Fields Institute Workshop on Nonlinear Wave Dynamics

Carleton University, August 20-22 2008


Gordon E. Swaters

Department of Mathematical and Statistical Sciences, University of Alberta

Direct Perturbation Theory for Modulated Solitons

Solitary waves or solitons (and we will use these terms interchangeably) typically arise in "wave problems" where the underlying dynamics is linear to leading order and nonlinearity plays a secondary but cumulatively non-negligible role. Frequently, solitary wave equations are derived via a straight forward weakly-nonlinear multiple time-scale asymptotic expansion of the underlying field equations exploiting the time separation between the linear wave dynamics and the cumulative effects of nonlinearity. The role of dispersion in the problem determines the type of solitary wave equation that is typically derived. For example, if the leading order linear problem is nondispersive (the phase speed does not explicitly depend on the wave number(s)), one typically obtains an equation from the Korteweg-de Vries (KdV) family to determine the evolution of the wave. On the other hand, if the leading order linear problem is dispersive (the phase speed explicitly depends on the wave number(s)), one typically obtains an equation from the nonlinear Schrodinger (NLS) family to determine the wave amplitude. There are other canonical soliton equations as well, of course, (e.g., sine-Gordon (SG) equation) that can arise.

In most "real world" situations there are many other physical process besides nonlinearity and dispersion (e.g., dissipation or spatially and temporally varying environments, etc.) that act to modulate the evolution of the solitary wave. It is often the case that one must take into account these other physical processes to properly interpret experimental and direct observations of solitary waves. It is of interest, therefore, to develop mathematical and/or computational machinery to determine the behavior of solitons that are being subjected to forces that areacting to modulate their propagation properties.

The principal goal of my lectures is to give an introduction (at a graduate student level) to the direct perturbation theory for perturbed solitons. We will examine in some detail the perturbed KdV equation. Time permitting, we shall also present the perturbation theory for the breather solutions to the NLS, SG and the modon solutions to the Charney-Hasegawa-Mima (CHM) equations, respectively.