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

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.

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

“…To generate the system for the unknown coefficients a ij and wave parameters c j , substitute (7) into (6) and set the coefficients of T j to zero. The resulting nonlinear algebraic system for the unknown a ij is parameterized by the wave parameters c j and the parameters in (1), if any.…”

“…Baldwin et al [7] implemented the tanh-method (and also sech, cn, and sn-methods) in Mathematica. Liu and Li implemented the tanhmethod [22,23] in Maple.…”

“…Although the tanh method is easy to apply, it leads to fairly cumbersome algebra without guarantee that a tanh solution can be found [6]. We therefore present [8] the fully automated software package, DDESpecialSolutions.m in Mathematica, which implements the algorithm.…”

Dispersive shock wave interactions and asymptotics

“…where cn(z, k) is the Jacobian elliptic 'cosine' (see [41]); it can be found using the methods in [42]. If we neglect the right-hand side of (13), κ, k, and V are arbitrary constants but vary slowly in general.…”

Exact Solutions Expressible in Hyperbolic and Jacobi Elliptic Functions of Some Important Equations of Ion-Acoustic Waves

“…The tanh-function method is the well known method as a direct selection of the function F(ξ)=tanh( ξ). Recently, many exact solutions expressed by various Jacobi elliptic functions (JEFs) of many NLEEs have been obtained by Jacobi elliptic function expansion method [7][8][9][10], mapping method [11,12], F-expansion method [13], extended F-expansion method [14], the generalized Jacobi elliptic function method [15] and other methods [16][17][18][19][20]. Various exact solutions were obtained by using these methods, including the solitary wave solutions, shock wave solutions and periodic wave solutions.…”