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

### March 22 @ 3:00 pm - 4:00 pm

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.