Semi-rational solutions for the (3+1)-dimensional Kadomtsev-Petviashvili equation in a plasma or fluid

Abstract In this paper, we investigate the ( 3 + 1 ) -dimensional Kadomtsev–Petviashvili equation in a plasma or fluid. For the amplitude of the electrostatic wave potential in the plasma or shallow-water wave in the fluid, via the Kadomtsev–Petviashvili hierarchy reduction, we obtain the semi-rational solutions in determinant form for such an equation. Interactions between the first-order lump (or rogue wave) and soliton are illustrated. We find that the lump arises and then separates from the soliton on the x – y and x – z planes; and that the rogue wave possesses a line profile and arises from the soliton (or constant background) on the y – z plane, where x , y and z are the scaled spatial coordinates. Interactions between the two lumps (or rogue waves) and two solitons are presented. Interactions between the second-order lump (or rogue wave) and one soliton are also presented.

[1]  Y. Kivshar,et al.  Self-focusing of plane dark solitons in nonlinear defocusing media. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[2]  T. Maimbourg,et al.  Oscillon dynamics and rogue wave generation in Faraday surface ripples. , 2012, Physical review letters.

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

[4]  Xin-Yi Gao,et al.  Looking at a nonlinear inhomogeneous optical fiber through the generalized higher-order variable-coefficient Hirota equation , 2017, Appl. Math. Lett..

[5]  K. Murawski,et al.  Solitary waves in plasmas and in the atmosphere , 1989 .

[6]  Yi-Tian Gao,et al.  Integrability and solitons for the higher-order nonlinear Schrödinger equation with space-dependent coefficients in an optical fiber , 2018 .

[7]  N. Akhmediev,et al.  Waves that appear from nowhere and disappear without a trace , 2009 .

[8]  Min Li,et al.  Dark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Efim Pelinovsky,et al.  Physical Mechanisms of the Rogue Wave Phenomenon , 2003 .

[10]  Lei Liu,et al.  Rogue waves for a variable-coefficient Kadomtsev-Petviashvili equation in fluid mechanics , 2018, Comput. Math. Appl..

[11]  Javier Villarroel,et al.  Dynamics of Lump Solutions in a 2 + 1 NLS Equation , 2009 .

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

[13]  Hui-Qin Hao,et al.  Multi-soliton solutions and Breathers for the generalized coupled nonlinear Hirota equations via the Hirota method , 2017 .

[14]  Zhaqilao.  A symbolic computation approach to constructing rogue waves with a controllable center in the nonlinear systems , 2018, Comput. Math. Appl..

[15]  V. Matveev,et al.  Darboux Transformations and Solitons , 1992 .

[16]  Xin-Yi Gao.  Bäcklund transformation and shock-wave-type solutions for a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev–Petviashvili equation in fluid mechanics , 2015 .

[17]  Yasuhiro Ohta,et al.  Rogue waves in the Davey-Stewartson I equation. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Bao-Feng Feng,et al.  Multi-dark soliton solutions of the two-dimensional multi-component Yajima-Oikawa systems , 2015, 1502.02759.

[19]  Yi-Tian Gao,et al.  Solitons for the (3+1)-dimensional variable-coefficient coupled nonlinear Schrödinger equations in an optical fiber , 2017 .

[20]  Meng Dexin,et al.  Rational Solitons in the Parity-Time-Symmetric Nonlocal Nonlinear Schrödinger Model , 2016 .

[21]  Guiqiong Xu,et al.  The soliton solutions, dromions of the Kadomtsev–Petviashvili and Jimbo–Miwa equations in (3 + 1)-dimensions , 2006 .

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

[23]  Jiguang Rao,et al.  Semi-rational solutions of the third-type Davey-Stewartson equation. , 2017, Chaos.

[24]  Yi-Tian Gao,et al.  Bilinear Bäcklund transformation, soliton and periodic wave solutions for a $$\varvec{(3 + 1)}$$(3+1)-dimensional variable-coefficient generalized shallow water wave equation , 2017 .

[25]  Zhaqilao.  Rogue waves and rational solutions of a (3+1)-dimensional nonlinear evolution equation , 2013 .

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

[27]  Lei Wang,et al.  Higher-order semirational solutions and nonlinear wave interactions for a derivative nonlinear Schrödinger equation , 2016, Commun. Nonlinear Sci. Numer. Simul..

[28]  Hong-Cai Ma,et al.  Lump Solution of (2+1)-Dimensional Boussinesq Equation* , 2016 .

[29]  Tao Xu,et al.  Generation mechanism of rogue waves for the discrete nonlinear Schrödinger equation , 2018, Appl. Math. Lett..