Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2002-01-08
Nonlinear Sciences
Exactly Solvable and Integrable Systems
39 pages. Software available from Willy Hereman's home page at http://www.mines.edu/fs_home/whereman/
Scientific paper
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.
Baldwin Douglas
Goktas Unal
Hereman Willy
Hong Linghui
Martino R. S.
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