In this paper, we prove the existence and uniqueness of the global smooth solution to the (2+1)-dimensional long wave–short wave resonance interaction equation.

1.
Bekiranov
,
D.
,
Ogawa
,
T.
, and
Ponce
,
G.
, “
On the well-posedness of Benney’s interaction equation for short and long waves
,”
Adv. Differ. Equ.
1
,
919
(
1996
).
2.
Bekiranov
,
D.
,
Ogawa
,
T.
, and
Ponce
,
G.
, “
Interaction equation for short and long dispersive waves
,”
J. Funct. Anal.
158
,
357
(
1998
).
3.
Brezis
,
H.
, and
Gallouet
,
T.
, “
Nonlinear Schrödinger evolution equations
,”
Nonlinear Anal. Theory, Methods Appl.
4
,
677
(
1980
).
4.
Guo
,
B.
, “
The global solution for one class of the system of LS nonlinear wave interaction
,”
J. Math. Res. Exposition
1,
69
(
1987
).
5.
Guo
,
B.
, “
The period initial value problems and initial value problems for one class of generalized LS type equations
,”
J. Eng. Math.
8
,
47
(
1991
) (in Chinese).
6.
Guo
,
B.
, and
Wang
,
B.
, “
The global solution and its longtime behavior for a class of generalized LS type equations
,”
Prog. Nat. Sci.
5
,
533
(
1996
).
7.
Huo
,
Z.
, and
Guo
,
B.
, “
Local well-posedness of interaction equations for short and long dispersive waves
,”
J. Partial Differ. Equ.
17
,
137
(
2004
).
8.
W. C. Lai
,
D.
, and
Chow
,
K. W.
,
, “
Positon’ and ‘dromion’ solutions of the (2+1) dimensional long wave-short wave resonance interaction equations
,”
J. Phys. Soc. Jpn.
68
,
1847
(
1999
).
9.
Li
,
Y.
, “
Long time behavior for the weakly damped driven long-wave-short-wave resonance equations
,”
J. Differ. Equations
223
,
261
(
2006
).
10.
Oikawa
,
M.
,
Okamura
,
M.
, and
Funakoshi
,
M.
, “
Two-dimensional resonant interaction between long and short waves
,”
J. Phys. Soc. Jpn.
58
,
4416
(
1989
).
11.
Tang
,
X.
, and
Lou
,
S.
, “
Periodic and localized solutions of the long wave-short wave resonance interaction equation
,”
J. Phys. A
38
,
9649
(
2005
).
12.
Tsutsumi
,
M.
, and
Hatano
,
S.
, “
Well-posedness of the Cauchy problem for Benney’s first equations of long wave short wave interactions
,”
Finkcialaj Ekvacioj
37,
289
(
1994
).
13.
Tsutsumi
,
M.
, and
Hatano
,
S.
, “
Well-posedness of the Cauchy problem for the long-wave-short-wave resonance equations
,”
Nonlinear Anal. Theory, Methods Appl.
22
,
155
(
1994
).
14.
Yurova
,
M.
, “
Application of dressing method for long wave-short wave resonance interaction equation
,”
J. Math. Phys.
48
,
053516
(
2007
).
15.
Zhang
,
R.
, and
Guo
,
B.
, “
Global solution and its long time behavior for the generalized long-short wave equations
,”
J. Partial Differ. Equ.
18
,
206
(
2005
).
16.
Zhou
,
Y.
, and
Guo
,
B.
, “
Period boundary value problem and initial value problem for the generalized Korteweg-de Vries equations of high order
,”
Acta Math. Sinica
27,
154
(
1984
) (in Chinese).
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