Júlio Cesar Santos Sampaio

Nonlinear self-adjointness of a generalized fifth-order KdV equation

The new concepts of self-adjoint equations formulated by Ibragimov and Gandarias are applied to a class of fifth-order evolution equations. Then, from Ibragimov?s theorem on conservation laws, conservation laws for the generalized Kawahara equation, simplified Kahawara equation and modified simplified Kawahara equation are established.

Nonlinear Self-Adjoint Classification of a Burgers-KdV Family of Equations

The concepts of strictly, quasi, weak, and nonlinearly self-adjoint differential equations are revisited. A nonlinear self-adjoint classification of a class of equations with second and third order is carried out.

Solução do tipo peakon para uma equação evolutiva de terceira ordem

Neste trabalho, mostramos que uma equacao evolutiva de terceira ordem que admite a solucao Soliton, admite tambem a solucao do tipo Peakon.

A family of wave equations with some remarkable properties

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 for 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 Korteweg–de Vries (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.

Sobre propriedades de invariância de uma nova equação dispersiva

Neste trabalho estudaremos propriedades de invariância de uma famiilia de equacoes dispersivas. Um dos principais objetivos e encontrar as condicoes para que essas equacoes sejam nao-linearmente auto-adjuntas e calcular suas leis de conservacao. Alem disso, encontraremos solucoes invariantes para essas equacoes via simetrias de Lie.