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arXiv · 1602.06525

A family of wave equations with some remarkable properties

Priscila Leal da Silva·Igor Leite Freire·Júlio Cesar Santos Sampaio
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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 Schrödinger 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.

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Priscila Leal da Silva, Igor Leite Freire, Júlio Cesar Santos Sampaio. 2018-02-14. A family of wave equations with some remarkable properties. https://doi.org/10.1098/rspa.2017.0763

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