American Institute of Mathematical Sciences

Advanced
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.   Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.   Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Jibin Li, Yi Zhang. Exact solitary wave and quasi-periodic wave solutions for four fifth-order nonlinear wave equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 623-631. doi: 10.3934/dcdsb.2010.13.623
[2]

Marina Chugunova, Dmitry Pelinovsky. Two-pulse solutions in the fifth-order KdV equation: Rigorous theory and numerical approximations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 773-800. doi: 10.3934/dcdsb.2007.8.773
[3]

Yanling Shi, Junxiang Xu, Xindong Xu. Quasi-periodic solutions of generalized Boussinesq equation with quasi-periodic forcing. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2501-2519. doi: 10.3934/dcdsb.2017104
[4]

Lei Jiao, Yiqian Wang. The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1585-1606. doi: 10.3934/cpaa.2009.8.1585
[5]

Timur Akhunov. Local well-posedness of quasi-linear systems generalizing KdV. Communications on Pure & Applied Analysis, 2013, 12 (2) : 899-921. doi: 10.3934/cpaa.2013.12.899
[6]

Yanling Shi, Junxiang Xu. Quasi-periodic solutions for a class of beam equation system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 31-53. doi: 10.3934/dcdsb.2019171
[7]

Siqi Xu, Dongfeng Yan. Smooth quasi-periodic solutions for the perturbed mKdV equation. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1857-1869. doi: 10.3934/cpaa.2016019
[8]

Meina Gao, Jianjun Liu. Quasi-periodic solutions for derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2101-2123. doi: 10.3934/dcds.2012.32.2101
[9]

Zhenguo Liang, Jiansheng Geng. Quasi-periodic solutions for 1D resonant beam equation. Communications on Pure & Applied Analysis, 2006, 5 (4) : 839-853. doi: 10.3934/cpaa.2006.5.839
[10]

Yanling Shi, Junxiang Xu. Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3479-3490. doi: 10.3934/dcdsb.2020241
[11]

Claudia Valls. On the quasi-periodic solutions of generalized Kaup systems. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 467-482. doi: 10.3934/dcds.2015.35.467
[12]

Osama Moaaz, Omar Bazighifan. Oscillation criteria for second-order quasi-linear neutral functional differential equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2465-2473. doi: 10.3934/dcdss.2020136
[13]

Qihuai Liu, Dingbian Qian, Zhiguo Wang. Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1537-1550. doi: 10.3934/dcdsb.2012.17.1537
[14]

Yongqin Liu, Shuichi Kawashima. Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1113-1139. doi: 10.3934/dcds.2011.29.1113
[15]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027
[16]

Tuhin Ghosh, Karthik Iyer. Cloaking for a quasi-linear elliptic partial differential equation. Inverse Problems & Imaging, 2018, 12 (2) : 461-491. doi: 10.3934/ipi.2018020
[17]

Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345
[18]

Vitali Liskevich, Igor I. Skrypnik. Pointwise estimates for solutions of singular quasi-linear parabolic equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1029-1042. doi: 10.3934/dcdss.2013.6.1029
[19]

Chengming Cao, Xiaoping Yuan. Quasi-periodic solutions for perturbed generalized nonlinear vibrating string equation with singularities. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1867-1901. doi: 10.3934/dcds.2017079
[20]

Hongzi Cong, Jianjun Liu, Xiaoping Yuan. Quasi-periodic solutions for complex Ginzburg-Landau equation of nonlinearity $|u|^{2p}u$. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 579-600. doi: 10.3934/dcdss.2010.3.579

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (168)
  • HTML views (134)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences