Stability of standing waves for a generalized Benney-Roskes system
Author:
José Raúl Quintero
Journal:
Quart. Appl. Math. 82 (2024), 65-79
MSC (2020):
Primary 35Q55, 35A15, 35B35
DOI:
https://doi.org/10.1090/qam/1654
Published electronically:
May 2, 2023
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We analyze the orbital stability of standing waves for a generalized Benney-Roskes system in spatial dimensions $N=2$, $3$. We establish stability of standing waves under certain conditions by reducing the system to a single nonlinear (nonlocal) Schrödinger equation, using the variational characterization of standing waves and a convexity argument.
References
- D. Beney and G. Roskes, Wave instability, Studies in Applied Math 48 (1969), 455–472.
- Henri Berestycki and Thierry Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), no. 9, 489–492 (French, with English summary). MR 646873
- T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), no. 4, 549–561. MR 677997, DOI 10.1007/BF01403504
- Rolci Cipolatti, On the existence of standing waves for a Davey-Stewartson system, Comm. Partial Differential Equations 17 (1992), no. 5-6, 967–988. MR 1177301, DOI 10.1080/03605309208820872
- Rolci Cipolatti, On the instability of ground states for a Davey-Stewartson system, Ann. Inst. H. Poincaré Phys. Théor. 58 (1993), no. 1, 85–104 (English, with English and French summaries). MR 1208793
- José R. Quintero and Juan C. Cordero, Instability of the standing waves for a Benney-Roskes/Zakharov-Rubenchik system and blow-up for the Zakharov equations, Discrete Contin. Dyn. Syst. Ser. B 25 (2020), no. 4, 1213–1240. MR 4063989, DOI 10.3934/dcdsb.2019217
- Juan Carlos Cordero Ceballos, Supersonic limit for the Zakharov-Rubenchik system, J. Differential Equations 261 (2016), no. 9, 5260–5288. MR 3542975, DOI 10.1016/j.jde.2016.07.022
- J. C. Cordero, Tesis de doctorado, Instituto de Matematica Pura e Aplicada (IMPA), 2010.
- A. Davey and K. Stewartson, On three-dimensional packets of surface waves, Proc. Roy. Soc. London Ser. A 338 (1974), 101–110. MR 349126, DOI 10.1098/rspa.1974.0076
- Jean-Michel Ghidaglia and Jean-Claude Saut, On the Zakharov-Schulman equations, Nonlinear dispersive wave systems (Orlando, FL, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 83–97. MR 1194632
- Jean-Michel Ghidaglia and Jean-Claude Saut, On the initial value problem for the Davey-Stewartson systems, Nonlinearity 3 (1990), no. 2, 475–506. MR 1054584, DOI 10.1088/0951-7715/3/2/010
- V. E. Zakharov and E. A. Kuznetsov, Hamiltonian formalism for systems of hydrodynamic type, Mathematical physics reviews, Vol. 4, Soviet Sci. Rev. Sect. C: Math. Phys. Rev., vol. 4, Harwood Academic Publ., Chur, 1984, pp. 167–220. Translated from the Russian. MR 768940
- David Lannes, The water waves problem, Mathematical Surveys and Monographs, vol. 188, American Mathematical Society, Providence, RI, 2013. Mathematical analysis and asymptotics. MR 3060183, DOI 10.1090/surv/188
- Masahito Ohta, Stability of standing waves for the generalized Davey-Stewartson system, J. Dynam. Differential Equations 6 (1994), no. 2, 325–334. MR 1280142, DOI 10.1007/BF02218533
- Masahito Ohta, Stability and instability of standing waves for the generalized Davey-Stewartson system, Differential Integral Equations 8 (1995), no. 7, 1775–1788. MR 1347979
- Gustavo Ponce and Jean-Claude Saut, Well-posedness for the Benney-Roskes/Zakharov-Rubenchik system, Discrete Contin. Dyn. Syst. 13 (2005), no. 3, 811–825. MR 2153145, DOI 10.3934/dcds.2005.13.811
- T. Passot, C. Sulem, and P. Sulem, Generalization of acoustic fronts by focusing wave packets, Physica D 94 (1996), 168–187.
- José R. Quintero, Stability and instability analysis for the standing waves for a generalized Zakharov-Rubenchik system, Proyecciones 41 (2022), no. 3, 663–682. MR 4448495, DOI 10.22199/issn.0717-6279-4547
- J. Quintero, Existence of positive solutions for a nonlinear elliptic class of equation in $\mathbb {R}^2$ and $\mathbb {R}^3$, JAM 20 (2021), 188–210.
- A. Rubenchik and V. Zakharov, Nonlinear interaction of high-frequency and low-frequency waves, Prikl. Mat. Techn. Phys. 5 (1972), 84–98.
- Jalal Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math. Phys. 91 (1983), no. 3, 313–327. MR 723756, DOI 10.1007/BF01208779
- Michael I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/83), no. 4, 567–576. MR 691044, DOI 10.1007/BF01208265
References
- D. Beney and G. Roskes, Wave instability, Studies in Applied Math 48 (1969), 455–472.
- Henri Berestycki and Thierry Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), no. 9, 489–492 (French, with English summary). MR 646873
- T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), no. 4, 549–561. MR 677997
- Rolci Cipolatti, On the existence of standing waves for a Davey-Stewartson system, Comm. Partial Differential Equations 17 (1992), no. 5-6, 967–988. MR 1177301, DOI 10.1080/03605309208820872
- Rolci Cipolatti, On the instability of ground states for a Davey-Stewartson system, Ann. Inst. H. Poincaré Phys. Théor. 58 (1993), no. 1, 85–104 (English, with English and French summaries). MR 1208793
- José R. Quintero and Juan C. Cordero, Instability of the standing waves for a Benney-Roskes/Zakharov-Rubenchik system and blow-up for the Zakharov equations, Discrete Contin. Dyn. Syst. Ser. B 25 (2020), no. 4, 1213–1240. MR 4063989, DOI 10.3934/dcdsb.2019217
- Juan Carlos Cordero Ceballos, Supersonic limit for the Zakharov-Rubenchik system, J. Differential Equations 261 (2016), no. 9, 5260–5288. MR 3542975, DOI 10.1016/j.jde.2016.07.022
- J. C. Cordero, Tesis de doctorado, Instituto de Matematica Pura e Aplicada (IMPA), 2010.
- A. Davey and K. Stewartson, On three-dimensional packets of surface waves, Proc. Roy. Soc. London Ser. A 338 (1974), 101–110. MR 349126, DOI 10.1098/rspa.1974.0076
- Jean-Michel Ghidaglia and Jean-Claude Saut, On the Zakharov-Schulman equations, Nonlinear dispersive wave systems (Orlando, FL, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 83–97. MR 1194632
- Jean-Michel Ghidaglia and Jean-Claude Saut, On the initial value problem for the Davey-Stewartson systems, Nonlinearity 3 (1990), no. 2, 475–506. MR 1054584
- V. E. Zakharov and E. A. Kuznetsov, Hamiltonian formalism for systems of hydrodynamic type, Mathematical physics reviews, Vol. 4, Soviet Sci. Rev. Sect. C: Math. Phys. Rev., vol. 4, Harwood Academic Publ., Chur, 1984, pp. 167–220. Translated from the Russian. MR 768940
- David Lannes, The water waves problem, Mathematical Surveys and Monographs, vol. 188, American Mathematical Society, Providence, RI, 2013. Mathematical analysis and asymptotics. MR 3060183, DOI 10.1090/surv/188
- Masahito Ohta, Stability of standing waves for the generalized Davey-Stewartson system, J. Dynam. Differential Equations 6 (1994), no. 2, 325–334. MR 1280142, DOI 10.1007/BF02218533
- Masahito Ohta, Stability and instability of standing waves for the generalized Davey-Stewartson system, Differential Integral Equations 8 (1995), no. 7, 1775–1788. MR 1347979
- Gustavo Ponce and Jean-Claude Saut, Well-posedness for the Benney-Roskes/Zakharov-Rubenchik system, Discrete Contin. Dyn. Syst. 13 (2005), no. 3, 811–825. MR 2153145, DOI 10.3934/dcds.2005.13.811
- T. Passot, C. Sulem, and P. Sulem, Generalization of acoustic fronts by focusing wave packets, Physica D 94 (1996), 168–187.
- José R. Quintero, Stability and instability analysis for the standing waves for a generalized Zakharov-Rubenchik system, Proyecciones 41 (2022), no. 3, 663–682. MR 4448495
- J. Quintero, Existence of positive solutions for a nonlinear elliptic class of equation in $\mathbb {R}^2$ and $\mathbb {R}^3$, JAM 20 (2021), 188–210.
- A. Rubenchik and V. Zakharov, Nonlinear interaction of high-frequency and low-frequency waves, Prikl. Mat. Techn. Phys. 5 (1972), 84–98.
- Jalal Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math. Phys. 91 (1983), no. 3, 313–327. MR 723756
- Michael I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/83), no. 4, 567–576. MR 691044
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2020):
35Q55,
35A15,
35B35
Retrieve articles in all journals
with MSC (2020):
35Q55,
35A15,
35B35
Additional Information
José Raúl Quintero
Affiliation:
Department of Mathematics, Universidad del Valle, Cali, Colombia
ORCID:
0000-0002-8762-3598
Email:
jose.quintero@correounivalle.edu.co
Keywords:
Standing waves,
variational approach,
stability
Received by editor(s):
October 11, 2022
Received by editor(s) in revised form:
January 2, 2023
Published electronically:
May 2, 2023
Additional Notes:
The author was supported by the Mathematics Department at Universidad del Valle under the project CI 71231 and MinCiencias-Colombia under the project MathAmSud 21-Math-03
Dedicated:
To Robert L. Pego
Article copyright:
© Copyright 2023
Brown University
