Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations

Lump-type solutions, rationally localized in many directions in the space, are analyzed for nonlinear differential equations derived from generalized bilinear differential equations. By symbolic computations with Maple, positive quadratic and quartic polynomial solutions to two classes of generalized bilinear differential equations on f are computed, and thus, lump-type solutions are presented to the corresponding nonlinear differential equations on u, generated from taking a transformation of dependent variables u = 2(ln f)x.

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

[2]  Kurt M. Berger,et al.  The Generation and Evolution of Lump Solitary Waves in Surface-Tension-Dominated Flows , 2000, SIAM J. Appl. Math..

[3]  Wen-Xiu Ma,et al.  AN APPLICATION OF THE CASORATIAN TECHNIQUE TO THE 2D TODA LATTICE EQUATION , 2008, 0804.0631.

[4]  Lena Vogler,et al.  The Direct Method In Soliton Theory , 2016 .

[5]  J. Satsuma.  Solitons and Rational Solutions of Nonlinear Evolution Equations (Theory of Nonlinear Waves) , 1978 .

[6]  Zhenyun Qin,et al.  Lump solutions to dimensionally reduced $$\varvec{p}$$p-gKP and $$\varvec{p}$$p-gBKP equations , 2016 .

[7]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

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

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

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

[11]  Jingsong He,et al.  The Darboux transformation of the derivative nonlinear Schrödinger equation , 2011, 1109.0674.

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

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

[14]  Peter A. Clarkson,et al.  THE DIRECT METHOD IN SOLITON THEORY (Cambridge Tracts in Mathematics 155) , 2006 .

[15]  Wenxiu Ma,et al.  Lump solutions to the Kadomtsev–Petviashvili equation , 2015 .

[16]  N. F. Smyth,et al.  Evolution of lump solutions for the KP equation , 1996 .

[17]  Jürgen Moser,et al.  On a class of polynomials connected with the Korteweg-deVries equation , 1978 .

[18]  Wen-Xiu Ma,et al.  Linear superposition principle of hyperbolic and trigonometric function solutions to generalized bilinear equations , 2016, Comput. Math. Appl..

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

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

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

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

[23]  Y. Stepanyants,et al.  Normal and anomalous scattering, formation and decay of bound states of two-dimensional solitons described by the Kadomtsev-Petviashvili equation , 1993 .

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

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

[26]  B. M. Fulk.  MATH , 1992 .

[27]  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.