Dynamics of kink solitary waves and lump waves with interaction phenomena in a generalized (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq equation

By using symbolic computation methods, we systematically study a generalized (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq (KP-Boussinesq) equation, which can be used to explain some interesting physical phenomena in the fields of fluids. We first obtain bilinear representation of the equation by employing the properties of Bell's polynomials. Based on that, its a series of exact solutions, such as one kink soliton solutions, two kink soliton solutions and lump wave solutions, are constructed in detail. It is worth mentioning that we also study the interaction solutions between lump waves and kink waves. Moreover, in order to describe the dynamic behaviours of those solutions, some graphical analyses are given by selecting different parameter values in such solutions. Those analyses are helpful for enriching the physical phenomena in KP-type equations.

[1]  Lakhveer Kaur,et al.  Kawahara equation and modified Kawahara equation with time dependent coefficients: symmetry analysis and generalized ‐expansion method , 2013 .

[2]  B. Guo,et al.  Long-time asymptotics of the focusing Kundu–Eckhaus equation with nonzero boundary conditions , 2019, Journal of Differential Equations.

[3]  M. Ruderman.  Freak waves in laboratory and space plasmas , 2010 .

[4]  Wei-Qi Peng,et al.  Breather waves and rational solutions in the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation , 2019, Comput. Math. Appl..

[5]  J. Fatome,et al.  Observation of Kuznetsov-Ma soliton dynamics in optical fibre , 2012, Scientific Reports.

[6]  Chun-Yan Qin.  Solitary Wave and Quasi-Periodic Wave Solutions to a (3+1)-Dimensional Generalized Calogero-Bogoyavlenskii-Schiff Equation , 2018, Advances in Applied Mathematics and Mechanics.

[7]  V. Konotop,et al.  Matter rogue waves , 2009 .

[8]  Abdul-Majid Wazwaz,et al.  General high-order breathers and rogue waves in the (3+1)-dimensional KP-Boussinesq equation , 2018, Commun. Nonlinear Sci. Numer. Simul..

[9]  Shou‐Fu Tian,et al.  On the breather waves, rogue waves and solitary waves to a generalized (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation , 2018 .

[10]  A. Wazwaz,et al.  Lump, breather and solitary wave solutions to new reduced form of the generalized BKP equation , 2019, International Journal of Numerical Methods for Heat & Fluid Flow.

[11]  Wen-Xiu Ma,et al.  Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations , 2016 .

[12]  Shou-Fu Tian,et al.  Characteristics of the breathers, rogue waves and solitary waves in a generalized (2+1)-dimensional Boussinesq equation , 2016 .

[13]  Abdul-Majid Wazwaz,et al.  New soliton and periodic solutions for the fifth-order forms of the Lax and Sawada–Kotera equations , 2007, Int. J. Comput. Math..

[14]  P. McClintock,et al.  Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium. , 2008, Physical review letters.

[15]  Shou-Fu Tian,et al.  On the Integrability of a Generalized Variable‐Coefficient Forced Korteweg‐de Vries Equation in Fluids , 2014 .

[16]  Abdul-Majid Wazwaz,et al.  Multiple-soliton solutions for a (3 + 1)-dimensional generalized KP equation , 2012 .

[17]  Shou‐Fu Tian,et al.  Nonlocal symmetries, conservation laws and interaction solutions for the classical Boussinesq–Burgers equation , 2018, Nonlinear Dynamics.

[18]  Jingsong He,et al.  Families of exact solutions of a new extended $$\varvec{(2+1)}$$(2+1)-dimensional Boussinesq equation , 2018 .

[19]  Shou‐Fu Tian,et al.  Solitary wave, breather wave and rogue wave solutions of an inhomogeneous fifth-order nonlinear Schrodinger equation from Heisenberg ferromagnetism , 2019, Rocky Mountain Journal of Mathematics.

[20]  L. Zou,et al.  Bilinear formalism, lump solution, lumpoff and instanton/rogue wave solution of a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation , 2019, Nonlinear Dynamics.

[21]  Anjan Biswas.  Multiple-scale analysis for solitons due to Langmuir waves in plasmas , 2010, Int. J. Comput. Math..

[22]  Shou-Fu Tian,et al.  Lump-type solutions and interaction solutions in the (3 + 1)-dimensional potential Yu–Toda–Sasa–Fukuyama equation , 2018, Analysis and Mathematical Physics.

[23]  Wen-Xiu Ma,et al.  Constructing lump solutions to a generalized Kadomtsev–Petviashvili–Boussinesq equation , 2016 .

[24]  Frédéric Dias,et al.  The Peregrine soliton in nonlinear fibre optics , 2010 .

[25]  Shou-Fu Tian,et al.  Bäcklund transformation, infinite conservation laws and periodic wave solutions to a generalized (2+1)-dimensional Boussinesq equation , 2016 .

[26]  Jin Li,et al.  Rogue waves, bright-dark solitons and traveling wave solutions of the (3+1)-dimensional generalized Kadomtsev-Petviashvili equation , 2018, Comput. Math. Appl..

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

[28]  Shou-Fu Tian,et al.  Solitons to rogue waves transition, lump solutions and interaction solutions for the (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation in fluid dynamics , 2018, Int. J. Comput. Math..

[29]  Jian-Ping Yu,et al.  A direct Bäcklund transformation for a (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like equation , 2017 .

[30]  Shou-Fu Tian,et al.  Initial–boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via the Fokas method☆ , 2017 .

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

[32]  Wen-Xiu Ma,et al.  Bäcklund transformation, multiple wave solutions and lump solutions to a (3 + 1)-dimensional nonlinear evolution equation , 2017, Nonlinear Dynamics.

[33]  Shou-Fu Tian,et al.  On the integrability of a generalized variable-coefficient Kadomtsev–Petviashvili equation , 2011, 1112.1499.

[34]  Shou‐Fu Tian,et al.  Lump wave and hybrid solutions of a generalized (3 + 1)-dimensional nonlinear wave equation in liquid with gas bubbles , 2019, Frontiers of Mathematics in China.

[35]  Ding Guo,et al.  Integrability, soliton solutions and modulation instability analysis of a (2+1)-dimensional nonlinear Heisenberg ferromagnetic spin chain equation , 2019, Comput. Math. Appl..

[36]  Shou‐Fu Tian,et al.  Long-time asymptotic behavior for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition , 2017 .

[37]  Xiaoen Zhang,et al.  Rogue wave and a pair of resonance stripe solitons to a reduced (3+1)-dimensional Jimbo-Miwa equation , 2016, Commun. Nonlinear Sci. Numer. Simul..

[38]  Wen-Xiu Ma,et al.  Lump solutions to nonlinear partial differential equations via Hirota bilinear forms , 2016, 1607.06983.

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

[40]  M. Shats,et al.  Capillary rogue waves. , 2010, Physical review letters.

[41]  Abdul-Majid Wazwaz,et al.  Solving the $$\mathbf{(3+1) }$$(3+1)-dimensional KP–Boussinesq and BKP–Boussinesq equations by the simplified Hirota’s method , 2017 .

[42]  Li Zhou,et al.  Characteristics of solitary wave, homoclinic breather wave and rogue wave solutions in a (2+1)-dimensional generalized breaking soliton equation , 2018, Comput. Math. Appl..

[43]  Wei-Qi Peng,et al.  Analysis on lump, lumpoff and rogue waves with predictability to the (2 + 1)-dimensional B-type Kadomtsev–Petviashvili equation , 2018, Physics Letters A.

[44]  Shou‐Fu Tian,et al.  Characteristics of the breather and rogue waves in a (2+1)-dimensional nonlinear Schrödinger equation , 2017 .

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

[46]  Li Zou,et al.  Solitary waves, homoclinic breather waves and rogue waves of the (3+1)-dimensional Hirota bilinear equation , 2017, Comput. Math. Appl..

[47]  Fabio Baronio,et al.  Rogue waves emerging from the resonant interaction of three waves. , 2013, Physical review letters.

[48]  Zhenya Yan.  Financial Rogue Waves , 2009, 0911.4259.

[49]  Wen-Xiu Ma,et al.  Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation , 2016 .

[50]  Abdul-Majid Wazwaz,et al.  Soliton solution for an inhomogeneous highly dispersive media with a dual-power nonlinearity law , 2010, Int. J. Comput. Math..

[51]  A. Wazwaz,et al.  Dynamical analysis of lump solutions for (3 + 1) dimensional generalized KP–Boussinesq equation and Its dimensionally reduced equations , 2018, Physica Scripta.

[52]  A. Wazwaz,et al.  Bright – dark optical solitons for Schrödinger-Hirota equation with variable coefficients , 2019, Optik.

[53]  L. Zou,et al.  Characteristics of the solitary waves and lump waves with interaction phenomena in a (2 + 1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation , 2018 .

[54]  Shou‐Fu Tian,et al.  Bäcklund Transformations, Nonlocal Symmetries and Soliton–Cnoidal Interaction Solutions of the (2 + 1)-Dimensional Boussinesq Equation , 2018, Bulletin of the Malaysian Mathematical Sciences Society.

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

[56]  Shou-Fu Tian,et al.  On periodic wave solutions with asymptotic behaviors to a (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation in fluid dynamics , 2016, Comput. Math. Appl..

[57]  Wen-Xiu Ma,et al.  Wronskian and Grammian solutions to a (3 + 1)-dimensional generalized KP equation , 2011, Appl. Math. Comput..

[58]  J. Y. Yang,et al.  Abundant lump-type solutions of the Jimbo-Miwa equation in (3+1)-dimensions , 2017, Comput. Math. Appl..

[59]  Xiaoen Zhang,et al.  Inverse scattering transformation for generalized nonlinear Schrödinger equation , 2019, Appl. Math. Lett..