A direct construction of a full family of Whitham solitary waves

TL;DR

This paper constructs a comprehensive family of solitary waves for the gravity Whitham equation by extending periodic wave solutions through limiting processes, providing new insights into wave behavior and structure.

Contribution

It introduces a direct method to generate a full family of solitary waves from periodic solutions, including the highest wave, using uniform estimates and limiting arguments.

Findings

Constructed a full family of solitary waves starting from zero solution.

Proved convergence of periodic waves to solitary waves with negative tails.

Established uniqueness and continuity properties for solutions.

Abstract

Starting with the periodic waves earlier constructed for the gravity Whitham equation, we parameterise the solution curves through relative wave height, and use a limiting argument to obtain a full family of solitary waves. The resulting branch starts from the zero solution, traverses unique points in the wave speed-wave height space, and reaches a singular highest wave at $\varphi(0) = \frac{\mu}{2}$. The construction is based on uniform estimates improved from earlier work on periodic waves for the same equation, together with limiting arguments and a Galilean transform to exclude vanishing waves and waves levelling off at negative surface depth. In fact, the periodic waves can be proved to converge locally uniformly to a wave with negative tails, which is then transformed to the desired branch of solutions. The paper also contains some proof concerning uniqueness and continuity for signed solutions (improved touching lemma).