×

Center manifold for the third-order nonlinear Schrödinger equation with critical lengths. (English) Zbl 1492.35253

Summary: This paper is concerned with the third-order nonlinear Schrödinger equation, a version of the center manifold theorem is established which is suitable for the third-order nonlinear Schrödinger equation with critical lengths.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A01 Existence problems for PDEs: global existence, local existence, non-existence
Full Text: DOI

References:

[1] Ball, J. M.; Dickey, R. W., Saddle point analysis for an ordinary differential equation in a Banach space, and an application to dynamic buckling of a beam, Nonlinear Elasticity, 93-160 (1973), New York: Academic Press, New York · Zbl 0288.34057
[2] Bates, P. W.; Jones, C. K., Invariant manifolds for semilinear partial differential equations, Dynamics Reported, vol. 2, 1-38 (1989), Chichester: Wiley, Chichester · Zbl 0674.58024 · doi:10.1007/978-3-322-96657-5_1
[3] Carr, J., Applications of Centre Manifold Theory (1981), New York: Springer, New York · Zbl 0464.58001 · doi:10.1007/978-1-4612-5929-9
[4] Chen, X. Y.; Hale, J. K.; Tan, B., Invariant foliations for \(C^1\) semigroups in Banach spaces, J. Differ. Equ., 139, 283-318 (1997) · Zbl 0994.34047 · doi:10.1006/jdeq.1997.3255
[5] Choi, Y.; Han, J., Dynamical bifurcation of the damped Kuramoto-Sivashinsky equation, J. Math. Anal. Appl., 421, 383-398 (2015) · Zbl 1333.37068 · doi:10.1016/j.jmaa.2014.07.009
[6] Choi, Y.; Han, J.; Hsia, C. H., Bifurcation analysis of the damped Kuramoto-Sivashinsky equation with respect to the period, Discrete Contin. Dyn. Syst., Ser. B, 20, 1993 (2015) · Zbl 1342.37056
[7] Chu, J.; Coron, J. M.; Shang, P., Asymptotic stability of a nonlinear Korteweg-de Vries equation with critical lengths, J. Differ. Equ., 259, 4045-4085 (2015) · Zbl 1320.35307 · doi:10.1016/j.jde.2015.05.010
[8] Da Prato, G.; Lunardi, A., Stability, instability and center manifold theorem for fully nonlinear autonomous parabolic equations, Arch. Ration. Mech. Anal., 101, 115-141 (1988) · Zbl 0661.35044 · doi:10.1007/BF00251457
[9] Haragus, M.; Iooss, G., Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems (2011), London: Springer, London · Zbl 1230.34002 · doi:10.1007/978-0-85729-112-7
[10] Hasegawa, A.; Kodama, Y., Nonlinear pulse propagation in a monomode dielectric guide, IEEE J. Quantum Electron., 23, 510-524 (1987) · doi:10.1109/JQE.1987.1073392
[11] Kelley, A., The stable, center-stable, center, center-unstable, unstable manifolds, J. Differ. Equ., 3, 546-570 (1967) · Zbl 0173.11001 · doi:10.1016/0022-0396(67)90016-2
[12] Khalil, H. K., Nonlinear Systems (1992), New York: Macmillan Publishing Company, New York · Zbl 0969.34001
[13] Kodama, Y., Optical solitons in a monomode fiber, J. Stat. Phys., 39, 597-614 (1985) · doi:10.1007/BF01008354
[14] Latushkin, Y., Prüss, J., Schnaubelt, R.: Center manifolds and dynamics near equilibria of quasilinear parabolic systems with fully nonlinear boundary conditions. Univ. Karlsruhe, Fak. für Mathematik (2007) · Zbl 1168.35021
[15] Lichtner, M.; Radziunas, M.; Recke, L., Well-posedness, smooth dependence and centre manifold reduction for a semilinear hyperbolic system from laser dynamics, Math. Methods Appl. Sci., 30, 931-960 (2007) · Zbl 1161.35033 · doi:10.1002/mma.816
[16] Lions, J. L.; Magenes, E., Non-homogeneous Boundary Value Problems and Applications (2012), Berlin: Springer, Berlin · Zbl 0227.35001
[17] Magal, P.; Ruan, S., Center Manifolds for Semilinear Equations with Non-dense Domain and Applications to Hopf Bifurcation in Age Structured Models (2009), Providence: Am. Math. Soc., Providence · Zbl 1191.35045
[18] Mielke, A., Locally invariant manifolds for quasilinear parabolic equations, Rocky Mt. J. Math., 21, 707-714 (1991) · Zbl 0747.35016 · doi:10.1216/rmjm/1181072962
[19] Ong, K. W., Dynamic transitions of generalized Kuramoto-Sivashinsky equation, Discrete Contin. Dyn. Syst., Ser. B, 21, 1225 (2016) · Zbl 1346.35164 · doi:10.3934/dcdsb.2016.21.1225
[20] Perko, L., Differential Equations and Dynamical Systems (2001), New York: Springer, New York · Zbl 0973.34001 · doi:10.1007/978-1-4613-0003-8
[21] Pillet, C. A.; Wayne, C. E., Invariant manifolds for a class of dispersive. Hamiltonian, partial differential equations, J. Differ. Equ., 141, 310-326 (1997) · Zbl 0890.35016 · doi:10.1006/jdeq.1997.3345
[22] Pliss, V. A., A reduction principle in the theory of stability of motion, Izv. Akad. Nauk SSSR, Ser. Mat., 28, 1297-1324 (1964) · Zbl 0131.31505
[23] Rosier, L.; Zhang, B-Y., Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM J. Control Optim., 45, 927-956 (2006) · Zbl 1116.35108 · doi:10.1137/050631409
[24] Scarpellini, B., Center manifolds of infinite dimensions, Z. Angew. Math. Phys., 42, 280-314 (1991) · Zbl 0731.58068 · doi:10.1007/BF00945799
[25] Shatah, J.; Zeng, C., Orbits homoclinic to centre manifolds of conservative PDEs, Nonlinearity, 16, 591 (2003) · Zbl 1026.35092 · doi:10.1088/0951-7715/16/2/314
[26] Tang, S.; Chu, J.; Shang, P., Asymptotic stability of a Korteweg-de Vries equation with a two-dimensional center manifold, Adv. Nonlinear Anal., 7, 497-515 (2018) · Zbl 1420.35327 · doi:10.1515/anona-2016-0097
[27] Vanderbauwhede, A.; Iooss, G., Center manifold theory in infinite dimensions, Dynamics Reported, 125-163 (1992), Berlin: Springer, Berlin · Zbl 0751.58025 · doi:10.1007/978-3-642-61243-5_4
[28] Van Minh, N.; Wu, J., Invariant manifolds of partial functional differential equations, J. Differ. Equ., 198, 381-421 (2004) · Zbl 1061.34056 · doi:10.1016/j.jde.2003.10.006
[29] Villagrán, O. P.V., Exact boundary controllability for higher order nonlinear Schrödinger equations with constant coefficients, Electron. J. Differ. Equ., 2005, 122, 1-31 (2005) · Zbl 1092.35100
[30] Xiao, M.; Basar, T., Center manifold of the viscous Moore-Greitzer PDE model, SIAM J. Appl. Math., 61, 855-869 (2000) · Zbl 0979.37039 · doi:10.1137/S0036139999354261
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.