Behavior of perturbations of solitary and periodic waves on the surface of a heavy liquid

Abstract

In [1] a system of equations was obtained for the case of a potential motion of an ideal incompressible homogeneous fluid; the system described the propagation of a train of waves in a medium with slowly varying properties, the motion in the train being characterized by a wave vector and a frequency. A solitary wave is a particular case of a wave train in which the length of the waves in the train is large. In [2, 3] a quasilinear system of partial differential equations was obtained which described two-dimensional and three-dimensional motion of a solitary wave in a layer of liquid of variable depth. It follows from this system that if the unperturbed state of the liquid is the quiescent state, then some integral quantity (the average wave energy [2–4]), referred to an element of the front, is preserved during the course of the motion. This fact is also valid for a train of waves, and can be demonstrated to be so upon applying the formalism of [1] to a Lagrangian similar to that used in [2]. In the present paper we obtain, for the case of a layer of liquid of constant depth, a solution in the form of simple waves for a system, equivalent to the system obtained in [3], describing the motion of a solitary wave and also the motion of a train of waves. We show that it is possible to have tilting of simple waves, leading in the case considered here to the formation of corner points on the wave front. We consider several examples of initial perturbations, and we obtain their asymptotics as t→∞. We make our presentation for the solitary wave case; however, in view of our statement above, the results automatically carry over to the case of a train of waves.