August  2017, 22(6): 2501-2519. doi: 10.3934/dcdsb.2017104

Quasi-periodic solutions of generalized Boussinesq equation with quasi-periodic forcing

1. 

College of Mathematics and Physics, Yancheng Institute of Technology, Yancheng 224051, China

2. 

Department of Mathematics, Southeast University, Nanjing 211189, China

E-mail address: shiyanling96998@163.com

Received  August 2014 Revised  November 2015 Published  March 2017

Fund Project: This work is supported by the Tian Yuan special Funds of the National Natural Science Foundation of China (Grant No. 11526178), NSFJS Grant (BK 20131285) and NSFC Grant(11371090,11301072).

In this paper, one-dimensional quasi-periodically forced generalized Boussinesq equation
$u_{tt}-u_{xx} + u_{xxxx} +\varepsilon \phi(t) ( u+u^3 )_{xx}=0$
with hinged boundary conditions is considered, where
$\varepsilon$
is a small positive parameter,
$\phi(t)$
is a real analytic quasi-periodic function in
$t$
with frequency vector
$\omega=( \omega_1,\omega_2,\cdots,\omega_m ).$
It is proved that, under a suitable hypothesis on
$\phi(t),$
there are many quasi-periodic solutions for the above equation via KAM theory.
Citation: Yanling Shi, Junxiang Xu, Xindong Xu. Quasi-periodic solutions of generalized Boussinesq equation with quasi-periodic forcing. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2501-2519. doi: 10.3934/dcdsb.2017104
References:
[1]

P. Baldi and M. Berti, Forced vibrations of a nonhomogeneous string, SIAM J. Math. Anal., 40 (2008), 382-412.

[2]

P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann., 359 (2014), 471-536.

[3]

M. Berti and M. Procesi, Quasi-periodic solutions of completely resonant forced wave equations, Comm. Partial Differential Equation, 31 (2006), 959-985.

[4]

J. Bona and R. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys., 118 (1988), 15-29.

[5]

M. Boussinesq, Théorie générale des mouvements qui sout propagés dans un canal rectangularire horizontal, C. R. Acad. Sci. Paris, 73 (1871), 256-260.

[6]

M. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pure Appl. Sect., 17 (1872), 55-108.

[7]

M. Boussinesq, Essai sur la théorie des eaux courantes, Mémoires présentés par divers savants á l'Académie des Sciences Inst. France, 2 (1877), 1-680.

[8]

P. Defit, C. Tomei and E. Trubowitz, Inverse scattering and the Boussinesq equation, Comm. Pure Appl. Math., 35 (1982), 567-628.

[9]

R. Feola and M. Procesi, Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, J. Differential Equations, 259 (2015), 3389-3447, arXiv: 1412.5786.

[10]

L. Jiao and Y. Wang, The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation, Commun. Pure Appl. Anal., 8 (2009), 1585-1606.

[11]

R. Johnson, A Morden Introduction of Mathematical Theory of Water Waves, Cambriadge Universty Press. , 2004.

[12]

S. Kuksin, Nearly Integrable Infinite-dimensional Hamiltonian Systems, Lecture Notes in Mathematics, 1556, Springer-Verlag, Berlin, 1993.

[13]

J. Liu and J. Si, Invariant tori for a derivative nonlinear Schrödinger equation with quasi periodic forcing, J. Math. Phys., 56 (2015), 032702, 25pp.

[14]

Y. Liu and R. Xu, Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation, Phys. D., 237 (2008), 721-731.

[15]

Y. Liu, Instability of solitary waves for generalized Boussinesq equations, J. Dynam. Differential Equations., 5 (1993), 537-558.

[16]

Y. Liu, Instability and blow-up of solutions to a generalized Boussinesq equation, SIAM J. Math. Anal., 26 (1995), 1527-1546.

[17]

Y. Liu, Strong instability of solitary-wave solutions of a generalized Boussinesq equation, J. Differential Equations., 164 (2000), 223-239.

[18]

Y. Liu, M. Ohta and G. Todorova, Instabilité forte d'ondes solitaires pour des équations de Klein-Gordon non linéaires et des équations généralisées de Boussinesq, Ann. Inst. H. Poincaré Anal. Non Linéaire., 24 (2007), 539-548.

[19]

L. Mi and K. Zhang, Invariant tori for Benjamin-Ono equation with unbounded quasiperiodically forced perturbation, Discrete Contin. Dyn. Syst., 34 (2014), 689-707.

[20]

J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 119-148.

[21]

P. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math., 20 (1967), 145-205.

[22]

P. Rabinowitz, Time periodic solutions of nonlinear wave equations, Manuscripta Math., 5 (1971), 165-194.

[23]

J. Rui and J. Si, Quasi-periodic solutions for quasi-periodically forced nonlinear Schrödinger equations with quasi-periodic inhomogeneous terms, Phys. D, 286 (2014), 1-31.

[24]

Y. Shi, J. Xu and X. Xu, On quasi-periodic solutions for a generalized Boussinesq equation, Nonlinear Anal., 105 (2014), 50-61.

[25]

Y. Shi, J. Xu and X. Xu, On the quasi-periodic solutions for generalized boussinesq equation with higher order nonlinearity, Applicable Analysis, 94 (2015), 1977-1996.

[26]

J. Si, Quasi-periodic solutions of a non-autonomous wave equations with quasi-periodic forcing, J. Differential Equations, 252 (2012), 5274-5360.

[27]

Y. Wang, Quasi-periodic solutions of a quasi-periodically forced nonlinear beam equation, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2682-2700.

[28]

E. Yusufoğlu, Blow-up solutions of the generalized Boussinesq equation obtained by variational iteration method, Nonlinear Dynam., 52 (2008), 395-402.

[29]

V. Zakharov, On the stochastization of one dimensional chains of nonlinear oscillators, Sov. Phys. JETP, 38 (1974), 108-110.

[30]

M. Zhang and J. Si, Quasi-periodic solutions of nonlinear wave equations with quasi-periodic forcing, Phys. D, 238 (2009), 2185-2215.

show all references

References:
[1]

P. Baldi and M. Berti, Forced vibrations of a nonhomogeneous string, SIAM J. Math. Anal., 40 (2008), 382-412.

[2]

P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann., 359 (2014), 471-536.

[3]

M. Berti and M. Procesi, Quasi-periodic solutions of completely resonant forced wave equations, Comm. Partial Differential Equation, 31 (2006), 959-985.

[4]

J. Bona and R. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys., 118 (1988), 15-29.

[5]

M. Boussinesq, Théorie générale des mouvements qui sout propagés dans un canal rectangularire horizontal, C. R. Acad. Sci. Paris, 73 (1871), 256-260.

[6]

M. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pure Appl. Sect., 17 (1872), 55-108.

[7]

M. Boussinesq, Essai sur la théorie des eaux courantes, Mémoires présentés par divers savants á l'Académie des Sciences Inst. France, 2 (1877), 1-680.

[8]

P. Defit, C. Tomei and E. Trubowitz, Inverse scattering and the Boussinesq equation, Comm. Pure Appl. Math., 35 (1982), 567-628.

[9]

R. Feola and M. Procesi, Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, J. Differential Equations, 259 (2015), 3389-3447, arXiv: 1412.5786.

[10]

L. Jiao and Y. Wang, The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation, Commun. Pure Appl. Anal., 8 (2009), 1585-1606.

[11]

R. Johnson, A Morden Introduction of Mathematical Theory of Water Waves, Cambriadge Universty Press. , 2004.

[12]

S. Kuksin, Nearly Integrable Infinite-dimensional Hamiltonian Systems, Lecture Notes in Mathematics, 1556, Springer-Verlag, Berlin, 1993.

[13]

J. Liu and J. Si, Invariant tori for a derivative nonlinear Schrödinger equation with quasi periodic forcing, J. Math. Phys., 56 (2015), 032702, 25pp.

[14]

Y. Liu and R. Xu, Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation, Phys. D., 237 (2008), 721-731.

[15]

Y. Liu, Instability of solitary waves for generalized Boussinesq equations, J. Dynam. Differential Equations., 5 (1993), 537-558.

[16]

Y. Liu, Instability and blow-up of solutions to a generalized Boussinesq equation, SIAM J. Math. Anal., 26 (1995), 1527-1546.

[17]

Y. Liu, Strong instability of solitary-wave solutions of a generalized Boussinesq equation, J. Differential Equations., 164 (2000), 223-239.

[18]

Y. Liu, M. Ohta and G. Todorova, Instabilité forte d'ondes solitaires pour des équations de Klein-Gordon non linéaires et des équations généralisées de Boussinesq, Ann. Inst. H. Poincaré Anal. Non Linéaire., 24 (2007), 539-548.

[19]

L. Mi and K. Zhang, Invariant tori for Benjamin-Ono equation with unbounded quasiperiodically forced perturbation, Discrete Contin. Dyn. Syst., 34 (2014), 689-707.

[20]

J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 119-148.

[21]

P. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math., 20 (1967), 145-205.

[22]

P. Rabinowitz, Time periodic solutions of nonlinear wave equations, Manuscripta Math., 5 (1971), 165-194.

[23]

J. Rui and J. Si, Quasi-periodic solutions for quasi-periodically forced nonlinear Schrödinger equations with quasi-periodic inhomogeneous terms, Phys. D, 286 (2014), 1-31.

[24]

Y. Shi, J. Xu and X. Xu, On quasi-periodic solutions for a generalized Boussinesq equation, Nonlinear Anal., 105 (2014), 50-61.

[25]

Y. Shi, J. Xu and X. Xu, On the quasi-periodic solutions for generalized boussinesq equation with higher order nonlinearity, Applicable Analysis, 94 (2015), 1977-1996.

[26]

J. Si, Quasi-periodic solutions of a non-autonomous wave equations with quasi-periodic forcing, J. Differential Equations, 252 (2012), 5274-5360.

[27]

Y. Wang, Quasi-periodic solutions of a quasi-periodically forced nonlinear beam equation, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2682-2700.

[28]

E. Yusufoğlu, Blow-up solutions of the generalized Boussinesq equation obtained by variational iteration method, Nonlinear Dynam., 52 (2008), 395-402.

[29]

V. Zakharov, On the stochastization of one dimensional chains of nonlinear oscillators, Sov. Phys. JETP, 38 (1974), 108-110.

[30]

M. Zhang and J. Si, Quasi-periodic solutions of nonlinear wave equations with quasi-periodic forcing, Phys. D, 238 (2009), 2185-2215.

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