Nonlinear modulations of solitary waves

The leading optical approximation to a slowly varying solitary crest on constant depth is the plane soliton solution with the local values of amplitude and orientation substituted. This leads to two nonlinear hyperbolic equations for the local amplitude and inclination of the crest that have been reported by several authors and predict the formation of progressive wave jumps, or shocks, from any initial perturbation of the crest. In comparison to numerical solutions of the Boussinesq equations we find that this optical approximation fails to reproduce essential properties of the crest dynamics, in particular that the crest modulations are damped and that well-defined wave jumps do not necessarily evolve. One purpose of the present work is to include such features in an amended optical approximation.

We obtain the leading correction to the ‘local soliton’ solution by a multiple scale technique. In addition to a modification to the wave profile the perturbation expansion also yields a diffracted wave system and a celerity that depends on the curvature of the crest. The principle of energy conservation then leads us to a second-order optical approximation consisting of transport equations of mixed hyperbolic/parabolic nature. Under additional assumptions the transport equations can be reduced to the well-known Burgers equation.

Numerical simulations of the Boussinesq equations are performed for modulations on otherwise straight crests and radially converging solitons. The improved optical, or ray, theory reproduces all essential features and agrees closely with the numerical solution in both cases. Contrary to purely hyperbolic optical descriptions the present theory also predicts wave jumps of finite width that are consistent with the triad solution of Miles (1977).

The present work indicates that while sinusoidal waves often are appropriately described by the lowest-order physical optics, higher-order corrections must be expected to be important for single crested waves.