Quasi-periodic solutions for class of Hamiltonian partial differential equations with fixed constant potential

doi:10.1007/s11464-017-0667-7

Front. Math. China

2018, Vol. 13

Issue (1) : 227-254 https://doi.org/10.1007/s11464-017-0667-7

RESEARCH ARTICLE

Quasi-periodic solutions for class of Hamiltonian partial differential equations with fixed constant potential

Xindong XU(

)

School of Mathematics, Southeast University, Nanjing 211189, China

Download: PDF(272 KB)
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

We consider Hamiltonian partial differential equations u_tt +|∂_x|u+ σu = f(u), x ∈ $T$ , t ∈ $?$ , with periodic boundary conditions, where f(u) is a real-analytic function of the form f(u) = u⁵ + o(u⁵) near u = 0, σ ∈ (0, 1) is a fixed constant, and $T = ? / 2 π Z$ T= R/2πZ. A family of quasi-periodic solutions with 2-dimensional are constructed for the equation above with σ ∈ (0, 1)\ $?$ . The proof is based on infinite-dimensional KAM theory and partial Birkhoff normal form.

Keywords Dense frequency quasi-periodic solution Birkhoff normal form

Corresponding Author(s): Xindong XU

Issue Date: 12 January 2018

Cite this article:

Xindong XU. Quasi-periodic solutions for class of Hamiltonian partial differential equations with fixed constant potential[J]. Front. Math. China, 2018, 13(1): 227-254.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-017-0667-7
https://academic.hep.com.cn/fmc/EN/Y2018/V13/I1/227

1	Bambusi D. On long time stability in Hamiltonian perturbations of non-resonant linear PDEs. Nonlinearity, 1999, 12: 823–850 https://doi.org/10.1088/0951-7715/12/4/305
2	Chierchia L, You J. KAM tori for 1D nonlinear wave equations with periodic boundary conditions. Comm Math Phys, 2000, 211: 498–525 https://doi.org/10.1007/s002200050824
3	Craig W, Worfolk P A. An integrable normal form for water waves in infinite depth. Phys D, 1995, 84: 513–531 https://doi.org/10.1016/0167-2789(95)00067-E
4	Craig W, Sulem C. Mapping properties of normal forms transformations for water waves. Boll Unione Mat Ital, 2016, 9(2): 289–318 https://doi.org/10.1007/s40574-016-0078-9
5	Eliasson L H, Grébert B, Kuksin S B. KAM for the nonlinear beam equation. Geom Funct Anal, 2016, 26: 1588–1715 https://doi.org/10.1007/s00039-016-0390-7
6	Geng J, Xu X, You J. An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation. Adv Math, 2011, 226: 5361–5402 https://doi.org/10.1016/j.aim.2011.01.013
7	Geng J, You J. A KAM theorem for one dimensional Schrödinger equation with periodic boundary conditions. J Differential Equations, 2005, 209: 1–56 https://doi.org/10.1016/j.jde.2004.09.013
8	Geng J, You J. A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces. Comm Math Phys, 2006, 262: 343–372 https://doi.org/10.1007/s00220-005-1497-0
9	Kuksin S B, Pöschel J. Invariant Cantor manifolds of quasiperiodic oscillations for a nonlinear Schrödinger equation. Ann Math, 1996, 143: 149–179 https://doi.org/10.2307/2118656
10	Liang Z. Quasi-periodic solutions for 1D Schrödinger equation with the nonlinearity \|u\|2pu. J Differential Equations, 2008, 244: 2185–2225 https://doi.org/10.1016/j.jde.2008.02.015
11	Liang Z, You J. Quasi-periodic solutions for 1D Schrödinger equation with higher nonlinearity. SIAM J Math Anal, 2005, 36(2): 1965–1990 https://doi.org/10.1137/S0036141003435011
12	Pöschel J.Quasi-periodic solutions for a nonlinear wave equation. Comment Math Helv, 1996, 71:269–296 https://doi.org/10.1007/BF02566420
13	Pöschel J. A KAM theorem for some nonlinear partial differential equations. Ann Sc Norm Super Pisa Cl Sci, 1996, 23: 119–148
14	Procesi C, Procesi M. A KAM algorithm for the resonant non-linear Schrödinger equation. Adv Math, 2015, 2 72: 399–470 https://doi.org/10.1016/j.aim.2014.12.004
15	Shi Y, Xu J, Xu X. On quasi-periodic solutions for generalized Boussinesq equation with quadratic nonlinearity. J Math Phys, 2015, 56(2): 022703 https://doi.org/10.1063/1.4906810
16	Whitney H. Analytical extensions of differentiable functions defined on closed set. Trans Amer Math Soc, 1934, 36: 63–89 https://doi.org/10.1090/S0002-9947-1934-1501735-3
17	Wu J, Xu X. A KAM theorem for some partial differential equations in one dimension. Proc Amer Math Soc, 2016, 144(5): 2149–2160 https://doi.org/10.1090/proc/12875
18	Xu X, Geng J. KAM tori for higher dimensional beam equation with a fixed constant potential. Sci China Ser A, 2009, 52(9): 2007–2018 https://doi.org/10.1007/s11425-008-0158-0
19	Yuan X. Quasi-periodic solutions of completely resonant nonlinear wave equations. J Differential Equations, 2006, 230: 213–274 https://doi.org/10.1016/j.jde.2005.12.012
20	Zakharov V E. Stability of periodic waves of finite amplitude on the surface of deep fluid. J Appl Mech Tech Phys, 1968, 2: 190–194

[1]	Yanmin NIU, Xiong LI. Dynamical behaviors for generalized pendulum type equations with p-Laplacian[J]. Front. Math. China, 2020, 15(5): 959-984.
[2]	Shiwen ZHANG, Zhiyan ZHAO. Diffusion bound and reducibility for discrete Schr?dinger equations with tangent potential[J]. Front Math Chin, 2012, 7(6): 1213-1235.

Viewed

Full text

Abstract

Cited

Shared

Discussed