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December  2018, 38(12): 6241-6285. doi: 10.3934/dcds.2018268

Quasi-periodic solution of quasi-linear fifth-order KdV equation

Yingte Sun and  Xiaoping Yuan

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received  September 2017 Revised  February 2018 Published  September 2018

Fund Project: Supported by NNSFC11421061.

We prove the existence of quasi-periodic small-amplitude solutions for quasi-linear Hamiltonian perturbation of the fifth order KdV equation on the torus in presence of a quasi-periodic forcing.

Keywords: KAM theory, quasi-linear PDEs, fifth-order KdV equation, quasi-periodic solutions.
Mathematics Subject Classification: Primary: 37K55; Secondary: 37F50, 35Q53.
Citation: Yingte Sun, Xiaoping Yuan. Quasi-periodic solution of quasi-linear fifth-order KdV equation. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 6241-6285. doi: 10.3934/dcds.2018268
References:
[1]

P. Baldi, Periodic solutions of fully nonlinear autonomous equations of Benjami-Ono type, Annales De Linstitut Henri Poincare Non Linear Analysis, 30 (2013), 33-77.  doi: 10.1016/j.anihpc.2012.06.001.

[2]

P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Mathematische Annalen, 359 (2014), 471-536.  doi: 10.1007/s00208-013-1001-7.

[3]

P. Baldi, M. Berti and R. Montalto, KAM for autonomous quasi-linear perturbations of KdV, Annales De Linstitut Henri Poincare Non Linear Analysis, 33 (2016), 1589-1638.  doi: 10.1016/j.anihpc.2015.07.003.

[4]

P. Baldi, M. Berti and R. Montalto, KAM for autonomous quasi-linear perturbations of mKdV, Bollettino dell'Unione Matematica Italiana, 9 (2016), 143-188.  doi: 10.1007/s40574-016-0065-1.

[5]

D. Bambusi and S. Graffi, Time quasi-periodic unbounded perturbations of schrödinger operators and KAM methods, Communications in Mathematical Physics, 219 (2001), 465-480.  doi: 10.1007/s002200100426.

[6]

M. Berti and R. Montalto, Quasi-periodic standing wave solutions of gravity-capillary water waves, preprint, arXiv: 1602.02411.

[7]

N. N. Bogoljubov, J. A. Mitropoliskii and A. M. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer-verlag, New York, 1976.

[8]

J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Annals of Mahematics Studies 158, Princeton University Press, 2005. doi: 10.1515/9781400837144.

[9]

H. Cong, L. Mi and X. Yuan, Positive quasi-periodic solutions to Lotka-Volterra system, Science China Mathematics, 53 (2010), 1151-1160.  doi: 10.1007/s11425-009-0217-1.

[10]

R. Feola and M. Procesi, Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, Journal of Differential Equations, 259 (2015), 3389-3447.  doi: 10.1016/j.jde.2015.04.025.

[11]

R. Feola, KAM for quasi-linear forced hamiltonian NLS, preprint, arXiv: 1602.01341.

[12]

T. Kappeler and J. Pöschel, KdV and KAM, Springer-verlag, New York, 2003. doi: 10.1007/978-3-662-08054-2.

[13]

S. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Springer-verlag, New York, 1993. doi: 10.1007/BFb0092243.

[14]

S. Kuksin, On small-denominators equations with large variable coefficients, Zeitschrift für angewandte Mathematik und Physik, 48 (1997), 262-271.  doi: 10.1007/PL00001476.

[15]

S. Kuksin, A KAM theorem for equations of the Korteweg-De Vries Type, Reviews in Mathematical Physics, 10 (1998), 1-64. 

[16]

S. Kuksin, Analysis of Hamiltonian PDEs, Oxford University Press, 2000.

[17]

S. Kuksin and J. Pöschel, Invariant Cantor Manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Annals of Mathematics, 143 (1996), 149-179.  doi: 10.2307/2118656.

[18]

P. D. Lax, Periodic solutions of the KdV equation, Communications on Pure & Applied Mathematics, 28 (1975), 141-188.  doi: 10.1002/cpa.3160280105.

[19]

J. Liu and X. Yuan, Spectrum for quantum duffing oscillator and small-divisor equation with large-variable coefficient, Communications on Pure & Applied Mathematics, 63 (2010), 1145-1172.  doi: 10.1002/cpa.20314.

[20]

J. Liu and X. Yuan, A KAM theorem for hamiltonian partial differential equations with unbounded perturbations, Communications in Mathematical Physics, 307 (2011), 629-673.  doi: 10.1007/s00220-011-1353-3.

[21]

R. Mcleod, Mean value theorems for vector valued functions, Proceedings of the Edinburgh Mathematical Society, 14 (1965), 197-209.  doi: 10.1017/S0013091500008786.

[22]

R. Montalto, Quasi-periodic solutions of forced Kirchhoff equation, Nonlinear Differential Equationsc and Applications, 24 (2017), Art. 9, 71 pp. doi: 10.1007/s00030-017-0432-3.

[23]

J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Annali Della Scuola Normale Superiore di Pisa-Classe di Scienze, 23 (1996), 119-148. 

[24]

J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Commentarii Mathematici Helvetici, 71 (1996), 269-296.  doi: 10.1007/BF02566420.

[25]

E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Communications in Mathematical Physics, 127 (1990), 479-528.  doi: 10.1007/BF02104499.

[26]

A. M. Wazwaz, Solitons and periodic solutions for the fifth-order KdV equation, Applied Mathematics Letters, 19 (2006), 1162-1167.  doi: 10.1016/j.aml.2005.07.014.

[27]

A. M. Wazwaz, The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, Applied Mathematics & Computation, 184 (2007), 1002-1014.  doi: 10.1016/j.amc.2006.07.002.

[28]

X. Yuan and K. Zhang, A reduction theorem for time dependent Schrödinger operator with finite differentiable unbounded perturbation, Journal of Mathematical Physics, 54 (2013), 052701, 23 pp. doi: 10.1063/1.4803852.

[29]

E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems, Ⅰ, Communications on Pure & Applied Mathematics, 28 (1975), 91-140.  doi: 10.1002/cpa.3160280104.

[30]

E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems, Ⅱ, Communications on Pure & Applied Mathematics, 29 (1976), 49-111.  doi: 10.1002/cpa.3160290104.

[31]

J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differential equations, Nonlinearity, 24 (2011), 1189-1228.  doi: 10.1088/0951-7715/24/4/010.

show all references

References:
[1]

P. Baldi, Periodic solutions of fully nonlinear autonomous equations of Benjami-Ono type, Annales De Linstitut Henri Poincare Non Linear Analysis, 30 (2013), 33-77.  doi: 10.1016/j.anihpc.2012.06.001.

[2]

P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Mathematische Annalen, 359 (2014), 471-536.  doi: 10.1007/s00208-013-1001-7.

[3]

P. Baldi, M. Berti and R. Montalto, KAM for autonomous quasi-linear perturbations of KdV, Annales De Linstitut Henri Poincare Non Linear Analysis, 33 (2016), 1589-1638.  doi: 10.1016/j.anihpc.2015.07.003.

[4]

P. Baldi, M. Berti and R. Montalto, KAM for autonomous quasi-linear perturbations of mKdV, Bollettino dell'Unione Matematica Italiana, 9 (2016), 143-188.  doi: 10.1007/s40574-016-0065-1.

[5]

D. Bambusi and S. Graffi, Time quasi-periodic unbounded perturbations of schrödinger operators and KAM methods, Communications in Mathematical Physics, 219 (2001), 465-480.  doi: 10.1007/s002200100426.

[6]

M. Berti and R. Montalto, Quasi-periodic standing wave solutions of gravity-capillary water waves, preprint, arXiv: 1602.02411.

[7]

N. N. Bogoljubov, J. A. Mitropoliskii and A. M. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer-verlag, New York, 1976.

[8]

J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Annals of Mahematics Studies 158, Princeton University Press, 2005. doi: 10.1515/9781400837144.

[9]

H. Cong, L. Mi and X. Yuan, Positive quasi-periodic solutions to Lotka-Volterra system, Science China Mathematics, 53 (2010), 1151-1160.  doi: 10.1007/s11425-009-0217-1.

[10]

R. Feola and M. Procesi, Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, Journal of Differential Equations, 259 (2015), 3389-3447.  doi: 10.1016/j.jde.2015.04.025.

[11]

R. Feola, KAM for quasi-linear forced hamiltonian NLS, preprint, arXiv: 1602.01341.

[12]

T. Kappeler and J. Pöschel, KdV and KAM, Springer-verlag, New York, 2003. doi: 10.1007/978-3-662-08054-2.

[13]

S. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Springer-verlag, New York, 1993. doi: 10.1007/BFb0092243.

[14]

S. Kuksin, On small-denominators equations with large variable coefficients, Zeitschrift für angewandte Mathematik und Physik, 48 (1997), 262-271.  doi: 10.1007/PL00001476.

[15]

S. Kuksin, A KAM theorem for equations of the Korteweg-De Vries Type, Reviews in Mathematical Physics, 10 (1998), 1-64. 

[16]

S. Kuksin, Analysis of Hamiltonian PDEs, Oxford University Press, 2000.

[17]

S. Kuksin and J. Pöschel, Invariant Cantor Manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Annals of Mathematics, 143 (1996), 149-179.  doi: 10.2307/2118656.

[18]

P. D. Lax, Periodic solutions of the KdV equation, Communications on Pure & Applied Mathematics, 28 (1975), 141-188.  doi: 10.1002/cpa.3160280105.

[19]

J. Liu and X. Yuan, Spectrum for quantum duffing oscillator and small-divisor equation with large-variable coefficient, Communications on Pure & Applied Mathematics, 63 (2010), 1145-1172.  doi: 10.1002/cpa.20314.

[20]

J. Liu and X. Yuan, A KAM theorem for hamiltonian partial differential equations with unbounded perturbations, Communications in Mathematical Physics, 307 (2011), 629-673.  doi: 10.1007/s00220-011-1353-3.

[21]

R. Mcleod, Mean value theorems for vector valued functions, Proceedings of the Edinburgh Mathematical Society, 14 (1965), 197-209.  doi: 10.1017/S0013091500008786.

[22]

R. Montalto, Quasi-periodic solutions of forced Kirchhoff equation, Nonlinear Differential Equationsc and Applications, 24 (2017), Art. 9, 71 pp. doi: 10.1007/s00030-017-0432-3.

[23]

J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Annali Della Scuola Normale Superiore di Pisa-Classe di Scienze, 23 (1996), 119-148. 

[24]

J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Commentarii Mathematici Helvetici, 71 (1996), 269-296.  doi: 10.1007/BF02566420.

[25]

E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Communications in Mathematical Physics, 127 (1990), 479-528.  doi: 10.1007/BF02104499.

[26]

A. M. Wazwaz, Solitons and periodic solutions for the fifth-order KdV equation, Applied Mathematics Letters, 19 (2006), 1162-1167.  doi: 10.1016/j.aml.2005.07.014.

[27]

A. M. Wazwaz, The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, Applied Mathematics & Computation, 184 (2007), 1002-1014.  doi: 10.1016/j.amc.2006.07.002.

[28]

X. Yuan and K. Zhang, A reduction theorem for time dependent Schrödinger operator with finite differentiable unbounded perturbation, Journal of Mathematical Physics, 54 (2013), 052701, 23 pp. doi: 10.1063/1.4803852.

[29]

E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems, Ⅰ, Communications on Pure & Applied Mathematics, 28 (1975), 91-140.  doi: 10.1002/cpa.3160280104.

[30]

E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems, Ⅱ, Communications on Pure & Applied Mathematics, 29 (1976), 49-111.  doi: 10.1002/cpa.3160290104.

[31]

J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differential equations, Nonlinearity, 24 (2011), 1189-1228.  doi: 10.1088/0951-7715/24/4/010.

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