Orbital stability of periodic traveling-wave solutions for the regularized Schamel equation

• We study the regularized Schamel equation.
• Existence of a one-parameter family of periodic traveling-wave solutions is proved.
• Orbital stability in the energy space is provided.
• Global well-posedness for the Cauchy problem in the energy space is established.

In this work we study the orbital stability of periodic traveling-wave solutions for dispersive models. The study of traveling waves started in the mid-18th century when John S. Russel established that the flow of water waves in a shallow channel has constant evolution. In recent years, the general strategy to obtain orbital stability consists in proving that the traveling wave in question minimizes a conserved functional restricted to a certain manifold. Although our method can be applied to other models, we deal with the regularized Schamel equation, which contains a fractional nonlinear term. We obtain a smooth curve of periodic traveling-wave solutions depending on the Jacobian elliptic functions and prove that such solutions are orbitally stable in the energy space. In our context, instead of minimizing the augmented Hamiltonian in the natural codimension two manifold, we minimize it in a “new” manifold, which is suitable to our purposes.