Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear partial differential and differential-difference equations

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). New algorithms are given to compute polynomial solutions of ODEs and PDEs in terms of the Jacobi elliptic functions. An adaptation of the tanh-method to nonlinear differential-difference equations (DDEs) is also presented. The new Mathematica packages, PDESpecialSolutions.m and DDESpecialSolutions.m, automatically compute closed-form solutions which are expressible as polynomials in the tanh, sech, sn and cn functions. For systems of ODEs, PDEs, or DDEs with constant parameters, the software finds the conditions on the parameters so that the given differential equations admit solutions involving tanh, sech, both, sn or cn. ∗Research supported in part by the National Science Foundation under Grants DMS9732069, DMS-9912293 and CCR-9901929. ⋆ Correspondence per email: [email protected]