On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type

TL;DR

This paper investigates the existence and stability of solitary-wave solutions in a class of Whitham-type evolution equations, using variational methods and nonlinear approximations, including the Korteweg-de Vries equation.

Contribution

It introduces a new approach to find solitary waves via constrained minimisation and demonstrates their conditional energetic stability.

Findings

Existence of solitary-wave solutions using variational methods.

Approximation of solutions by weakly nonlinear PDEs like KdV.

Conditional energetic stability of the solitary waves.

Abstract

We consider a class of pseudodifferential evolution equations of the form $$u_t + (n(u) + Lu)_x = 0,$$ in which $L$ is a linear smoothing operator and $n$ is at least quadratic near the origin; this class includes in particular the Whitham equation. A family of solitary-wave solutions is found using a constrained minimisation principle and concentration-compactness methods for noncoercive functionals. The solitary waves are approximated by (scalings of) the corresponding solutions to partial differential equations arising as weakly nonlinear approximations; in the case of the Whitham equation the approximation is the Korteweg-deVries equation. We also demonstrate that the family of solitary-wave solutions is conditionally energetically stable.