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Existence of quasiperiodic solutions of elliptic equations on the entire space with a quadratic nonlinearity
1. | School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States |
2. | Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada |
$\Delta u+{{u}_{yy}}+f(x,u) = 0,\quad (x,y)\in {{\mathbb{R}}^{N}}\times \mathbb{R}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)$ |
$ f $ |
$ x $ |
$ f(\cdot,0)\equiv 0 $ |
$ y $ |
$ |x|\to\infty $ |
$ y $ |
$ f_u(x,0) $ |
$ f_{uu}(x,0) $ |
$ f(x,\cdot) $ |
References:
[1] |
S. Agmon, Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-body Schrödinger Operators, volume 29 of Mathematical Notes, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1982. |
[2] |
D. Bambusi, An introduction to Birkhoff normal form, Università di Milano, 2014. Google Scholar |
[3] |
H. W. Broer, S. N. Chow, Y. Kim and G. Vegter,
A normally elliptic Hamiltonian bifurcation, Z. Angew. Math. Phys., 44 (1993), 389-432.
doi: 10.1007/BF00953660. |
[4] |
H. W. Broer and G. B. Huitema,
A proof of the isoenergetic KAM-theorem from the "ordinary" one, Journal of Differential Equations, 90 (1991), 52-60.
doi: 10.1016/0022-0396(91)90160-B. |
[5] |
B. Grébert, Birkhoff normal form and Hamiltonian PDEs, Séminaries and Congrès, 15 (2007), 1–46. |
[6] |
M. Haragus and G. Iooss, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems, Universitext. Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011.
doi: 10.1007/978-0-85729-112-7. |
[7] |
H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser, 1994.
doi: 10.1007/978-3-0348-8540-9. |
[8] |
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. |
[9] |
K. Kirchgässner,
Wave solutions of reversible systems and applications, Journal of Differential Equations, 45 (1982), 113-127.
doi: 10.1016/0022-0396(82)90058-4. |
[10] |
A. Mielke, Hamiltonian and Lagrangian Flows on Center Manifolds, Lecture Notes in Mathematics, 1489. Springer-Verlag, 1991.
doi: 10.1007/BFb0097544. |
[11] |
P. Poláčik and D. A. Valdebenito,
Existence of quasiperiodic solutions of elliptic equations on $\mathbb R^{N+1}$ via center manifold and KAM theorems, Journal of Differential Equations, 262 (2017), 6109-6164.
doi: 10.1016/j.jde.2017.02.027. |
[12] |
J. Pöschel,
Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696.
doi: 10.1002/cpa.3160350504. |
[13] |
M. Reed and B. Simon, Methods of Mathematical Physics, Volume Ⅳ, Academic Press, New York-London, 1979.
![]() |
[14] |
J. Scheurle,
Bifurcation of quasiperiodic solutions from equilibrium points of reversible dynamical systems, Arch. Rational Mech. Anal., 97 (1987), 103-139.
doi: 10.1007/BF00251911. |
[15] |
Y. Shi, J. Xu and X. Xu, On quasi-periodic solutions for generalized Boussinesq equation with quadratic nonlinearity, J. Math. Phys., 56 (2015), 022703, 15pp.
doi: 10.1063/1.4906810. |
[16] |
J. M. Tuwankotta and F. Verhulst,
Hamiltonian systems with widely separated frequencies, Nonlinearity, 16 (2003), 689-706.
doi: 10.1088/0951-7715/16/2/319. |
[17] |
C. Valls,
Existence of quasi-periodic solutions for elliptic equations on a cylindrical domain, Comentarii Mathematici Helvetici, 81 (2006), 783-800.
doi: 10.4171/CMH/73. |
[18] |
A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, in Dynamics Reported, Springer-Verlag, 1 (1992), 125–163. |
show all references
References:
[1] |
S. Agmon, Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-body Schrödinger Operators, volume 29 of Mathematical Notes, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1982. |
[2] |
D. Bambusi, An introduction to Birkhoff normal form, Università di Milano, 2014. Google Scholar |
[3] |
H. W. Broer, S. N. Chow, Y. Kim and G. Vegter,
A normally elliptic Hamiltonian bifurcation, Z. Angew. Math. Phys., 44 (1993), 389-432.
doi: 10.1007/BF00953660. |
[4] |
H. W. Broer and G. B. Huitema,
A proof of the isoenergetic KAM-theorem from the "ordinary" one, Journal of Differential Equations, 90 (1991), 52-60.
doi: 10.1016/0022-0396(91)90160-B. |
[5] |
B. Grébert, Birkhoff normal form and Hamiltonian PDEs, Séminaries and Congrès, 15 (2007), 1–46. |
[6] |
M. Haragus and G. Iooss, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems, Universitext. Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011.
doi: 10.1007/978-0-85729-112-7. |
[7] |
H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser, 1994.
doi: 10.1007/978-3-0348-8540-9. |
[8] |
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. |
[9] |
K. Kirchgässner,
Wave solutions of reversible systems and applications, Journal of Differential Equations, 45 (1982), 113-127.
doi: 10.1016/0022-0396(82)90058-4. |
[10] |
A. Mielke, Hamiltonian and Lagrangian Flows on Center Manifolds, Lecture Notes in Mathematics, 1489. Springer-Verlag, 1991.
doi: 10.1007/BFb0097544. |
[11] |
P. Poláčik and D. A. Valdebenito,
Existence of quasiperiodic solutions of elliptic equations on $\mathbb R^{N+1}$ via center manifold and KAM theorems, Journal of Differential Equations, 262 (2017), 6109-6164.
doi: 10.1016/j.jde.2017.02.027. |
[12] |
J. Pöschel,
Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696.
doi: 10.1002/cpa.3160350504. |
[13] |
M. Reed and B. Simon, Methods of Mathematical Physics, Volume Ⅳ, Academic Press, New York-London, 1979.
![]() |
[14] |
J. Scheurle,
Bifurcation of quasiperiodic solutions from equilibrium points of reversible dynamical systems, Arch. Rational Mech. Anal., 97 (1987), 103-139.
doi: 10.1007/BF00251911. |
[15] |
Y. Shi, J. Xu and X. Xu, On quasi-periodic solutions for generalized Boussinesq equation with quadratic nonlinearity, J. Math. Phys., 56 (2015), 022703, 15pp.
doi: 10.1063/1.4906810. |
[16] |
J. M. Tuwankotta and F. Verhulst,
Hamiltonian systems with widely separated frequencies, Nonlinearity, 16 (2003), 689-706.
doi: 10.1088/0951-7715/16/2/319. |
[17] |
C. Valls,
Existence of quasi-periodic solutions for elliptic equations on a cylindrical domain, Comentarii Mathematici Helvetici, 81 (2006), 783-800.
doi: 10.4171/CMH/73. |
[18] |
A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, in Dynamics Reported, Springer-Verlag, 1 (1992), 125–163. |
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