Symbolic Computation of Solutions for Three Generalized Nonlinear Partial Differential eQuations by Using the Tanh Method

Three nonlinear partial differential equations, namely, the standard KdV equation, the Boussinesq equation and the generalized fifthorder KdV equation are considered here from of point the view of construct exact solutions for them. The equations that we consider here are in its most general form. New exact solutions which include periodic and soliton solutions are formally derived by using the tanh method. The programming language Matematica is used.

[1]  César A. Gómez Sierra,et al.  New exact solutions for the combined sinh-cosh-Gordon equation , 2006 .

[2]  Mark J. Ablowitz,et al.  Solitons and the Inverse Scattering Transform , 1981 .

[3]  Alvaro Salas.  Some exact solutions for the Caudrey-Dodd-Gibbon equation , 2008 .

[4]  Cesar Augusto Gómez Sierra,et al.  New periodic and soliton solutions for the Generalized BBM and Burgers-BBM equations , 2010, Appl. Math. Comput..

[5]  Hongyan Zhi,et al.  A new Riccati equation expansion method with symbolic computation to construct new travelling wave solution of nonlinear differential equations , 2006, Appl. Math. Comput..

[6]  Jairo Ernesto Castillo Hernández,et al.  Exact solutions for a nonlinear model , 2010, Appl. Math. Comput..

[7]  P. Drazin.  SOLITONS, NONLINEAR EVOLUTION EQUATIONS AND INVERSE SCATTERING (London Mathematical Society Lecture Note Series 149) , 1993 .

[8]  Alvaro Salas,et al.  About the seventh-order Kaup-Kupershmidt equation and its solutions , 2008 .

[9]  Cesar A. Gómez.  A NEW TRAVELLING WAVE SOLUTION OF THE MIKHAILOV-NOVIKOV-WANG SYSTEM USING THE EXTENDED TANH METHOD , 2007 .

[10]  Cesar Augusto Gómez Sierra,et al.  The variational iteration method combined with improved generalized tanh-coth method applied to Sawada-Kotera equation , 2010, Appl. Math. Comput..

[11]  Alvaro H. Salas,et al.  EXACT SOLUTIONS FOR A REACTION DIFFUSION EQUATION BY USING THE GENERALIZED TANH METHOD Soluciones exactas para una ecuación de reacción difusión, usando el método generalizado de la tanh , 2007 .

[12]  A S Cesar Gómez,et al.  Special forms of the fifth-order KdV equation with new periodic and soliton solutions , 2007 .

[13]  Alvaro Salas Salas.  Two standard methods for solving the Ito equation , 2008 .

[14]  A. Wazwaz.  Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method , 2001 .

[15]  Alvaro H. Salas,et al.  SOLUTIONS FOR A CLASS OF FIFTH-ORDER NONLINEAR PARTIAL DIFFERENTIAL SYSTEM , 2008 .

[16]  Alvaro H. Salas,et al.  A PRACTICAL APPROACH TO SOLVE COUPLED SYSTEMS OF NONLINEAR PDE's , 2009 .

[17]  Abdul-Majid Wazwaz,et al.  The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations , 2007, Appl. Math. Comput..

[18]  Cesar Augusto Gómez Sierra,et al.  The Cole-Hopf transformation and improved tanh-coth method applied to new integrable system (KdV6) , 2008, Appl. Math. Comput..

[19]  J. Gibbon,et al.  Solitons and Nonlinear Wave Equations , 1982 .

[20]  Zhao Qiang,et al.  EXPANSION METHOD ABOUT THE JACOBI ELLIPTIC FUNCTION ANDITS APPLICATIONS TO NONLINEAR WAVE EQUATIONS , 2001 .

[21]  A. G. Bratsos.  The solution of the Boussinesq equation using the method of lines , 1998 .

[22]  Willy Hereman,et al.  Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs , 2002, J. Symb. Comput..

[23]  G. Lamb.  Elements of soliton theory , 1980 .

[24]  Cesar Augusto Gómez Sierra,et al.  The generalized tanh-coth method to special types of the fifth-order KdV equation , 2008, Appl. Math. Comput..

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

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

[27]  Engui Fan,et al.  Generalized tanh Method Extended to Special Types of Nonlinear Equations , 2002 .

[28]  Alvaro Salas,et al.  Exact solutions for the general fifth order KdV equation by the extended tanh method , 2008 .

[29]  Cesar Augusto Gómez Sierra,et al.  Computing exact solutions for some fifth KdV equations with forcing term , 2008, Appl. Math. Comput..

[30]  Zhenya Yan,et al.  The Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear differential equations , 2003 .

[31]  Ji-Huan He,et al.  Generalized solitary solution and compacton-like solution of the Jaulent–Miodek equations using the Exp-function method , 2008 .

[32]  Cesar Augusto Gómez Sierra,et al.  New solutions for the modified generalized Degasperis-Procesi equation , 2008, 0809.2862.

[33]  S. Novikov,et al.  Theory of Solitons: The Inverse Scattering Method , 1984 .

[34]  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 .

[35]  M. Wadati,et al.  A Soliton and Two Solitons in an Exponential Lattice and Related Equations , 1973 .

[36]  Alvaro H. Salas,et al.  EXACT SOLUTIONS FOR A NEW INTEGRABLE SYSTEM (KDV6) , 2008 .

[37]  Cesar Augusto Gómez Sierra,et al.  New abundant solutions for the Burgers equation , 2009, Comput. Math. Appl..

[38]  S Gómez,et al.  EXACT SOLUTIONS FOR A NEW FIFTH-ORDER INTEGRABLE SYSTEM , 2006 .

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