On Lie symmetries, optimal systems and explicit solutions to the Kudryashov-Sinelshchikov equation

Under investigation in this paper is the Kudryashov-Sinelshchikov equation, which describes influence of viscosity and heat transfer on propagation of the pressure waves. The Lie symmetry method is used to study its vector fields and optimal systems, respectively. Furthermore, the symmetry reductions and exact solutions of the equation are obtained on the basic of the optimal systems. Finally, based on the power series theory, a kind of explicit power series solutions for the equation is well constructed with a detailed derivation.

[1]  Pavel N. Ryabov,et al.  Exact solutions of the Kudryashov-Sinelshchikov equation , 2010, Appl. Math. Comput..

[2]  D. Levi,et al.  Integrability test of discrete nonlinear Schrödinger equations via multiscale reduction , 2010 .

[3]  Yong Chen,et al.  Weierstrass semi-rational expansion method and new doubly periodic solutions of the generalized Hirota-Satsuma coupled KdV system , 2006, Appl. Math. Comput..

[4]  N. M. Ivanova,et al.  Conservation Laws and Potential Symmetries of Linear Parabolic Equations , 2007, 0706.0443.

[5]  M. Younis,et al.  Solitons for compound KdV–Burgers equation with variable coefficients and power law nonlinearity , 2015 .

[6]  R. Hirota.  Direct Methods in Soliton Theory (非線形現象の取扱いとその物理的課題に関する研究会報告) , 1976 .

[7]  E. Fan,et al.  Soliton solutions for a generalized Hirota–Satsuma coupled KdV equation and a coupled MKdV equation , 2001 .

[8]  Zhi Hongyan.  Lie point symmetry and some new soliton-like solutions of the Konopelchenko-Dubrovsky equations , 2008, Appl. Math. Comput..

[9]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[10]  N. Asmar,et al.  Partial Differential Equations with Fourier Series and Boundary Value Problems , 2004 .

[11]  P. Clarkson,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering: References , 1991 .

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

[13]  Shou-Fu Tian,et al.  Lie symmetries and nonlocally related systems of the continuous and discrete dispersive long waves system by geometric approach , 2015 .

[14]  Muhammad Younis,et al.  Travelling wave solutions to some time-space nonlinear evolution equations , 2014, Appl. Math. Comput..

[15]  S. Lou,et al.  Infinitely many Lax pairs and symmetry constraints of the KP equation , 1997 .

[16]  N. Ibragimov.  Transformation groups applied to mathematical physics , 1984 .

[17]  Nikolai A. Kudryashov,et al.  Nonlinear waves in bubbly liquids with consideration for viscosity and heat transfer , 2010, 1112.5436.

[18]  Nikolai A. Kudryashov,et al.  Nonlinear evolution equations for describing waves in bubbly liquids with viscosity and heat transfer consideration , 2010, Appl. Math. Comput..

[19]  Yi Zhang,et al.  Hirota bilinear equations with linear subspaces of solutions , 2012, Appl. Math. Comput..

[20]  G. Bluman,et al.  Symmetry and Integration Methods for Differential Equations , 2002 .

[21]  W. Miller,et al.  Group analysis of differential equations , 1982 .

[22]  M. C. Nucci,et al.  The nonclassical method is more general than the direct method for symmetry reductions. An example of the Fitzhugh-Nagumo equation , 1992 .

[23]  Conservation laws and hierarchies of potential symmetries for certain diffusion equations , 2009 .

[24]  Changzheng Qu,et al.  Classification of coupled systems with two-component nonlinear diffusion equations by the invariant subspace method , 2009 .

[25]  Syed Tahir Raza Rizvi,et al.  Multiple travelling wave solutions for electrical transmission line model , 2015 .

[26]  Engui Fan,et al.  Double periodic solutions with Jacobi elliptic functions for two generalized Hirota–Satsuma coupled KdV systems , 2002 .

[27]  Dmitry I. Sinelshchikov,et al.  Equation for the three-dimensional nonlinear waves in liquid with gas bubbles , 2012, 1201.4583.

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

[29]  Xing-Biao Hu,et al.  An integrable symmetric (2+1)-dimensional Lotka–Volterra equation and a family of its solutions , 2005 .

[30]  Wolfgang K. Schief,et al.  Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory , 2002 .

[31]  M. Younis,et al.  Bright, dark, and singular solitons in magneto-electro-elastic circular rod , 2015 .

[32]  G. Bluman,et al.  Nonlocal symmetries and nonlocal conservation laws of Maxwell’s equations , 1997 .

[33]  Lei Liu,et al.  Lie group classifications and exact solutions for two variable-coefficient equations , 2009, Appl. Math. Comput..

[34]  P. Lax.  INTEGRALS OF NONLINEAR EQUATIONS OF EVOLUTION AND SOLITARY WAVES. , 1968 .

[35]  M. Younis,et al.  Traveling wave solutions for nonlinear dispersive water-wave systems with time-dependent coefficients , 2015 .