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Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains
On the quasi-periodic solutions of generalized Kaup systems
1. | Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa |
References:
[1] |
D. Bambusi, Lyapunov center theorem for some nonlinear PDE's: A simple proof, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 823-837. |
[2] |
D. Bambusi and S. Paleari, Normal form and exponential stability for some nonlinear string equations, Z. Angew. Math. Phys., 52 (2001), 1033-1052.
doi: 10.1007/PL00001582. |
[3] |
M. Berti and P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities, Comm. Math. Phys., 243 (2003), 315-328.
doi: 10.1007/s00220-003-0972-8. |
[4] |
M. Berti and P. Bolle, Multiplicity of periodic solutions of nonlinear wave equations, Nonlinear Anal., 56 (2004), 1011-1046.
doi: 10.1016/j.na.2003.11.001. |
[5] |
J. Bona, M. Chen and J. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318.
doi: 10.1007/s00332-002-0466-4. |
[6] |
J. Bourgain, Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal., 5 (1995), 629-639.
doi: 10.1007/BF01902055. |
[7] |
C. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys., 211 (2000), 497-525.
doi: 10.1007/s002200050824. |
[8] |
W. Craig and C. Wayne, Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46 (1993), 1409-1498.
doi: 10.1002/cpa.3160461102. |
[9] |
G. El, R. Grimshaw and M. Pavlov, Integrable shallow-water equations and undular bores, Stud. Appl. Math., 106 (2001), 157-186.
doi: 10.1111/1467-9590.00163. |
[10] |
G. Gentile and V. Mastropietro, Construction of periodic solutions of nonlinear wave equations with Dirichlet boundary conditions by the Lindstedt series method, J. Math. Pures Appl., 83 (2004), 1019-1065.
doi: 10.1016/j.matpur.2004.01.007. |
[11] |
A. Kamchatnov, R. Kraenkel and B. Umarov, Asymptotic soliton train solutions of Kaup-Boussinesq equations, Wave Motion, 38 (2003), 355-365.
doi: 10.1016/S0165-2125(03)00062-3. |
[12] |
D. Kaup, A higher-order water-wave equation and the method for solving it, Prog. Theor. Phys., 54 (1975), 396-408.
doi: 10.1143/PTP.54.396. |
[13] |
S. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funct. Anal. Appl., 21 (1987), 22-37. |
[14] |
S. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math., 143 (1996), 149-179.
doi: 10.2307/2118656. |
[15] |
V. Matveev and M. Yavor, Solutions presque périodiques et à $N$-solitons de l'équation hydrodynamique non linéaire de Kaup, Ann. Inst. H. Poincaré Sect. A (N.S.), 31 (1979), 25-41. |
[16] |
J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 119-148. |
[17] |
C. Valls, The Boussinesq system: Dynamics on the center manifold, Comm. Pure Appl. Anal., 4 (2005), 839-860.
doi: 10.3934/cpaa.2005.4.839. |
[18] |
G. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, 1974. |
show all references
References:
[1] |
D. Bambusi, Lyapunov center theorem for some nonlinear PDE's: A simple proof, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 823-837. |
[2] |
D. Bambusi and S. Paleari, Normal form and exponential stability for some nonlinear string equations, Z. Angew. Math. Phys., 52 (2001), 1033-1052.
doi: 10.1007/PL00001582. |
[3] |
M. Berti and P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities, Comm. Math. Phys., 243 (2003), 315-328.
doi: 10.1007/s00220-003-0972-8. |
[4] |
M. Berti and P. Bolle, Multiplicity of periodic solutions of nonlinear wave equations, Nonlinear Anal., 56 (2004), 1011-1046.
doi: 10.1016/j.na.2003.11.001. |
[5] |
J. Bona, M. Chen and J. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318.
doi: 10.1007/s00332-002-0466-4. |
[6] |
J. Bourgain, Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal., 5 (1995), 629-639.
doi: 10.1007/BF01902055. |
[7] |
C. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys., 211 (2000), 497-525.
doi: 10.1007/s002200050824. |
[8] |
W. Craig and C. Wayne, Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46 (1993), 1409-1498.
doi: 10.1002/cpa.3160461102. |
[9] |
G. El, R. Grimshaw and M. Pavlov, Integrable shallow-water equations and undular bores, Stud. Appl. Math., 106 (2001), 157-186.
doi: 10.1111/1467-9590.00163. |
[10] |
G. Gentile and V. Mastropietro, Construction of periodic solutions of nonlinear wave equations with Dirichlet boundary conditions by the Lindstedt series method, J. Math. Pures Appl., 83 (2004), 1019-1065.
doi: 10.1016/j.matpur.2004.01.007. |
[11] |
A. Kamchatnov, R. Kraenkel and B. Umarov, Asymptotic soliton train solutions of Kaup-Boussinesq equations, Wave Motion, 38 (2003), 355-365.
doi: 10.1016/S0165-2125(03)00062-3. |
[12] |
D. Kaup, A higher-order water-wave equation and the method for solving it, Prog. Theor. Phys., 54 (1975), 396-408.
doi: 10.1143/PTP.54.396. |
[13] |
S. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funct. Anal. Appl., 21 (1987), 22-37. |
[14] |
S. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math., 143 (1996), 149-179.
doi: 10.2307/2118656. |
[15] |
V. Matveev and M. Yavor, Solutions presque périodiques et à $N$-solitons de l'équation hydrodynamique non linéaire de Kaup, Ann. Inst. H. Poincaré Sect. A (N.S.), 31 (1979), 25-41. |
[16] |
J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 119-148. |
[17] |
C. Valls, The Boussinesq system: Dynamics on the center manifold, Comm. Pure Appl. Anal., 4 (2005), 839-860.
doi: 10.3934/cpaa.2005.4.839. |
[18] |
G. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, 1974. |
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