.

*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

*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 4*γv* _{ z } do not kill the integrability of the mKP equation (Eq. 125), but will change the dispersion relation of the mKP equation.

### Multiple-Front Wave Solutions

*f*(

*x*,

*y*,

*z*,

*t*) for the single-front wave solution is given by

*R*and

*r*

_{1},

*s*

_{1}we find

*v*(

*x*,

*y*,

*z*,

*t*).

*θ*

_{ i },

*i*= 1, 2 are given in Eq. 138. For the three-front wave solutions, we set

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.

### Multiple–Singular Front Wave Solutions

**7**below shows the obtained singular-kink solution

*v*(

*x*,

*y*,

*z*,

*t*).

*θ*

_{ i },

*i*= 1, 2 are given in Eq. 138.

### 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).

*u*

_{ 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

*ω*

_{ i }, the dispersion relation

*θ*

_{ i }becomes

*R*, where the auxiliary function

*f*(

*x*,

*y*,

*z*,

*t*) is given by

*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

*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

*s*

_{ i }=

*k*

_{ i. }For the first case, the dispersion relation and the dispersion variable read

*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*(

*x*,

*y*,

*z*,

*t*) for this case for

*n*= 3 and

*n*= 7 respectively.

*θ*

_{1}and

*θ*

_{2}are given in Eq. 170, to obtain the phase shift

*a*

_{12}by

*N*.

### Second Case

*r*

_{ i }=

*k*

_{ i }, \( {s}_i={k}_i^n. \) For the second case, the dispersion relation and the dispersion variable read

*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

*θ*

_{1}and

*θ*

_{2}are given in Eq. 186, to obtain the phase shift

*a*

_{12}by

*N*.

### Third Case

*θ*

_{1}and

*θ*

_{2}are given in Eq. 186, to obtain the phase shift

*a*

_{12}by

*N*.

### Fourth Case

*s*

_{ i }=

*k*

_{ i. }For the fourth case, the dispersion relation and the dispersion variable read

*s*

_{1}=

*k*

_{1}(Eq. 161) the single-soliton solution for the extended (3 + 1)-dimensional nonlinear evolution equation (Eq. 157) is given by

*θ*

_{1}and

*θ*

_{2}are defined earlier, to obtain the phase shift

*a*

_{12}by

*N*.

### Fifth Case

*θ*

_{1}and

*θ*

_{2}are given before, to obtain the phase shift

*a*

_{12}by

*N*.

### Discussion

Parameters and dispersion relations of the five case models

Case | The coefficients | 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 | \( {k}_i,\;{r}_i=\frac{1}{k_i^n}, {s}_i=\frac{1}{k_i^n} \) | \( {\omega}_i=3\left({k}_i^{n+2}+2{k}_i^{-n}\right)-{k}_i^3 \) |

*a*

_{ ij }, 1 ≤

*i*<

*j <*3. Table 3 below shows the distinct phase shifts for all five cases.

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

Case | The phase shifts |
---|---|

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.

*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 u

_{3x}and u

_{5x}. 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.

- (i)The Sawada-Kotera (SK) equation is given bycharacterized by$$ {u}_t+5{u}^2{u}_x+5{u}_x{u}_{xx}+5u{u}_{3x}+{u}_{5x}=0, $$(208)where$$ \beta =\gamma, \alpha =\frac{1}{5}{\gamma}^2, $$(209)
*γ*= 5 is selected. - (ii)The Caudrey-Dodd-Gibbon equation (CDG) is given bycharacterized by$$ {u}_t+180{u}^2ux+30{u}_x{u}_{xx}+30u{u}_{xxx}+{u}_{xxxxx}=0, $$(210)where$$ \beta =\gamma, \alpha =\frac{1}{5}{\gamma}^2, $$(211)
*γ*= 30 is selected. - (iii)The Lax equation readscharacterized by$$ ut+30{u}^2{u}_x+20{u}_x{u}_{xx}+10u{u}_{3x}+{u}_{5x}=0, $$(212)where$$ \beta =2\gamma, \alpha =\frac{3}{10}{\gamma}^2, $$(213)
*γ*= 10 is selected. - (iv)The Kaup-Kuperschmidt (KK) equation readscharacterized by$$ {u}_t+20{u}^2{u}_x+25{u}_x{u}_{xx}+10u{u}_{3x}+{u}_{5x}=0, $$(214)where$$ \beta =\frac{5}{2}\gamma, \alpha =\frac{1}{5}{\gamma}^2, $$(215)
*γ*= 10 is selected. - (v)The Ito equation is given ascharacterized by$$ {u}_t+2{u}^2ux+6{u}_x{u}_{xx}+3u{u}_{3x}+{u}_{5x}=0, $$(216)where$$ \beta =2\gamma, \alpha =\frac{2}{9}{\gamma}^2, $$(217)
*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.

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

_{i}are the basis vectors, then

*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

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

*as*given by the characterization of the Sawada-Kotera equation given earlier. Using the transformation

*i*is odd and antisoliton solution if

*i*is even but with the same amplitude.

*a*

_{12}by

*f*in Eq. 247.

### Couplings of the Fifth-Order Lax Equation

*i*is odd and antisoliton solution if

*i*is even but with the same amplitude.

*a*

_{12}by

### 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.

## Bibliography

## Primary Literature

- Blaszak M, Szablikowski B, Silindir B (2012) Construction and separability of nonlinear soliton integrable couplings. Appl Math Comput 219(2012):1866–1873zbMATHMathSciNetCrossRefGoogle Scholar
- Geng X (2003) Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations. J Phys A Math Gen 36:2289–2303zbMATHCrossRefADSGoogle Scholar
- Hereman W, Nuseir A (1997) Symbolic methods to construct exact solutions of nonlinear partial differential equations. Math Comput Simul 43:13–27zbMATHMathSciNetCrossRefGoogle Scholar
- Hietarinta J (1987) A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations. J Math Phys 28(8):1732–1742zbMATHMathSciNetCrossRefADSGoogle Scholar
- Hirota R (1971) Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons. Phys Rev Lett 27(18):1192–1194zbMATHCrossRefADSGoogle Scholar
- Hirota R (1972) Exact solutions of the modified Korteweg-de Vries equation for multiple collisions of solitons. J Phys Soc Jpn 33(5):1456–1458CrossRefADSGoogle Scholar
- Hirota R, Ito M (1983) Resonance of solitons in one dimension. J Phys Soc Jpn 52(3):744–748MathSciNetCrossRefADSGoogle Scholar
- Kadomtsev BB, Petviashvili VI (1970) On the stability of solitary waves in weakly dispersive media. Sov Phys Dokl 15:539–541zbMATHADSGoogle Scholar
- Lenells J (2005) Travelling wave solutions of the Camassa-Holm equation. J Differ Equ 217:393–430zbMATHMathSciNetCrossRefADSGoogle Scholar
- Liu Z, Wang R, Jing Z (2004) Peaked wave solutions of Camassa–Holm equation. Chaos Solitons Fractals 19:77–92zbMATHMathSciNetCrossRefADSGoogle Scholar
- Ma WX, Fuchssteiner B (1996) Integrable theory of the perturbation equations. Chaos Solitons Fractals 7:1227–1250zbMATHMathSciNetCrossRefADSGoogle Scholar
- Ma WX, Abdeljabbar A, Asaad MG (2011) Wronskian and Grammian solutions to a (3 + 1)-dimensional generalized KP equation. Appl Math Comput 217:10016–10023zbMATHMathSciNetCrossRefGoogle Scholar
- Malfliet W (1992) Solitary wave solutions of nonlinear wave equations. Am J Phys 60(7):650–654zbMATHMathSciNetCrossRefADSGoogle Scholar
- Malfliet W, Hereman W (1996a) The tanh method:I. Exact solutions of nonlinear evolution and wave equations. Phys Scr 54:563–568zbMATHMathSciNetCrossRefADSGoogle Scholar
- Malfliet W, Hereman W (1996b) The tanh method:II. Perturbation technique for conservative systems. Phys Scr 54:569–575zbMATHMathSciNetCrossRefADSGoogle Scholar
- Shen H-F, Tu M-H (2011) On the constrained B-type Kadomtsev-Petviashvili equation: Hirota bilinear equations and Virasoro symmetry. J Math Phys 52:032704MathSciNetCrossRefADSGoogle Scholar
- Veksler A, Zarmi Y (2005) Wave interactions and the analysis of the perturbed Burgers equation. Phys D 211:57–73zbMATHMathSciNetCrossRefGoogle Scholar
- Wadati M (1972) The exact solution of the modified Korteweg-de Vries equation. J Phys Soc Jpn 32:1681–1687CrossRefADSGoogle Scholar
- Wadati M (2001) Introduction to solitons. Pramana J Phys 57(5/6):841–847CrossRefADSGoogle Scholar
- Wazwaz AM (2007a) Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh-coth method. Appl Math Comput 190:633–640zbMATHMathSciNetCrossRefGoogle Scholar
- Wazwaz AM (2007b) Multiple-front solutions for the Burgers equation and the coupled Burgers equations. Appl Math Comput 190:1198–1206zbMATHMathSciNetCrossRefGoogle Scholar
- Wazwaz AM (2007c) New solitons and kink solutions for the Gardner equation. Commun Nonlinear Sci Numer Simul 12(8):1395–1404zbMATHMathSciNetCrossRefADSGoogle Scholar
- Wazwaz AM (2007d) Multiple-soliton solutions for the Boussinesq equation. Appl Math Comput 192:479–486zbMATHMathSciNetCrossRefGoogle Scholar
- Wazwaz AM (2008a) The Hirota’s direct method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Ito seventh-order equation. Appl Math Comput 199(1):133–138zbMATHMathSciNetCrossRefGoogle Scholar
- Wazwaz AM (2008b) Multiple-front solutions for the Burgers-Kadomtsev-Petvisahvili equation. Appl Math Comput 200:437–443zbMATHMathSciNetCrossRefGoogle Scholar
- Wazwaz AM (2008c) Multiple-soliton solutions for the Lax-Kadomtsev-Petvisahvili (Lax- KP) equation. Appl Math Comput 201(1/2):168–174zbMATHMathSciNetCrossRefGoogle Scholar
- Wazwaz AM (2008d) The Hirota’s direct method for multiple-soliton solutions for three model equations of shallow water waves. Appl Math Comput 201(1/2):489–503zbMATHMathSciNetCrossRefGoogle Scholar
- Wazwaz AM (2008e) Multiple-soliton solutions of two extended model equations for shallow water waves. Appl Math Comput 201(1/2):790–799zbMATHMathSciNetCrossRefGoogle Scholar
- Wazwaz AM (2008f) Single and multiple-soliton solutions for the (2 + 1)-dimensional KdV equation. Appl Math Comput 204:20–26zbMATHMathSciNetCrossRefGoogle Scholar
- Wazwaz AM (2008g) Solitons and singular solitons for the Gardner-KP equation. Appl Math Comput 204:162–169zbMATHMathSciNetCrossRefGoogle Scholar
- Wazwaz AM (2008h) Regular soliton solutions and singular soliton solutions for the modified Kadomtsev-Petviashvili equations. Appl Math Comput 204:817–823zbMATHMathSciNetCrossRefGoogle Scholar
- Wazwaz AM (2009a) Adomian decomposition method applied to nonlinear evolution equations in solitons theory. In: Meyers RA (ed) Encyclopedia of complexity and systems science. Springer, HeibelbergGoogle Scholar
- Wazwaz AM (2011a) Multi-front waves for extended form of modified Kadomtsev–Petviashvili equation. Appl Math Mech 32(7):875–880Google Scholar
- Wazwaz AM (2011b) Distinct kinds of multiple soliton solutions for a (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili equation. Phys Scr 84:055006Google Scholar
- Wazwaz AM (2011c) A new fifth order nonlinear integrable equation: multiple soliton solutions. Physica Scripta 83:015012Google Scholar
- Wazwaz AM (2011d) A new generalized fifth-order nonlinear integrable equation. Phys Scr 83:035003Google Scholar
- Wazwaz AM (2012) Multiple soliton solutions for some (3 + 1)-dimensional nonlinear models generated by the Jaulent-Miodek hierarchy. Appl Math Lett 25(1):1936–1940zbMATHMathSciNetCrossRefGoogle Scholar
- Zabusky NJ, Kruskal MD (1965) Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15:240–243zbMATHCrossRefADSGoogle Scholar
- Zhang Y, Tam H (2010) Three kinds of coupling integrable couplings of the Kortewegde Vries hierarchy of evolution equations. J Math Phys 51:043510MathSciNetCrossRefADSGoogle Scholar
- Zhaqilao (2012) A generalized AKNS hierarchy, bi-Hamiltonian structure, and Darboux transformation. Commun Nonlinear Sci Numer Simul 17:2319–2332zbMATHMathSciNetCrossRefADSGoogle Scholar

## Books and Reviews

- Ablowitz MJ, Clarkson PA (1991) Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
- Drazin PG, Johnson RS (1996) Solitons: an introduction. Cambridge University Press, CambridgeGoogle Scholar
- Hirota R (2004) The direct method in soliton theory. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
- Wazwaz AM (2002) Partial differential equations: methods and applications. Balkema Publishers, LisseGoogle Scholar
- Wazwaz AM (2009b) Partial differential equations: methods and solitary waves theory. Springer/HEP, Berlin/BeijingGoogle Scholar
- Wazwaz AM (1997) A first course in integral equations. World Scientific, SingaporezbMATHCrossRefGoogle Scholar
- Wazwaz AM (2011e) Linear and nonlinear integral equations. Springer/HEP, Berlin/BeijingGoogle Scholar
- Whitham GB (1999) Linear and nonlinear waves. Wiley–Interscience Series, New YorkzbMATHCrossRefGoogle Scholar