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Quasi-periodic solution of quasi-linear fifth-order KdV equation
School of Mathematical Sciences, Fudan University, Shanghai 200433, China |
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.
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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