Stability and instability analysis for the standing waves for a generalized Zakharov-Rubenchik system

Quintero, José R.; Quintero, José R.

doi:10.22199/issn.0717-6279-4547

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Proyecciones (Antofagasta)

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Proyecciones (Antofagasta) vol.41 no.3 Antofagasta June 2022

http://dx.doi.org/10.22199/issn.0717-6279-4547

Artículos

Stability and instability analysis for the standing waves for a generalized Zakharov-Rubenchik system

José R. Quintero¹

^¹Department of Mathematics, Universidad del Valle, Calle 13, 100-00, Cali, Colombia. e-mail: jose.quintero@correounivalle.edu.co

Abstract

In this paper, we analyze the stability and instability of standing waves for a generalized Zakharov-Rubenchik system (or the Benney-Roskes system) in spatial dimensions N = 2, 3. We show that the standing waves generated by the set of minimizers for the associated variational problem are stable, for N = 2 and σ(p − 2) > 0. We also show that the standing waves are strongly unstable, for N = 3 and if either σ < 0 and 4/3 <p< 4, or σ > 0 and 0 <p< 2. Results follow by using the variational characterization of standing waves, the concentration compactness principle due to J. Lions and the compactness lemma due to E. Lieb to solve the associated minimization problem.

Keywords: standing waves; virial identity; stability; blow up

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Acknowledgment

J. Quintero was supported by the Mathematics Department at Universidad del Valle, under the research project CI 71231. J. Quintero was supported by Math AmSud and Minciencias-Colombia under the project MathAmSud 21-Math-03.

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Received: October 30, 2020; Accepted: October 30, 2021

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