SciELO - Scientific Electronic Library Online

vol.41 issue3A new refinement of the generalized Hölder’s inequality with applicationsSpectral operation in locally convex algebras author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand




Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google


Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.41 no.3 Antofagasta June 2022 


Stability and instability analysis for the standing waves for a generalized Zakharov-Rubenchik system

José R. Quintero1 

1Department of Mathematics, Universidad del Valle, Calle 13, 100-00, Cali, Colombia. e-mail:


In this paper, we analyze the stability and instability of standing waves for a generalized Zakharov-Rubenchik system (or the Benney-Roskes system) in spatial dimensions N = 2, 3. We show that the standing waves generated by the set of minimizers for the associated variational problem are stable, for N = 2 and σ(p − 2) > 0. We also show that the standing waves are strongly unstable, for N = 3 and if either σ < 0 and 4/3 <p< 4, or σ > 0 and 0 <p< 2. Results follow by using the variational characterization of standing waves, the concentration compactness principle due to J. Lions and the compactness lemma due to E. Lieb to solve the associated minimization problem.

Keywords: standing waves; virial identity; stability; blow up

Texto completo disponible sólo en PDF

Full text available only in PDF format.


J. Quintero was supported by the Mathematics Department at Universidad del Valle, under the research project CI 71231. J. Quintero was supported by Math AmSud and Minciencias-Colombia under the project MathAmSud 21-Math-03.


[1] D. Benney and G. Roskes, “Wave Instability”, Studies in Applied Mathematics, vol. 48, pp. 455-472, 1969. [ Links ]

[2] H. Brezis and E. Lieb, “A relation between pointwise convergence of functions and convergence of functionals”, Proceedings of the American Mathematical Society, vol. 88, no. 3, pp. 486-490, 1983. doi: 10.2307/2044999 [ Links ]

[3] T. Cazenave, An introduction to nonlinear Schrodinger equations. Universidade Federal do Rio de Janeiro, 1996. [ Links ]

[4] R. Cipolatti, “On the existence of standing waves for a Davey-Stewartson System”, Communications in Partial Differential Equations, vol. 17, no. 5-6, pp. 967-988, 1992. doi: 10.1080/03605309208820872 [ Links ]

[5] R. Cipolatti, “On the instability of ground states for a Davey- Stewartson system”, Annales de l'Institut Henri Poincaré, physique théorique, vol. 58, pp. 84-104, 1993. [ Links ]

[6] J. Cordero, "Sudsonic and supersonic limits for the Zakharov-Rubenchik system", Tese de Doutorado, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 2010. [ Links ]

[7] J. C. Cordero, “Supersonic limit for the Zakharov-Rubenchik System”, Journal of Differential Equations, vol. 261, no. 9, pp. 5260-5288, 2016. doi: 10.1016/j.jde.2016.07.022 [ Links ]

[8] J. C. Cordero and J. R. Quintero, “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, vol. 25, no. 4, pp. 1213-1240, 2020. doi: 10.3934/dcdsb.2019217 [ Links ]

[9] A. Davey and K. Stewartson, “On three-dimensional packets of surface waves”, Proceedings of the Royal Society of London A, vol. 338, no. 1613, pp. 101-110, 1974. doi: 10.1098/rspa.1974.0076 [ Links ]

[10] J. Fröhlich, E. H. Lieb, and M. Loss, “Stability of coulomb systems with magnetic fields”, Communications in Mathematical Physics, vol. 104, no. 2, pp. 251-270, 1986. doi: 10.1007/bf01211593 [ Links ]

[11] J.-M. Ghidaglia and J.-C. Saut, “On the initial value problem for the Davey-Stewartson Systems”, Nonlinearity, vol. 3, no. 2, pp. 475-506, 1990. doi: 10.1088/0951-7715/3/2/010 [ Links ]

[12] J. Ghidaglia and J. Saut, “On the Zakharov-Schulman equations”, In: Nonlinear Dispersive Waves, L. Debnath Ed., World Scientific, 1992. [ Links ]

[13] E. Kuznetsov and V. Zakharov, “Hamiltonian formalism for systems of hydrodynamics type”, Mathematical Physics Review, Soviet Scientific Reviews, vol. 4, pp. 167-220, 1984. [ Links ]

[14] D. Lannes, Water waves: mathematical theory and asymptotics. Providence: AMS, 2013. [ Links ]

[15] E. H. Lieb , “On the lowest eigenvalue of the laplacian for the intersection of two domains”, Inventiones Mathematicae, vol. 74, no. 3, pp. 441-448, 1983. doi: 10.1007/bf01394245 [ Links ]

[16] P. L. Lions, “The concentration-compactness principle in the calculus of variations. the locally compact case, part 1”, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, vol. 1, no. 2, pp. 109-145, 1984. doi: 10.1016/s0294-1449(16)30428-0 [ Links ]

[17] P. L. Lions, “The concentration-compactness principle in the calculus of variations. the locally compact case, part 2”, Annales de l'Institut Henri Poincaré C, Analyse non linéaire , vol. 1, no. 4, pp. 223-283, 1984. doi: 10.1016/s0294-1449(16)30422-x [ Links ]

[18] H. Nawa, “Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity”, Journal of the Mathematical Society of Japan, vol. 46, no. 4, pp. 557-586, 1994. doi: 10.2969/jmsj/04640557 [ Links ]

[19] M. Ohta, “Stability of standing waves for the generalized Davey-Stewartson System”, Journal of Dynamics and Differential Equations, vol. 6, no. 2, pp. 325-334, 1994. doi: 10.1007/bf02218533 [ Links ]

[20] M. Ohta, “Stability and instability of standing vaves for the generalized Davey-Stewartson System”, Differential and Integral Equations, vol. 8, no. 7, pp. 1775-1788, 1995. [ Links ]

[21] G. Ponce and J. C. Saut, “Well-posedness for the Benney-Roskes/Zakharov-Rubenchik system”, Discrete and Continuous Dynamical Systems, vol. 13, no. 3, pp. 811-825, 2005. doi: 10.3934/dcds.2005.13.811 [ Links ]

[22] J. Quintero, “On the existence of positive solutions for a nonlinear elliptic class of equations in R² and R³”, Journal of advances in mathematics, vol. 20, pp. 188-210, 2021. doi: 10.24297/jam.v20i.9044 [ Links ]

[23] J. Quintero, “Stability of Standing Waves for a generalized Zakharov-Rubenchik System”. Preprint, 2020. [Online]. Available: ]

[24] A. Rubenchik and V. Zakharov , “Nonlinear Interaction of High-Frequency and Low-Frequency Waves”, Journal of Applied Mechanics and Technical Physics, vol. 13, pp. 669-681, 1972. [ Links ]

[25] M. I. Weinstein, “Nonlinear schrödinger equations and sharp interpolation estimates,” Communications in Mathematical Physics , vol. 87, no. 4, pp. 567-576, 1983. doi: 10.1007/bf01208265 [ Links ]

Received: October 30, 2020; Accepted: October 30, 2021

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License