×

Weakly compact uniform attractor for the nonautonomous long-short wave equations. (English) Zbl 1291.35339

Summary: Solutions and weakly compact uniform attractor for the nonautonomous long-short wave equations with translation compact forces were studied in a bounded domain. We first established the existence and the uniqueness of the solution to the system by using Galerkin method and then obtained the uniform absorbing set and the weakly compact uniform attractor of the problem by applying techniques of constructing skew product flow in the extended phase space.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B41 Attractors
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Grimshaw, R. H. J., The modulation of an internal gravity-wave packet, and the resonance with the mean motion, 56, 3, 241-266 (1976/77) · Zbl 0361.76029
[2] Nicholson, D. R.; Goldman, M. V., Damped nonlinear Schrödinger equation, The Physics of Fluids, 19, 10, 1621-1625 (1976) · doi:10.1063/1.861368
[3] Benney, D. J., A general theory for interactions between short and long waves, 56, 1, 81-94 (1976/77) · Zbl 0358.76011
[4] Guo, B. L., The global solution for one class of the system of LS nonlinear wave interaction, Journal of Mathematical Research and Exposition, 7, 1, 69-76 (1987)
[5] Guo, B., The periodic initial value problems and initial value problems for one class of generalized long-short type equations, Journal of Engineering Mathematics, 8, 47-53 (1991)
[6] Du, X.; Guo, B., The global attractor for LS type equation in \(\mathbb{R}^1\), Acta Mathematicae Applicatae Sinica, 28, 723-734 (2005)
[7] Zhang, R., Existence of global attractor for LS type equations, Journal of Mathematical Research and Exposition, 26, 4, 708-714 (2006) · Zbl 1156.35335
[8] Li, Y., Long time behavior for the weakly damped driven long-wave-short-wave resonance equations, Journal of Differential Equations, 223, 2, 261-289 (2006) · Zbl 1094.35101 · doi:10.1016/j.jde.2005.07.006
[9] Guo, B.; Chen, L., Orbital stability of solitary waves of the long wave-short wave resonance equations, Mathematical Methods in the Applied Sciences, 21, 10, 883-894 (1998) · Zbl 0919.35126 · doi:10.1002/(SICI)1099-1476(19980710)21:10<883::AID-MMA974>3.0.CO;2-B
[10] Guo, B.; Wang, B., Attractors for the long-short wave equations, Journal of Partial Differential Equations, 11, 4, 361-383 (1998) · Zbl 1126.35337
[11] Guo, B.; Wang, B., The global solution and its long time behavior for a class of generalized LS type equations, Progress in Natural Science, 6, 5, 533-546 (1996)
[12] Bekiranov, D.; Ogawa, T.; Ponce, G., On the well-posedness of Benney’s interaction equation of short and long waves, Advances in Differential Equations, 1, 6, 919-937 (1996) · Zbl 0861.35104
[13] Bekiranov, D.; Ogawa, T.; Ponce, G., Interaction equations for short and long dispersive waves, Journal of Functional Analysis, 158, 2, 357-388 (1998) · Zbl 0909.35123 · doi:10.1006/jfan.1998.3257
[14] Huo, Z.; Guo, B., Local well-posedness of interaction equations for short and long dispersive waves, Journal of Partial Differential Equations, 17, 2, 137-151 (2004) · Zbl 1057.35062
[15] Tsutsumi, M.; Hatano, S., Well-posedness of the Cauchy problem for Benney’s first equations of long wave short wave interactions, Funkcialaj Ekvacioj, 37, 2, 289-316 (1994) · Zbl 0841.35107
[16] Tsutsumi, M.; Hatano, S., Well-posedness of the Cauchy problem for the long wave-short wave resonance equations, Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods, 22, 2, 155-171 (1994) · Zbl 0818.35116 · doi:10.1016/0362-546X(94)90032-9
[17] Zhang, R.; Guo, B., Global solution and its long time behavior for the generalized long-short wave equations, Journal of Partial Differential Equations, 18, 3, 206-218 (2005) · Zbl 1085.35040
[18] Chepyzhov, V. V.; Vishik, M. I., Non-autonomous evolutionary equations with translation-compact symbols and their attractors, Comptes Rendus de l’Académie des Sciences, 321, 2, 153-158 (1995) · Zbl 0837.35059
[19] Chepyzhov, V. V.; Vishik, M. I., Attractors for Equations of Mathematical Physics. Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49 (2002), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 0986.35001
[20] Chepyzhov, V. V.; Vishik, M. I., Attractors of nonautonomous dynamical systems and their dimension, Journal de Mathématiques Pures et Appliquées, 73, 3, 279-333 (1994) · Zbl 0838.58021
[21] Bergh, J.; Laöfstraöm, J., Interpolation Spaces (1976), Berlin, Germany: Springer-Verlag, Berlin, Germany · Zbl 0344.46071
[22] Xin, J.; Guo, B.; Han, Y.; Huang, D., The global solution of the \((2 + 1)\)-dimensional long wave-short wave resonance interaction equation, Journal of Mathematical Physics, 49, 7, 073504-07350413 (2008) · Zbl 1152.81635 · doi:10.1063/1.2940320
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.