Two-level solitary waves as generalized solutions of the KdV equation

Abstract

An analytical class of standing wave solutions to the Korteweg–de Vries (KdV) equation is obtained in the framework of continuous finite generalized functions. This paper shows how this class can be used to describe a great variety of wave shapes, especially bores and jumps. These new solutions are built by appropriately combining parts of two ordinary KdV waves. These represent a system with a steep transition between different energy levels of two potential wells. A number of specific cases of the generalized solutions are identical to those obtained recently within the theory of the KdV equation forced by a Dirac delta function. Numerical simulations of both stationary and transient KdV equations are carried out in a few cases. The weak formulation used in the numerical scheme is equivalent to the analytical generalized functions approach. Simulations of initially perturbed wave fronts prove the high degree of stability of many of these solutions.