A family of wave equations with some remarkable properties
Abstract
We consider a family of homogeneous nonlinear dispersive equations with two arbitrary parameters. Conservation laws are established from the point symmetries and imply that the whole family admits square integrable solutions. Recursion operators are found to two members of the family investigated. For one of them, a Lax pair is also obtained, proving its complete integrability. From the Lax pair we construct a Miura-type transformation relating the original equation to the KdV equation. This transformation, on the other hand, enables us to obtain solutions of the equation from the kernel of a Schr\"odinger operator with potential parametrized by the solutions of the KdV equation. In particular, this allows us to exhibit a kink solution to the completely integrable equation from the 1-soliton solution of the KdV equation. Finally, peakon-type solutions are also found for a certain choice of the parameters, although for this particular case the equation is reduced to a homogeneous second order nonlinear evolution equation.
Keywords
Cite
@article{arxiv.1602.06525,
title = {A family of wave equations with some remarkable properties},
author = {Priscila Leal da Silva and Igor Leite Freire and Júlio Cesar Santos Sampaio},
journal= {arXiv preprint arXiv:1602.06525},
year = {2018}
}
Comments
This is a previous version of the published paper. There are some differences between this version and the one published. However the scientific content is just the same