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

José R. Quintero; Juan C. Cordero

doi:10.3934/dcdsb.2019217

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2020, Volume 25, Issue 4: 1213-1240. Doi: 10.3934/dcdsb.2019217

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

José R. Quintero^1, , and
Juan C. Cordero^2,

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
^* Corresponding author: José R. Quintero

Received: December 31, 2018

Revised: May 31, 2019

Early access: September 2019

Published: April 2020

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 76B25, 35Q51, 35B35; Secondary: 35B60.

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