D.E. Pelinovsky

Instability of double-periodic solutions in the nonlinear Schrodinger equation

Frontiers in Physics 9 (2021) 599146 (10 pages)

Abstract:
It is shown how to compute the instability rates for the double-periodic solutions to the cubic NLS (nonlinear Schrodinger) equation by using the Lax linear equations. The wave function modulus of the double-periodic solutions is periodic both in space and time coordinates; such solutions generalize the standing waves which have the time-independent and space-periodic wave function modulus. Similar to other waves in the NLS equation, the double-periodic solutions are spectrally unstable and this instability is related to the bands of the Lax spectrum outside the imaginary axis. A simple numerical method is used to compute the unstable spectrum and to compare the instability rates of the double-periodic solutions with those of the standing periodic waves.

Keywords:
nonlinear Schrodinger equation, double-periodic solutions, Lax spectrum, modulational instability.