A Maximisation Technique for Solitary Waves: The Case of the Nonlocally Dispersive Whitham Equation

Abstract

Recently, two different proofs for large and intermediate-size solitary waves of the nonlocally dispersive Whitham equation have been presented, using either global bifurcation theory or the limit of waves of large period. We give here a different approach by maximising directly the dispersive part of the energy functional, while keeping the remaining nonlinear terms fixed with an Orlicz-space constraint. This method is, to the best of our knowledge new in the setting of water waves. The constructed solutions are bell-shaped in the sense that they are even, one-sided monotone, and attain their maximum at the origin. The method initially considers weaker solutions than in earlier works, and is not limited to small waves: a family of solutions is obtained, along which the dispersive energy is continuous and increasing. In general, our construction admits more than one solution for each energy level, and waves with the same energy level may have different heights. Although a transformation in the construction hinders us from concluding the family with an extreme wave, we give a quantitative proof that the set reaches ‘large’ or ‘intermediate-sized’ waves.