Instability of double-periodic waves in the nonlinear Schrodinger equation

2020/12/30 by Pelinovsky, Dmitry E.
#Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Pattern Formation and Solitons (nlin.PS)

paper · doi:10.48550/arxiv.2012.15344

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.

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