Aspects of the modified regularized long-wave equation

Abstract

We study various properties of the soliton solutions of the modified regularized long-wave equation. This model possesses exact one- and two-soliton solutions but no other solutions are known. We show that numerical three-soliton configurations, for which the initial conditions were taken in the form of a linear superposition of three single-soliton solutions, evolve in time as three-soliton solutions of the model and in their scatterings each individual soliton experiences a total phase-shift that is the sum of pairwise phase-shifts. We also investigate the soliton resolution conjecture for this equation, and find that individual soliton-like lumps initially evolve very much like lumps for integrable models but eventually (at least) some blow-up, suggesting basic instability of the model.