Article Contents
Article Contents

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

• * Corresponding author: José R. Quintero

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.

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

 Citation:

•  [1] D. Beney and G. Roskes, Wave instability, Studies in Applied Math, 48 (1969), 455-472. [2] R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system, Communications in Partial Differential Equations, 17 (1992), 967-988.  doi: 10.1080/03605309208820872. [3] R. Cipolatti, On the instability of ground states for a Davey-Stewartson system, Annales de L'I.H.P, A, 58 (1993), 84-104. [4] J. Cordero, Subsonic and Supersonic Limits for the Zakharov-Rubenchik System, Ph.D thesis, Instituto de Matemática Pura e Aplicada - IMPA, Rio de Janeiro, 2010. [5] J. C. Cordero, Supersonic limits for the Zakharov-Rubenchik system, Journal of Differential Equations, 261 (2016), 5260-5288.  doi: 10.1016/j.jde.2016.07.022. [6] A. Davey and K. Stewartson, On three-dimensional packets of surface waves, Proc. R. Soc. A, 338 (1974), 101-110.  doi: 10.1098/rspa.1974.0076. [7] J. Ghidaglia and J. Saut, On the initial value problem for the Davey-Stewartson systems, Nonlinearity, 3 (1990), 475-506.  doi: 10.1088/0951-7715/3/2/010. [8] J. Ghidaglia and J. Saut, On the Zakharov-Schulman equations, in Nonlinear Dispersive Waves (L. Debnath Ed.), World Scientific, (1992), 83–97. [9] R. Glassey, On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation, J. Math. Phys, 18 (1977), 1794-1797.  doi: 10.1063/1.523491. [10] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in presence of symmetry, I, Functional Anal, 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9. [11] E. Kuznetsov and V. Zakharov, Hamiltonian formalism for systems of hydrodynamics type, Mathematical Physics Review, Soviet Scientific Reviews, 4 (1984), 167-220. [12] D. Lannnes, Water Waves: Mathematical Theory and Asymptotics, Mathematical Surveys and Monographs, vol. 188, AMS, Providence, 2013. doi: 10.1090/surv/188. [13] F. Merle, Blow-up Results of Viriel type for Zakharov Equations, Comunications in Mathematical Physics, 175 (1996), 433-455.  doi: 10.1007/BF02102415. [14] F. Oliveira, Stability of the solitons for the one-dimensional Zakharov-Rubenchik equation, Physica D, 175 (2003), 220-240.  doi: 10.1016/S0167-2789(02)00722-4. [15] F. Oliveira, Adiabatic limit of the Zakharov-Rubenchik equation, Reports on Mathematical Physics, 61 (2008), 13-27.  doi: 10.1016/S0034-4877(08)00006-2. [16] T. Passot, C. Sulem and P. Sulem, Generalization of acoustic fronts by focusing wave packets, Physic D, 94 (1996), 168-187. [17] A. Rubenchik and V. Zakharov, Nonlinear interaction of high-frequency and low-frequency waves, Prikl. Mat. Techn. Phys, 5 (1972), 84-98. [18] J. C. Saut and G. Ponce, Wellposedness for the Benney-Roskes/Zakharov-Rubenchik system, Discrete and Continuous Dynamical Systems, 13 (2005), 811-825.  doi: 10.3934/dcds.2005.13.811. [19] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton university press, Princeton, New Jersey, 1970. [20] M. Tsutsumi, Nonexistence of global solutions to nonlinear Schrödinger, (unpublished manuscript), 1982. [21] M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys, 87 (1983), 567-576.