Bibliography
Primary Literature
Blaszak M, Szablikowski B, Silindir B (2012) Construction and separability of nonlinear soliton integrable couplings. Appl Math Comput 219(2012):1866–1873
Geng X (2003) Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations. J Phys A Math Gen 36:2289–2303
Hereman W, Nuseir A (1997) Symbolic methods to construct exact solutions of nonlinear partial differential equations. Math Comput Simul 43:13–27
Hietarinta J (1987) A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations. J Math Phys 28(8):1732–1742
Hirota R (1971) Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons. Phys Rev Lett 27(18):1192–1194
Hirota R (1972) Exact solutions of the modified Korteweg-de Vries equation for multiple collisions of solitons. J Phys Soc Jpn 33(5):1456–1458
Hirota R, Ito M (1983) Resonance of solitons in one dimension. J Phys Soc Jpn 52(3):744–748
Kadomtsev BB, Petviashvili VI (1970) On the stability of solitary waves in weakly dispersive media. Sov Phys Dokl 15:539–541
Lenells J (2005) Travelling wave solutions of the Camassa-Holm equation. J Differ Equ 217:393–430
Liu Z, Wang R, Jing Z (2004) Peaked wave solutions of Camassa–Holm equation. Chaos Solitons Fractals 19:77–92
Ma WX, Fuchssteiner B (1996) Integrable theory of the perturbation equations. Chaos Solitons Fractals 7:1227–1250
Ma WX, Abdeljabbar A, Asaad MG (2011) Wronskian and Grammian solutions to a (3 + 1)-dimensional generalized KP equation. Appl Math Comput 217:10016–10023
Malfliet W (1992) Solitary wave solutions of nonlinear wave equations. Am J Phys 60(7):650–654
Malfliet W, Hereman W (1996a) The tanh method:I. Exact solutions of nonlinear evolution and wave equations. Phys Scr 54:563–568
Malfliet W, Hereman W (1996b) The tanh method:II. Perturbation technique for conservative systems. Phys Scr 54:569–575
Shen H-F, Tu M-H (2011) On the constrained B-type Kadomtsev-Petviashvili equation: Hirota bilinear equations and Virasoro symmetry. J Math Phys 52:032704
Veksler A, Zarmi Y (2005) Wave interactions and the analysis of the perturbed Burgers equation. Phys D 211:57–73
Wadati M (1972) The exact solution of the modified Korteweg-de Vries equation. J Phys Soc Jpn 32:1681–1687
Wadati M (2001) Introduction to solitons. Pramana J Phys 57(5/6):841–847
Wazwaz AM (2007a) Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh-coth method. Appl Math Comput 190:633–640
Wazwaz AM (2007b) Multiple-front solutions for the Burgers equation and the coupled Burgers equations. Appl Math Comput 190:1198–1206
Wazwaz AM (2007c) New solitons and kink solutions for the Gardner equation. Commun Nonlinear Sci Numer Simul 12(8):1395–1404
Wazwaz AM (2007d) Multiple-soliton solutions for the Boussinesq equation. Appl Math Comput 192:479–486
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):133–138
Wazwaz AM (2008b) Multiple-front solutions for the Burgers-Kadomtsev-Petvisahvili equation. Appl Math Comput 200:437–443
Wazwaz AM (2008c) Multiple-soliton solutions for the Lax-Kadomtsev-Petvisahvili (Lax- KP) equation. Appl Math Comput 201(1/2):168–174
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):489–503
Wazwaz AM (2008e) Multiple-soliton solutions of two extended model equations for shallow water waves. Appl Math Comput 201(1/2):790–799
Wazwaz AM (2008f) Single and multiple-soliton solutions for the (2 + 1)-dimensional KdV equation. Appl Math Comput 204:20–26
Wazwaz AM (2008g) Solitons and singular solitons for the Gardner-KP equation. Appl Math Comput 204:162–169
Wazwaz AM (2008h) Regular soliton solutions and singular soliton solutions for the modified Kadomtsev-Petviashvili equations. Appl Math Comput 204:817–823
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, Heibelberg
Wazwaz AM (2011a) Multi-front waves for extended form of modified Kadomtsev–Petviashvili equation. Appl Math Mech 32(7):875–880
Wazwaz AM (2011b) Distinct kinds of multiple soliton solutions for a (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili equation. Phys Scr 84:055006
Wazwaz AM (2011c) A new fifth order nonlinear integrable equation: multiple soliton solutions. Physica Scripta 83:015012
Wazwaz AM (2011d) A new generalized fifth-order nonlinear integrable equation. Phys Scr 83:035003
Wazwaz AM (2012) Multiple soliton solutions for some (3 + 1)-dimensional nonlinear models generated by the Jaulent-Miodek hierarchy. Appl Math Lett 25(1):1936–1940
Zabusky NJ, Kruskal MD (1965) Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15:240–243
Zhang Y, Tam H (2010) Three kinds of coupling integrable couplings of the Kortewegde Vries hierarchy of evolution equations. J Math Phys 51:043510
Zhaqilao (2012) A generalized AKNS hierarchy, bi-Hamiltonian structure, and Darboux transformation. Commun Nonlinear Sci Numer Simul 17:2319–2332
Books and Reviews
Ablowitz MJ, Clarkson PA (1991) Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, Cambridge
Drazin PG, Johnson RS (1996) Solitons: an introduction. Cambridge University Press, Cambridge
Hirota R (2004) The direct method in soliton theory. Cambridge University Press, Cambridge
Wazwaz AM (2002) Partial differential equations: methods and applications. Balkema Publishers, Lisse
Wazwaz AM (2009b) Partial differential equations: methods and solitary waves theory. Springer/HEP, Berlin/Beijing
Wazwaz AM (1997) A first course in integral equations. World Scientific, Singapore
Wazwaz AM (2011e) Linear and nonlinear integral equations. Springer/HEP, Berlin/Beijing
Whitham GB (1999) Linear and nonlinear waves. Wiley–Interscience Series, New York
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Wazwaz, AM. (2015). (3+1)-Dimensional Nonlinear Equations and Couplings of Fifth-Order Equations in the Solitary Waves Theory: Multiple Soliton Solutions. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_5-7
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