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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

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

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.

Keywords: Standing waves, virial identity, instability, blow up.
Mathematics Subject Classification: Primary: 76B25, 35Q51, 35B35; Secondary: 35B60.
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
References:
[1]

D. Beney and G. Roskes, Wave instability, Studies in Applied Math, 48 (1969), 455-472.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. Ghidaglia and J. Saut, On the Zakharov-Schulman equations, in Nonlinear Dispersive Waves (L. Debnath Ed.), World Scientific, (1992), 83–97.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

E. Kuznetsov and V. Zakharov, Hamiltonian formalism for systems of hydrodynamics type, Mathematical Physics Review, Soviet Scientific Reviews, 4 (1984), 167-220.   Google Scholar

[12]

D. Lannnes, Water Waves: Mathematical Theory and Asymptotics, Mathematical Surveys and Monographs, vol. 188, AMS, Providence, 2013. doi: 10.1090/surv/188.  Google Scholar

[13]

F. Merle, Blow-up Results of Viriel type for Zakharov Equations, Comunications in Mathematical Physics, 175 (1996), 433-455.  doi: 10.1007/BF02102415.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

T. Passot, C. Sulem and P. Sulem, Generalization of acoustic fronts by focusing wave packets, Physic D, 94 (1996), 168-187.   Google Scholar

[17]

A. Rubenchik and V. Zakharov, Nonlinear interaction of high-frequency and low-frequency waves, Prikl. Mat. Techn. Phys, 5 (1972), 84-98.   Google Scholar

[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.  Google Scholar

[19]

E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton university press, Princeton, New Jersey, 1970.  Google Scholar

[20]

M. Tsutsumi, Nonexistence of global solutions to nonlinear Schrödinger, (unpublished manuscript), 1982. Google Scholar

[21]

M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys, 87 (1983), 567-576.   Google Scholar

show all references

References:
[1]

D. Beney and G. Roskes, Wave instability, Studies in Applied Math, 48 (1969), 455-472.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. Ghidaglia and J. Saut, On the Zakharov-Schulman equations, in Nonlinear Dispersive Waves (L. Debnath Ed.), World Scientific, (1992), 83–97.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

E. Kuznetsov and V. Zakharov, Hamiltonian formalism for systems of hydrodynamics type, Mathematical Physics Review, Soviet Scientific Reviews, 4 (1984), 167-220.   Google Scholar

[12]

D. Lannnes, Water Waves: Mathematical Theory and Asymptotics, Mathematical Surveys and Monographs, vol. 188, AMS, Providence, 2013. doi: 10.1090/surv/188.  Google Scholar

[13]

F. Merle, Blow-up Results of Viriel type for Zakharov Equations, Comunications in Mathematical Physics, 175 (1996), 433-455.  doi: 10.1007/BF02102415.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

T. Passot, C. Sulem and P. Sulem, Generalization of acoustic fronts by focusing wave packets, Physic D, 94 (1996), 168-187.   Google Scholar

[17]

A. Rubenchik and V. Zakharov, Nonlinear interaction of high-frequency and low-frequency waves, Prikl. Mat. Techn. Phys, 5 (1972), 84-98.   Google Scholar

[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.  Google Scholar

[19]

E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton university press, Princeton, New Jersey, 1970.  Google Scholar

[20]

M. Tsutsumi, Nonexistence of global solutions to nonlinear Schrödinger, (unpublished manuscript), 1982. Google Scholar

[21]

M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys, 87 (1983), 567-576.   Google Scholar

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