Integrability and exact solutions of the nonautonomous mixed mKdV-sinh-Gordon equation

Abstract In this paper, a nonautonomous mixed mKdV–sinh–Gordon equation with one arbitrary time-dependent variable coefficient is discussed in detail. It is proved that the equation passes the Painleve test in the case of positive and negative resonances, respectively. Furthermore, a dependent variable transformation is introduced to get its bilinear form. Then, soliton, negaton, positon and interaction solutions are introduced by means of the Wronskian representation. Velocities are found to depend on the time-dependent variable coefficient appearing in the equation and this leads to a wide range of interesting behaviours. The singularities and asymptotic estimate of these solutions are discussed. At last, the superposition formulae for these solutions are also constructed.

[1]  Abdul-Majid Wazwaz,et al.  Travelling wave solutions for the MKdV-sine-Gordon and the MKdV-sinh-Gordon equations by using a variable separated ODE method , 2006, Appl. Math. Comput..

[2]  J. F. Gomes,et al.  A class of mixed integrable models , 2009, 0903.0579.

[3]  Kimiaki Konno,et al.  Effect of Weak Dislocation Potential on Nonlinear Wave Propagation in Anharmonic Crystal , 1974 .

[4]  刘长河,et al.  mKdV-sine-Gordon方程丰富的相互作用解 , 2010 .

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

[6]  Andrew Pickering,et al.  Analysing negative resonances in the Painlevé test , 1991 .

[7]  J. Nimmo,et al.  A method of obtaining the N-soliton solution of the Boussinesq equation in terms of a wronskian , 1983 .

[8]  R. Beutler,et al.  Positon solutions of the sine‐Gordon equation , 1993 .

[9]  Sergey V. Sazonov,et al.  Few-cycle nonlinear optics of multicomponent media , 2006 .

[10]  Yuqin Yao,et al.  The new integrable deformations of a short pulse equation and sine-Gordon equation, and their solutions , 2010, 1012.0462.

[11]  Ji Lin,et al.  Painlevé Properties And Exact Solutions Of The Generalized Coupled Kdv Equations , 2005 .

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

[13]  Wei Xu,et al.  Wronskian determinant solutions of the (3 + 1)-dimensional Jimbo-Miwa equation , 2011, Appl. Math. Comput..

[14]  Andrew Pickering,et al.  A perturbative Painleve´ approach to nonlinear differential equations , 1993 .

[15]  Dumitru Mihalache,et al.  Few-optical-cycle solitons: Modified Korteweg-de Vries sine-Gordon equation versus other non-slowly-varying-envelope-approximation models , 2009 .

[16]  François Sanchez,et al.  Models for optical solitons in the two-cycle regime , 2004, nlin/0411010.

[17]  M. Tabor,et al.  The Painlevé property for partial differential equations , 1983 .

[18]  Shu-fang Deng,et al.  The Novel Multi-Soliton Solutions of the MKdV–Sine Gordon Equations , 2002 .

[19]  J. Nimmo,et al.  Soliton solution of three differential-difference equations in wronskian form , 1983 .

[20]  Wen-Xiu Ma,et al.  Wronskian solutions of the Boussinesq equation—solitons, negatons, positons and complexitons , 2007 .

[21]  Alfred Ramani,et al.  The Painlevé property and singularity analysis of integrable and non-integrable systems , 1989 .

[22]  Hongxia Wu,et al.  Negaton and complexiton solutions of sine-Gordon equation , 2007 .

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

[24]  Sun Jiong,et al.  A direct method for solving sine-Gordon type equations , 2002 .

[25]  Kimiaki Konno,et al.  Effect of Weak Dislocation Potential on Nonlinear Wave Propagation in Anharmonic Crystal : An Infinite Set of Conservation Laws , 1974 .

[26]  J. F. Gomes,et al.  Nonautonomous mixed mKdV–sinh–Gordon hierarchy , 2010, 1004.3488.