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January  2015, 35(1): 467-482. doi: 10.3934/dcds.2015.35.467

## On the quasi-periodic solutions of generalized Kaup systems

 1 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa

Received  November 2013 Revised  May 2014 Published  August 2014

In this paper we analyze the behavior of small amplitude solutions of the variant of the classical Kap system given by $\partial_t u = \partial_x v - 2 \partial_x(v^3), \quad \partial_t v = \partial_x u - \frac 1 3 \partial_{xxx} u.$ It is proved that the above equation admits small-amplitude solutions that are quasiperiodic in time and that correspond to finite dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof relies on the Hamiltonian formulation of the problem, the study of its Birkhoff normal form and an infinite dimensional KAM theorem. This is the abstract of your paper and it should not exceed.
Citation: Claudia Valls. On the quasi-periodic solutions of generalized Kaup systems. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 467-482. doi: 10.3934/dcds.2015.35.467
##### 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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##### 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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