Applied math. talk: Periodic travelling waves in nonlinear wave equations: modulation instability and rogue waves by Dmitry Pelinovsky, McMaster University, Canada

Abstract:     I will overview the following different wave phenomena in
integrable nonlinear wave equations:

(1) universal patterns in the dynamics of fluxon condensates in the
semi-classical limit;
(2) modulational instability of periodic travelling waves;
(3) rogue waves on the background of periodic and double-periodic waves.

Main examples include the sine-Gordon equation, the nonlinear
Schroedinger equation, and the derivative nonlinear Schroedinger
equation. For the latter equation, in collaboration with Jinbing Chen
(South East University, China) and Jeremy Upsal (University of
Washington, USA), we adapted the method of nonlinearization of the Lax
system in order to characterize the existence and modulation stability
of periodic travelling waves. We give precise information on the
location of Lax and stability spectra, with assistance of numerical
package based on the so-called Hill’s method. Particularly interesting
outcome is the explicit relation between the onset of modulation
instability and the existence of a rogue wave (localized solution in
space and time) on the background of periodic travelling waves.