A systemized version of the tanh method is used to solve particular evolution and wave equations. If one deals with conservative systems, one seeks travelling wave solutions in the form of a h i t e series in tanh. If present, boundary conditions are implemented in this expansion. The associated velocity can then be determined a priori, provided the solution vanishes at in6nity. Hence, exact closed form solutions can be obtained easily in various cases.

No abstract

With the aid of the tanh method, nonlinear wave equations are solved in a perturbative way. First, the KdVBurgers equation is investigated in the limit of weak dispersion. As a result, a general shock wave profile, with a perturbative solitary-wave contribution superposed, emerges. For a particular choice of the parameters, a comparison with the exact solution is made. Further, the MKdVBurgers is investigated in the same limit and similar results are obtained.Colorado School of Mines. Their hospitality is greatly appreciated. Thanks are due to Professor Frank Verheest for fruitful discussions.

A new algorithm is presented to find exact traveling wave solutions of differentialdifference 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. Program summary

A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (P∆Es) is reviewed. The method assumes that the P∆Es are defined on a quadrilateral, and consistent around the cube. Next, the method is extended to systems of P∆Es where one has to carefully account for equations defined on edges of the quadrilateral. Lax pairs are presented for scalar integrable P∆Es classified by Adler, Bobenko, and Suris and systems of P∆Es including the integrable 2-component potential Korteweg-de Vries lattice system, as well as nonlinear Schrödinger and Boussinesq-type lattice systems. Previously unknown Lax pairs are presented for P∆Es recently derived by Hietarinta (J. Phys. A: Math. Theor., 44 (2011) Art. No. 165204). The method is algorithmic and is being implemented in Mathematica.

No abstract

No abstract

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 PDESpecialSolutions.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.