American Institute of Mathematical Sciences

April  2020, 25(4): 1213-1240. doi: 10.3934/dcdsb.2019217

Instability of the standing waves for a Benney-Roskes/Zakharov-Rubenchik system and blow-up for the Zakharov equations

 1 Mathematics Department, Universidad del Valle, Cali, Valle del Cauca, Colombia 2 Mathematics and Statistics Department, Universidad Nacional de Colombia, Manizales, Caldas, Colombia

* Corresponding author: José R. Quintero

Received  January 2019 Revised  June 2019 Published  September 2019

Fund Project: JRQ is supported by the Mathematics Department at Universidad del Valle and JCC is supported by the Mathematics Department at Universidad Nacional de Colombia.

In this paper we establish the nonlinear orbital instability of ground state standing waves for a Benney-Roskes/Zakharov-Rubenchik system that models the interaction of low amplitude high frequency waves, acustic type waves in $N = 2$ and $N = 3$ spatial directions. For $N = 2$, we follow M. Weinstein's approach used in the case of the Schrödinger equation, by establishing a virial identity that relates the second variation of a momentum type functional with the energy (Hamiltonian) on a class of solutions for the Benney-Roskes/Zakharov-Rubenchik system. From this identity, it is possible to show that solutions for the Benney-Roskes/Zakharov-Rubenchik system blow up in finite time, in the case that the energy (Hamiltonian) of the initial data is negative, indicating a possible blow-up result for non radial solutions to the Zakharov equations. For $N = 3$, we establish the instability by using a scaling argument and the existence of invariant regions under the flow due to a concavity argument.

Citation: José R. Quintero, Juan C. Cordero. Instability of the standing waves for a Benney-Roskes/Zakharov-Rubenchik system and blow-up for the Zakharov equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1213-1240. doi: 10.3934/dcdsb.2019217
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