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.

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 Td 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 Td
,”
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 Td
,”
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 D(r,s),O for scalar functions is defined in (2.2). The vector function G:D(r,s)×OCm, (m < ∞) is similarly defined as GD(r,s),O=i=1mGiD(r,s),O.

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
).
You do not currently have access to this content.