# 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  April 2020 Early access  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 and Continuous Dynamical Systems - B, 2020, 25 (4) : 1213-1240. doi: 10.3934/dcdsb.2019217
##### References:
 [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.

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##### References:
 [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.
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