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Existence of a global attractor for fractional differential hemivariational inequalities
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 |
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.
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.
|
show all references
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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