Symbolic algorithms for the Painlevé test, special solutions, and recursion operators for nonlinear PDEs

This paper discusses the algorithms and implementations of three MATHEMATICA packages for the study of integrability and the computation of closed-form solutions of nonlinear polynomial PDEs. The first package, PainleveTest.m, symbolically performs the Painlevé integrability test. The second package, PDESpecialSolutions.m, computes exact solutions expressible in hyperbolic or elliptic functions. The third package, PDERecursionOperator.m, generates and tests recursion operators

Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs

Algorithms are presented for the tanh- and sech-methods, which lead to closed-form solutions of nonlinear ordinary and partial differential equations (ODEs and PDEs). New algorithms are given to find exact polynomial solutions of ODEs and PDEs in terms of Jacobi's elliptic functions. For systems with parameters, the algorithms determine the conditions on the parameters so that the differential equations admit polynomial solutions in tanh, sech, combinations thereof, Jacobi's sn or cn functions. Examples illustrate key steps of the algorithms. The new algorithms are implemented in Mathematica. The package DDESpecialSolutions.m can be used to automatically compute new special solutions of nonlinear PDEs. Use of the package, implementation issues, scope, limitations, and future extensions of the software are addressed. A survey is given of related algorithms and symbolic software to compute exact solutions of nonlinear differential equations.Comment: 39 pages. Software available from Willy Hereman's home page at http://www.mines.edu/fs_home/whereman

Symbolic computation of hyperbolic tangent solutions for nonlinear differential-difference equations

A new algorithm is presented to find exact traveling wave solutions of differential-difference equations in terms of tanh functions. For systems with parameters, the algorithm determines the conditions on the parameters so that the equations might admit polynomial solutions in tanh. Examples illustrate the key steps of the algorithm. Parallels are drawn through discussion and example to the tanh-method for partial differential equations. The new algorithm is implemented in Mathematica. The package DDESpecialSolutions.m can be used to automatically compute traveling wave solutions of nonlinear polynomial differential-difference equations. Use of the package, implementation issues, scope, and limitations of the software are addressed.Comment: 19 pages submitted to Computer Physics Communications. The software can be downloaded at http://www.mines.edu/fs_home/wherema

Superposition in nonlinear wave and evolution equations

Real and bounded elliptic solutions suitable for applying the Khare-Sukhatme superposition procedure are presented and used to generate superposition solutions of the generalized modified Kadomtsev-Petviashvili equation (gmKPE) and the nonlinear cubic-quintic Schroedinger equation (NLCQSE).Comment: submitted to International Journal of Theoretical Physics, 23 pages, 2 figures, style change

New Exact Solutions of a Generalized Shallow Water Wave Equation

In this work an extended elliptic function method is proposed and applied to the generalized shallow water wave equation. We systematically investigate to classify new exact travelling wave solutions expressible in terms of quasi-periodic elliptic integral function and doubly-periodic Jacobian elliptic functions. The derived new solutions include rational, periodic, singular and solitary wave solutions. An interesting comparison with the canonical procedure is provided. In some cases the obtained elliptic solution has singularity at certain region in the whole space. For such solutions we have computed the effective region where the obtained solution is free from such a singularity.Comment: A discussion about singularity and some references are added. To appear in Physica Script

Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear partial differential and . . .

The Mathematica implementation of the tanh and sech-methods for computing exact travelling wave solutions of nonlinear partial differential equations (PDEs) is presented. These methods also apply to ordinary differential equations (ODEs). Ne

Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs

Algorithms are presented for the tanh- and sech-methods, which lead to closed-form solutions of nonlinear ordinary and partial dierential equations (ODEs and PDEs). New algorithms are given to nd exact polynomial solutions of ODEs and PDEs in terms of Jacobi's elliptic functions. Fo