Modified method of simplest equation and its applications to the Bogoyavlenskii equation

In this paper, we construct the exact traveling wave solutions of the Bogoyavlenskii equation using modified method of simplest equation. The simplest equation herewith is a special Riccati equation. In the meantime, the proposed method in this work is proved to be a powerful mathematical tool for obtaining exact solutions of other nonlinear partial differential equations in mathematical physics and other research fields.

[1]  Anjan Biswas,et al.  Solitons and other solutions to quantum Zakharov–Kuznetsov equation in quantum magneto-plasmas , 2013 .

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

[3]  Anjan Biswas,et al.  Topological solitons and cnoidal waves to a few nonlinear wave equations in theoretical physics , 2013 .

[4]  E. H. Doha,et al.  A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled KdV equations , 2016 .

[5]  Houria Triki,et al.  SOLITONS AND OTHER SOLUTIONS TO THE (3+1)- DIMENSIONAL EXTENDED KADOMTSEV-PETVIASHVILI EQUATION WITH POWER LAW NONLINEARITY , 2013 .

[6]  Dmitry I. Sinelshchikov,et al.  Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations , 2011, Appl. Math. Comput..

[7]  Wen-Xiu Ma,et al.  Computers and Mathematics with Applications Constructing Nonlinear Discrete Integrable Hamiltonian Couplings , 2022 .

[8]  Sheng Zhang.  Exp-function method for Riccati equation and new exact solutions with two arbitrary functions of (2 + 1)-dimensional Konopelchenko-Dubrovsky equations , 2010, Appl. Math. Comput..

[9]  Xin Zeng,et al.  Exact vortex solitons in a quasi-two-dimensional Bose–Einstein condensate with spatially inhomogeneous cubic–quintic nonlinearity , 2012 .

[10]  Mingliang Wang,et al.  Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics , 1996 .

[11]  O I Bogoyavlenskii,et al.  Breaking solitons in 2+1-dimensional integrable equations , 1990 .

[12]  Holger Kantz,et al.  Application of the method of simplest equation for obtaining exact traveling-wave solutions for the extended Korteweg-de Vries equation and generalized Camassa-Holm equation , 2013, Appl. Math. Comput..

[13]  Chen Li-li.  FORMALLY VARIABLE SEPARATION APPROACH AND NEW EXACT SOLUTIONS OF GENERALIZED HIROTA-SATSUMA EQUATIONS , 1999 .

[14]  Wen-Xiu Ma,et al.  Computers and Mathematics with Applications Linear Superposition Principle Applying to Hirota Bilinear Equations , 2022 .

[15]  Nikolay K. Vitanov,et al.  Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics , 2010 .

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

[17]  Shaoyong Li,et al.  Kink-like wave and compacton-like wave solutions for generalized KdV equation , 2015 .

[18]  Fuhong Lin,et al.  Analytical study on a two-dimensional Korteweg–de Vries model with bilinear representation, Bäcklund transformation and soliton solutions , 2015 .

[19]  Liu Na.  Bäcklund transformation and multi-soliton solutions for the (3+1)-dimensional BKP equation with Bell polynomials and symbolic computation , 2015 .

[20]  Mingliang Wang.  Exact solutions for a compound KdV-Burgers equation , 1996 .

[21]  M. A. Abdou.  The extended F-expansion method and its application for a class of nonlinear evolution equations , 2007 .

[22]  Chaudry Masood Khalique,et al.  Rational solutions to an extended Kadomtsev-Petviashvili-like equation with symbolic computation , 2016, Comput. Math. Appl..

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

[24]  Cao Li-na,et al.  Symbolic Computation and q-Deformed Function Solutions of (2+1)-Dimensional Breaking Soliton Equation , 2007 .

[25]  Sirendaoreji.  Auxiliary equation method and new solutions of Klein-Gordon equations , 2007 .

[26]  A. Biswas.  1-Soliton solution of the generalized Zakharov–Kuznetsov equation with nonlinear dispersion and time-dependent coefficients , 2009 .

[27]  Jibin Li,et al.  Exact traveling wave solutions and bifurcations of the dual Ito equation , 2015 .

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

[29]  Wen-Xiu Ma,et al.  Solving the (3 + 1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm , 2012, Appl. Math. Comput..

[30]  Xing Lü,et al.  Soliton behavior for a generalized mixed nonlinear Schrödinger model with N-fold Darboux transformation. , 2013, Chaos.

[31]  E. Alaidarous,et al.  Erratum to: Exact solutions for a perturbed nonlinear Schrödinger equation by using Bäcklund transformations , 2013 .

[32]  E. Fan,et al.  A note on the homogeneous balance method , 1998 .

[33]  Nikolai A. Kudryashov,et al.  One method for finding exact solutions of nonlinear differential equations , 2011, 1108.3288.

[34]  Holger Kantz,et al.  Modified method of simplest equation and its application to nonlinear PDEs , 2010, Appl. Math. Comput..

[35]  Y. Huang.  Exact multi-wave solutions for the KdV equation , 2014 .

[36]  Hassan A. Zedan,et al.  Exact solutions for a perturbed nonlinear Schrödinger equation by using Bäcklund transformations , 2013 .

[37]  A. Biswas,et al.  Shock wave solutions to the Bogoyavlensky–Konopelchenko equation , 2014 .

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

[39]  R. K. Gupta,et al.  Exact and numerical solutions of coupled short pulse equation with time-dependent coefficients , 2015 .

[40]  Mingshu Peng,et al.  Nonautonomous motion study on accelerated and decelerated solitons for the variable-coefficient Lenells-Fokas model. , 2013, Chaos.

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

[42]  Anand Malik,et al.  Exact solutions of the Bogoyavlenskii equation using the multiple (G'/G)-expansion method , 2012, Comput. Math. Appl..

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

[44]  N. Kudryashov,et al.  EXACT SOLITON SOLUTIONS OF THE GENERALIZED EVOLUTION EQUATION OF WAVE DYNAMICS , 1988 .

[45]  Yi Cheng,et al.  The constraint of the Kadomtsev-Petviashvili equation and its special solutions , 1991 .

[46]  P. Pereira,et al.  Soliton-type and other travelling wave solutions for an improved class of nonlinear sixth-order Boussinesq equations , 2015 .

[47]  W. M. Liu,et al.  Quantized vortices in a rotating Bose-Einstein condensate with spatiotemporally modulated interaction , 2011, 1108.2097.

[48]  Deng-Shan Wang,et al.  Integrable properties of the general coupled nonlinear Schrödinger equations , 2010 .

[49]  Wenxiu Ma,et al.  Explicit and exact solutions to a Kolmogorov-Petrovskii-Piskunov equation , 1995, solv-int/9511005.

[50]  Xing Lü,et al.  Soliton excitations and shape-changing collisions in alpha helical proteins with interspine coupling at higher order , 2016, Commun. Nonlinear Sci. Numer. Simul..

[51]  E. Abdi Aghdam,et al.  Modified Kudryashov method for finding exact solitary wave solutions of higher‐order nonlinear equations , 2011 .

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

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

[54]  P. R. Gordoa,et al.  Multicomponent equations associated to non-isospectral scattering problems , 1997 .

[55]  Yunbo Zeng.  An approach to the deduction of the finite-dimensional integrability from the infinite-dimensional integrability , 1991 .

[56]  Zhifei Zhang,et al.  Integrability and equivalence relationships of six integrable coupled Korteweg–de Vries equations , 2016 .

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

[58]  Bao-Zhu Zhao,et al.  Exact rational solutions to a Boussinesq-like equation in (1+1)-dimensions , 2015, Appl. Math. Lett..

[59]  Deng-Shan Wang,et al.  AUTO-BÄCKLUND TRANSFORMATION AND NEW EXACT SOLUTIONS OF THE (2 + 1)-DIMENSIONAL NIZHNIK–NOVIKOV–VESELOV EQUATION , 2005 .

[60]  Anjan Biswas,et al.  Application of Tanh Method to Complex Coupled Nonlinear Evolution Equations , 2016 .

[61]  Anjan Biswas,et al.  Soliton Solution and Conservation Law of Gear-Grimshaw Model for Shallow Water Waves , 2014 .

[62]  A. H. Bhrawy,et al.  An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system , 2014, Appl. Math. Comput..

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

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

[65]  Wen-Xiu Ma,et al.  A Study on Rational Solutions to a KP-like Equation , 2015 .

[66]  Anjan Biswas,et al.  Jacobi spectral collocation approximation for multi-dimensional time-fractional Schrödinger equations , 2016 .

[67]  A. H. Bhrawy.  A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations , 2015, Numerical Algorithms.

[68]  A. Biswas,et al.  Solving nonlinear fractional differential equations using exp-function and (G/G′) -expansion methods , 2015 .

[69]  M. M. Hassan,et al.  Explicit exact solutions of some nonlinear evolution equations with their geometric interpretations , 2015, Appl. Math. Comput..

[70]  Zuntao Fu,et al.  New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations , 2001 .

[71]  Nikolay K. Vitanov,et al.  Modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations: further development of the methodology with applications , 2015, Appl. Math. Comput..

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

[73]  J. Prada,et al.  A Generalization of the Sine-Gordon Equation to 2 + 1 Dimensions , 2004 .

[74]  Nikolai A. Kudryashov,et al.  On types of nonlinear nonintegrable equations with exact solutions , 1991 .

[75]  Nikolay K. Vitanov,et al.  Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity , 2010 .

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

[77]  Abdul-Majid Wazwaz,et al.  The extended tanh method for abundant solitary wave solutions of nonlinear wave equations , 2007, Appl. Math. Comput..

[78]  Nikolay K. Vitanov,et al.  Solitary wave solutions for nonlinear partial differential equations that contain monomials of odd and even grades with respect to participating derivatives , 2014, Appl. Math. Comput..

[79]  Wen-Xiu Ma,et al.  A note on rational solutions to a Hirota-Satsuma-like equation , 2016, Appl. Math. Lett..

[80]  Deng-Shan Wang,et al.  Integrability and exact solutions of a two-component Korteweg-de Vries system , 2016, Appl. Math. Lett..

[81]  X. Feng,et al.  Exploratory Approach to Explicit Solution ofNonlinear Evolution Equations , 2000 .

[82]  Anjan Biswas,et al.  1-Soliton solution of the generalized Zakharov equation in plasmas by He's variational principle , 2010, Appl. Math. Comput..

[83]  Saudi Arabia,et al.  DYNAMICS OF TWO-LAYERED SHALLOW WATER WAVES WITH COUPLED KdV EQUATIONS , 2014 .

[84]  Wen-Xiu Ma,et al.  Complexiton solutions to the Korteweg–de Vries equation , 2002 .

[85]  M. Khater.  On the New Solitary Wave Solution of the Generalized Hirota-Satsuma Couple KdV System , 2015 .

[86]  Mingliang Wang,et al.  Periodic wave solutions to a coupled KdV equations with variable coefficients , 2003 .

[87]  Anjan Biswas,et al.  Cnoidal and snoidal wave solutions to coupled nonlinear wave equations by the extended Jacobi's elliptic function method , 2013, Commun. Nonlinear Sci. Numer. Simul..

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

[89]  Robert Conte,et al.  Link between solitary waves and projective Riccati equations , 1992 .

[90]  Chaudry Masood Khalique,et al.  Solitary waves with the Madelung fluid description: A generalized derivative nonlinear Schrödinger equation , 2016, Commun. Nonlinear Sci. Numer. Simul..

[91]  Min Chen,et al.  Direct search for exact solutions to the nonlinear Schrödinger equation , 2009, Appl. Math. Comput..

[92]  W. M. Liu,et al.  Quantized quasi-two-dimensional Bose-Einstein condensates with spatially modulated nonlinearity , 2009, 0912.1432.

[93]  Ye Tian,et al.  Integrability and bright soliton solutions to the coupled nonlinear Schrödinger equation with higher-order effects , 2014, Appl. Math. Comput..

[94]  Anjan Biswas,et al.  Integration of Complex-Valued Klein-Gordon Equation in Φ-4 Field Theory , 2015 .

[95]  A. H. Bhrawy,et al.  A Jacobi-Gauss-Lobatto collocation method for solving generalized Fitzhugh-Nagumo equation with time-dependent coefficients , 2013, Appl. Math. Comput..

[96]  S. Lou,et al.  Special solutions from the variable separation approach: the Davey - Stewartson equation , 1996 .