Lump-Type Solutions to the (3+1)-Dimensional Jimbo-Miwa Equation

Abstract: Taking advantage of the Hirota bilinear form, four classes of lump-type solutions to the (3+1)-dimensional Jimbo-Miwa equation are presented through symbolic computation with Maple. Special choices of the involved parameters guaranteeing analyticity of the fourth solution are given, together with two particular lump-type solutions.

[1]  David J. Kaup,et al.  The lump solutions and the Bäcklund transformation for the three‐dimensional three‐wave resonant interaction , 1981 .

[2]  Alberto Tacchella.  On rational solutions of multicomponent and matrix KP hierarchies , 2011 .

[3]  Wenxiu Ma,et al.  Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions , 2004, nlin/0503001.

[4]  Wenxiu Ma.  Generalized Wronskians and Solutions to the Korteweg-de Vries Equation , 2003 .

[5]  Alan S. Osborne,et al.  THE FOURTEENTH 'AHA HULIKO' A HAWAIIAN WINTER WORKSHOP , 2005 .

[6]  F. Gesztesy,et al.  Rational KP and mKP-solutions in Wronskian form , 1991 .

[7]  D. Sinelshchikov.  Comment on: New exact traveling wave solutions of the (3 + 1)-dimensional Kadomtsev–Petviashvili (KP) equation , 2010 .

[8]  Mingliang Wang,et al.  The (G' G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics , 2008 .

[9]  Kenji Imai,et al.  DROMION AND LUMP SOLUTIONS OF THE ISHIMORI-I EQUATION , 1997 .

[10]  Wenxiu Ma,et al.  A second Wronskian formulation of the Boussinesq equation , 2009 .

[11]  M. Jimbo,et al.  Solitons and Infinite Dimensional Lie Algebras , 1983 .

[12]  Wenxiu Ma,et al.  Bilinear equations, Bell polynomials and linear superposition principle , 2013 .

[13]  J. Nimmo,et al.  Lump solutions of the BKP equation , 1990 .

[14]  W. Malfliet.  Solitary wave solutions of nonlinear wave equations , 1992 .

[15]  J. Nimmo,et al.  On the combinatorics of the Hirota D-operators , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[16]  Wenxiu Ma.  Wronskians, generalized Wronskians and solutions to the Korteweg–de Vries equation , 2003, nlin/0303068.

[17]  M. Khalfallah.  New exact traveling wave solutions of the (3 + 1) dimensional Kadomtsev–Petviashvili (KP) equation , 2009 .

[18]  B. Jalali,et al.  Optical rogue waves , 2007, Nature.

[19]  Abdul-Majid Wazwaz.  Two B-type Kadomtsev–Petviashvili equations of (2 + 1) and (3 + 1) dimensions: Multiple soliton solutions, rational solutions and periodic solutions , 2013 .

[20]  Xiao-yan Tang,et al.  Variable separation solutions for the (3 + 1)-dimensional Jimbo-Miwa equation , 2006 .

[21]  M. Senthilvelan,et al.  On the extended applications of Homogenous Balance Method , 2001, Appl. Math. Comput..

[22]  Wenxiu Ma,et al.  Bilinear Equations and Resonant Solutions Characterized by Bell Polynomials , 2013 .

[23]  Bo Tian,et al.  Beyond travelling waves: a new algorithm for solving nonlinear evolution equations , 1996 .

[24]  P. Winternitz,et al.  Point symmetries of conditionally integrable nonlinear evolution equations , 1990 .

[25]  P. Caudrey,et al.  Memories of Hirota’s method: application to the reduced Maxwell–Bloch system in the early 1970s† , 2011, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[26]  J. Satsuma,et al.  Two‐dimensional lumps in nonlinear dispersive systems , 1979 .

[27]  Wen-Xiu Ma,et al.  A bilinear Bäcklund transformation of a (3+1) -dimensional generalized KP equation , 2012, Appl. Math. Lett..

[28]  Wenxiu Ma,et al.  Comment on the 3+1 dimensional Kadomtsev–Petviashvili equations , 2011 .

[29]  Q. P. Liu,et al.  Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  Wenxiu Ma,et al.  Trilinear equations, Bell polynomials, and resonant solutions , 2013 .

[31]  W. Ma.  Generalized Bilinear Differential Equations , 2012 .

[32]  Abdul-Majid Wazwaz,et al.  New solutions of distinct physical structures to high-dimensional nonlinear evolution equations , 2008, Appl. Math. Comput..

[33]  広田 良吾,et al.  The direct method in soliton theory , 2004 .

[34]  C. Garrett.  Rogue waves , 2012 .

[35]  Alfred Ramani,et al.  Are all the equations of the Kadomtsev–Petviashvili hierarchy integrable? , 1986 .

[36]  Wenxiu Ma,et al.  Rational solutions of the Toda lattice equation in Casoratian form , 2004 .

[37]  Ljudmila A. Bordag,et al.  Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction , 1977 .

[38]  Yi Zhang,et al.  Rational solutions to a KdV-like equation , 2015, Appl. Math. Comput..

[39]  Wenxiu Ma,et al.  A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation , 2009, 0903.5337.