In this paper, we prove an infinite dimensional Kolmogorov-Arnold-Moser theorem. As an application, it is shown that there are many small-amplitude linearly-stable quasi-periodic solutions for higher dimensional wave equations with a real Fourier multiplier, which are under nonlocal and forced perturbations with a special structure in space and short range property.
REFERENCES
1.
S. B.
Kuksin
, Nearly Integrable Infinite Dimensional Hamiltonian Systems
, Lecture Notes in Mathematics Vol. 1556 (Springer
, Berlin
, 1993
).2.
E.
Wayne
, “Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory
,” Commun. Math. Phys.
127
, 473
–528
(1990
).3.
L. H.
Eliasson
and S.
Kuksin
, “KAM for non-linear Schrödinger equation
,” Ann. Math.
172
, 371
–435
(2010
).4.
J.
Geng
, X.
Xu
, and J.
You
, “An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation
,” Adv. Math.
226
, 5361
–5402
(2011
).5.
J.
Geng
and J.
You
, “A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces
,” Commun. Math. Phys.
262
, 343
–372
(2006
).6.
M.
Procesi
and X.
Xu
, “Quasi-töplitz functions in KAM theorem
,” SIAM J. Math. Anal.
45
(4
), 2148
–2181
(2013
).7.
C.
Procesi
and M.
Procesi
, “A KAM algorithm for the resonant non-linear Schrödinger equation
,” Adv. Math.
272
, 399
–470
(2015
).8.
W.
Craig
and C. E.
Wayne
, “Newton’s method and periodic solutions of nonlinear wave equations
,” Commun. Pure Appl. Math.
46
, 1409
–1498
(1993
).9.
J.
Bourgain
, “Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE
,” Int. Math. Res. Not.
1994
(11
), 475
–497
(1994
).10.
J.
Bourgain
, “Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations
,” Ann. Math.
148
, 363
–439
(1998
).11.
J.
Bourgain
, Green’s Function Estimates for Lattice Schrödinger Operators and Applications
, Annals of Mathematics Studies Vol. 158 (Princeton University Press
, Princeton, NJ
, 2005
).12.
M.
Berti
and P.
Bolle
, “Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential
,” Nonlinearity
25
, 2579
–2613
(2012
).13.
M.
Berti
and P.
Bolle
, “Quasi-periodic solutions with Sobolev regularity of NLS on with a multiplicative potential
,” Eur. J. Math.
15
, 229
–286
(2013
).14.
M.
Berti
, L.
Corsi
, and M.
Procesi
, “An abstract Nash-Moser theorem and quasi-periodic solutions for NLW and NLS on compact Lie groups and homogeneous manifolds
,” Commun. Math. Phys.
334
(3
), 1413
–1454
(2015
).15.
W.
Wang
, “Energy supercritical nonlinear schrödinger equations: Quasi-periodic solutions
,” Duke Math. J.
165
(6
), 1129
–1192
(2016
).16.
R.
Montalto
, “A reducibility result for a class of linear wave equations on
,” Int. Math. Res. Not.
2019
(6
), 1788
–1862
.17.
Y.
Chen
and J.
Geng
, “A KAM theorem for higher dimensional wave equations under nonlocal perturbation
,” J. Dyn. Differ. Equations
32
, 419
–440
(2020
).18.
L.
Corsi
and R.
Montalto
, “Quasi-periodic solutions for the forced Kirchhoff equation on
,” Nonlinearity
31
, 5075
–5109
(2018
).19.
J.
Geng
, J.
Viveros
, and Y.
Yi
, “Quasi-periodic breathers in Hamiltonian newworks of long-range coupling
,” Physica D
237
, 2866
–2892
(2008
).20.
The norm for scalar functions is defined in (2.2). The vector function , (m < ∞) is similarly defined as .
21.
L.
Chierchia
and J.
You
, “KAM tori for 1D nonlinear wave equations with periodic boundary conditions
,” Commun. Math. Phys.
211
, 497
–525
(2000
).22.
J.
You
, “Perturbation of lower dimensional tori for Hamiltonian systems
,” J. Differ. Equations
152
, 1
–29
(1999
).© 2020 Author(s).
2020
Author(s)
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