New Aspects of Exponential Asymptotics in Multiple-Scale Nonlinear Wave Problems

摘要

It is known that standard multiple-scale perturbation techniques fail to pinpoint the soliton solution branches that bifurcate at edges of bandgaps in periodic media, owing to the appearance of exponentially small growing wave tails when the soliton's envelope is not properly positioned. When the bifurcation is from a single wave mode of a band edge, this difficulty has been handled in recent work by computing these tails via an exponential asymptotics technique in the wave number domain. However, the same approach is not directly applicable to the bifurcation of solitons near the opening of a bandgap, where wave modes from two nearby band edges interact with each other. Here, we discuss two nontrivial extensions of the exponential asymptotics technique that enable resolving this issue. For simplicity, the analysis focuses on two model problems, namely, a steady-state forced Korteweg-de Vries equation and a steady-state forced nonlinear Schrodinger equation, with the precise form of forcing and balance between nonlinear and dispersive terms chosen so as to mimic the situation encountered in the bifurcation of solitons near a bandgap opening. Our analysis exhibits a number of new features that are significantly different from previous exponential asymptotics procedures, such as the treatments when the nonlinearity dominates dispersion and when the decay rates of the Fourier-transformed solution are asymmetric. In addition, the analysis reveals new, and in some cases rather unexpected, functional forms for exponentially small wave tails, which are also confirmed by numerical results.