Numerical Studies of the Regularized Long Wave Equation

Abstract

The Regularized Long Wave (RLW) equation, u t + u x + u u x -µ u x x t =0, has similar properties to the Korteweg-de Vries (KdV) equation. For example, the RLW equation has a stable solitary solution and dispersive property, However, the RLW equation has been found to have only two invariants, while the KdV equation has an infinite number of invariants. The present numerical studies show that the so-called recurrence property (almost-periodicity) for the KdV equation solution does not hold in the case of the RLW equation solution. However, the energy of the RLW solution is shared only among the lower modes of the system (no-thermalization). If the coefficient µ value is large (µ≧1.0), it is found by numerical computations that the recurrence property approximately holds. Some discussions are made on almost-periodicity of the RLW equation with a large value µ. Though the µ-dependence of almost-periodicity of the RLW equation does not become clear, we conjecture that collisions among many solitary waves are...