Spontaneously generated waves and obstacles to integrability in perturbed evolution equations

Abstract

The perturbed Burgers and KdV equations are considered. Often, the perturbation excites waves that are different from the solution one is seeking. In the case of the Burgers equation, the spontaneously generated wave is also a solution of the equation. In contrast, in the case of the KdV equation, this wave is constructed from new (non-KdV) solitons that undergo an elastic collision around the origin. Their amplitudes have opposite signs, which they exchange upon collision. The perturbation then contains terms, which represent coupling between the solution and these spontaneously generated waves. Whereas the unperturbed equations describe gradient systems, these coupling terms may be non-gradient. In that case, they turn out to be obstacles to asymptotic integrability, encountered in the analysis of the solutions of the perturbed evolution equations.