Part I: Soliton on a Beach and Related Problems. Part II: Modulated Capillary Waves

Abstract

I. SOLITON ON A SLOPING BEACH AND RELATED PROBLEMS The problem of the behaviour of a soliton on a slowly varying beach is considered. It is shown that for a correct description, the full Boussinesq equations rather than a Korteweg-de Vries type approximation must be used. Using both energy conservation and two-timing expansions, the behaviour of the soliton is analysed. The slowly varying soliton is found not to conserve mass and momentum and it has been suggested that to conserve these quantities, both forward and reflected waves must be added behind the soliton, these waves being solutions of the linear shallow water equations. It is shown that to the order of approximation of the Boussinesq equations, only a forward wave (or tail) behind the soliton is necessary to fulfill mass and momentum conservation. A perturbed Korteweg-de Vries equation for which the perturbation adds energy to the soliton is considered. It is found that a tail is formed behind the soliton. The development of this tail into new solitons is analysed. II. MODULATED CAPILLARY WAVES An exact hodograph solution for symmetric and antisymmetric capillary waves on a fluid sheet (of possibly infinite thickness) has been previously found. Using this solution, an exact averaged Lagrangian for slowly varying capillary waves is calculated. Modulation equations can be found from this averaged Lagrangian, but due to the algebraic complexity of the equations, the limit of waves on a thin fluid sheet is considered. From the modulation equations, the stability of symmetric and antisymmetric capillary waves on a thin fluid sheet is found. The modulation equations for antisymmetric waves form a hyperbolic system and the simple wave solutions for this system are calculated. These simple wave solutions are interpreted physically.