Analytical solutions to a generalized Drinfel'd-Sokolov equation related to DSSH and KdV6

Abstract Analytical solutions to the generalized Drinfel’d–Sokolov (GDS) equations u t + α 1 uu x + β 1 u xxx + γ ( v δ ) x = 0 and v t + α 2 uv x + β 2 v xxx = 0 are obtained for various values of the model parameters. In particular, we provide perturbation solutions to illustrate the strong influence of the parameters β 1 and β 2 on the behavior of the solutions. We then consider a Miura-type transform which reduces the gDS equations into a sixth-order nonlinear differential equation under the assumption that δ  = 1. Under such a transform the GDS reduces to the sixth-order Drinfel’d–Sokolov–Satsuma–Hirota (DSSH) equation (also known as KdV6) in the very special case α 1  = − α 2 . The method of homotopy analysis is applied in order to obtain analytical solutions to the resulting equation for arbitrary α 1 and α 2 . An error analysis of the obtained approximate analytical solutions is provided.

[1]  Ahmet Bekir,et al.  Applications of the extended tanh method for coupled nonlinear evolution equations , 2008 .

[2]  何斌,et al.  BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS FOR GENERALIZED DRINFELD-SOKOLOV EQUATIONS ∗ , 2006 .

[3]  Abdul-Majid Wazwaz,et al.  Solitary wave solutions of the generalized shallow water wave (GSWW) equation by Hirota's method, tanh-coth method and Exp-function method , 2008, Appl. Math. Comput..

[4]  M. A. Semenov-Tian-Shansky,et al.  Drinfeld–Sokolov Reduction for Difference Operators and Deformations of W-Algebras¶ II. The General Semisimple Case , 1998 .

[5]  MalflietW..  The tanh method , 2004 .

[6]  Marco Pedroni,et al.  Drinfeld-Sokolov reduction on a simple lie algebra from the bihamiltonian point of view , 1992 .

[7]  Robert A. Van Gorder,et al.  Exponential-type solutions to a generalized Drinfel'd–Sokolov equation , 2010 .

[8]  W. Hereman,et al.  The tanh method: II. Perturbation technique for conservative systems , 1996 .

[9]  M. Gurses,et al.  Integrable KdV systems: Recursion operators of degree four☆ , 1998, solv-int/9811013.

[10]  Huaitang Chen.  New Double Periodic Solutions of the Classical Drinfel’d‐Sokolov‐Wilson Equation , 2008 .

[11]  Michele Sciacca,et al.  A direct method to find solutions of some type of coupled Korteweg-de Vries equations using hyperelliptic functions of genus two , 2008 .

[12]  V. V. Sokolov,et al.  Lie algebras and equations of Korteweg-de Vries type , 1985 .

[13]  David J. Evans,et al.  The tanh function method for solving some important non-linear partial differential equations , 2005, Int. J. Comput. Math..

[14]  Y. Tan,et al.  A General Approach to Obtain Series Solutions of Nonlinear Differential Equations , 2007 .

[15]  Alfred Ramani,et al.  Soliton structure of the Drinfel’d–Sokolov–Wilson equation , 1986 .

[16]  Qi Wang,et al.  Elliptic equation rational expansion method and new exact travelling solutions for Whitham-Broer-Kaup equations , 2005 .

[17]  Jianlan Hu,et al.  A new method of exact travelling wave solution for coupled nonlinear differential equations , 2004 .

[18]  Abdul-Majid Wazwaz,et al.  Compactons and solitary patterns structures for variants of the KdV and the KP equations , 2003, Appl. Math. Comput..

[19]  I. Marshall,et al.  Generalized Drinfeld-Sokolov reductions and KdV type hierarchies , 1992, hep-th/9210037.

[20]  Dogan Kaya,et al.  Exact and numerical solutions of generalized Drinfeld-Sokolov equations , 2008 .

[21]  Abdul-Majid Wazwaz,et al.  Exact and explicit travelling wave solutions for the nonlinear Drinfeld–Sokolov system , 2006 .

[22]  Alfred Ramani,et al.  BILINEARIZATION AND SOLUTIONS OF THE KdV6 EQUATION , 2008 .

[23]  S. Liao.  Notes on the homotopy analysis method: Some definitions and theorems , 2009 .

[24]  W. Hereman,et al.  The tanh method: I. Exact solutions of nonlinear evolution and wave equations , 1996 .

[25]  Robert A. Van Gorder,et al.  Trigonometric and hyperbolic type solutions to a generalized Drinfel'd-Sokolov equation , 2010, Appl. Math. Comput..

[26]  Edward Frenkel,et al.  Characters and fusion rules forW-algebras via quantized Drinfeld-Sokolov reduction , 1992 .

[27]  L. Fehér,et al.  Extensions of the matrix Gelfand-Dickey hierarchy from generalized Drinfeld-Sokolov reduction , 1995, hep-th/9503217.

[28]  Hong-qing Zhang,et al.  Improved Jacobin elliptic function method and its applications , 2003 .

[29]  S. Liao,et al.  Beyond Perturbation: Introduction to the Homotopy Analysis Method , 2003 .

[30]  Masaki Kashiwara,et al.  Operator Approach to the Kadomtsev-Petviashvili Equation —Transformation Groups for Soliton Equations III— , 1981 .

[31]  Senfa Chen,et al.  Traveling wave solutions for Generalized Drinfeld―Sokolov equations , 2009 .

[32]  Shijun Liao,et al.  On the homotopy analysis method for nonlinear problems , 2004, Appl. Math. Comput..

[33]  S. Liao.  An explicit, totally analytic approximate solution for Blasius’ viscous flow problems , 1999 .

[34]  S. Abbasbandy.  The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV equation , 2007 .

[35]  L. Fehér,et al.  Nonstandard Drinfeld-Sokolov reduction , 1997, solv-int/9708002.

[36]  K. Vajravelu,et al.  On the selection of auxiliary functions, operators, and convergence control parameters in the application of the Homotopy Analysis Method to nonlinear differential equations: A general approach , 2009 .

[37]  Zuntao Fu,et al.  Periodic solutions for a class of coupled nonlinear partial differential equations , 2005 .

[38]  Adem C. Cevikel,et al.  New exact travelling wave solutions of nonlinear physical models , 2009 .

[39]  何斌,et al.  BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS FOR GENERALIZED DRINFELD-SOKOLOV EQUATIONS , 2006 .

[40]  W. Malfliet,et al.  The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations , 2004 .

[41]  M. A. Semenov-Tian-Shansky,et al.  Drinfeld–Sokolov Reduction for Difference Operators and Deformations of -Algebras¶I. The Case of Virasoro Algebra , 1998 .

[42]  A. Bekir.  New Exact Travelling Wave Solutions for Regularized Long-wave, Phi-Four and Drinfeld-Sokolov Equations , 2008 .

[43]  Abdul-Majid Wazwaz,et al.  The Cole-Hopf transformation and multiple soliton solutions for the integrable sixth-order Drinfeld-Sokolov-Satsuma-Hirota equation , 2009, Appl. Math. Comput..

[44]  Boris Feigin,et al.  Quantization of the Drinfeld-Sokolov reduction , 1990 .