155 research outputs found
Solitary Wave and Shock Wave Solutions of the Variants of Boussinesq Equations
This paper obtains the solitary wave as well as the shock wave solutions of the variants of the Boussinesq equations in both (1+1) and (1+2) dimensions. The domain restrictions are also identified in the process
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
Adding integrable defects to the Boussinesq equation
The purpose of this paper is to extend the store of models able to support
integrable defects by investigating the two-dimensional Boussinesq nonlinear
wave equation. As has been previously noted in many examples, insisting that a
defect contributes to energy and momentum to ensure their conservation, despite
the presence of discontinuities and the explicit breaking of translation
invariance, leads to sewing conditions relating the two fields and their
derivatives on either side of the defect. The manner in which several types of
soliton solutions to the Boussinesq equation are affected by the defect is
explored and reveals new effects that have not been observed in other
integrable systems, such as the possibility of a soliton reflecting from a
defect or of a defect decaying into one or two solitons.Comment: 28 pages, 9 figure
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