621
Internat. J. Math. & Math. Sci.
Vol. 8 No. 3 (1985) 621-623
RESEARCH NOTES
LOCAL ENERGY DECAY FOR WA..VES GOVERNED BY A SYSTEM
OF NONLINEAR SCHRODINGER EQUATIONS
IN A NONUNIFORM MEDIUM
J. E. LIN
l)eprtment of Hntlem,atlcal Sclences
George Mason University
22030
Fairfax, Virginia
U. S. A.
(Received June 21, 1985)
We show that tlm local energy of a smooth locallzed solution to a system
of coupled nonlinear SchrSdlnger equations in a certaln nonunlform medlum decays to
ABSTRACT.
zero as the time approaches inflnlty.
K’Y WORDS AND PIIIAS.’S.
1980 MA]’IIF;MA2’ICS
1.
Local energy, waves, nonlineo_e ehrOdin.qer eqnat{ona.
SUBJECT CLA,gSIFICA]’ION CODF;. 35Q$O
INTRODUCTION.
Conslder a system of m coupled nonlinear Schrdlnger equatlons in a nonuniform
meduum
t(O/Ot)On
where n
(c32/Ox2)Un + Fn (1112’
lUn 12
)Un
+
kn(X)Un
0
(1.1)
are real-valued functions of x only and F’s are
m, k’s
n
n
I, 2,
We will show that under certain condltlons o[F’s and k
real-valued functions.
n
n ’s0
namely,
Fn(IU112
n-I
lUn 12’
IUml2)=CnlUn 12 + h’z=1 Iuh 12 +
with positive constant C
for all n
n
I/(I +
k (x)
n
for all n
I, 2
a2x 2)
1, 2
(UI,
Urn)
z
h =n +l
1,11,12
(1.2)
m, and
wlth 0 < a <_ (2/3)
m, the local energy
and localized solution
m
Z
(1.3)
r IUnI2 (x,
t) dx for the smooth
n --1 -r
decays to zero as t approaches Inflnlty.
Eq. (I.I) with one component and in a llnear type of nonunlform medlum was derived by Chen and Llu [1-3] in the study of so]Irons In
See
nonuniform medium.
also Newell [4].
Gupta et al [5], Gupta [6] and upta and Ray [7] studied Eq. (I.I)
with one component and a parabollc type of nonunlformlty for Its exact solution and
the inverse scattering method. Eq. (1.1) with two components and k
O, for all n,
n
was derived for envelope waves with different circular polarizations in an isotroplc
nonlinear medium by Berkhoer and Zakharov [8] and also used by Eiphlck [9] for the
quantum version of the one-component nonlinear Scltr6dinger
model.
Kaiser [I0] discussed the well-posedness of it for an Inltlal-value and boundary-value problem.
Our method in this work conslsts of nn exploration of the conservation laws
which Eq.
(1.1) possesses and is
a genera]izatlon of
the one-component nonlinear Scltrdlngpr equation,
(O/OX)Wn
by w
n’ x’
etc.
and the solution (U
1,
autltor’s previous work ill] for
in tile loll’owing, we shall denote
U
m)
will be assumed to be
622
J.E. LIN
i.e., U
smooth and localized
and all its partial derivatives approach zero as
approaches infinity, for each t and for all n
1,
]x
m.
HETIIOD.
2.
Hultiplying Eq. (I.I) by V
where V
n’
is the complex conjugate of U
n
n
and taking
the imaginary part, we get
(lun 12) t
i(v
u
u n, x v n
n, x n
(2.1)
x
Hence
IUnlZ(x, t) dx
Next, multiplying Eq. (1.1) by
(2.2)
constant
Vn, t’
taking the real part of it, making the
-,
use of (1.2) and integrating in x from- to
we get
</z)c.lo.l 4)
12 knlUnl 2
dx < constant
+
z
+
nX
n
i
where the constant on the right-hand side is independent of t.
(Iu
f
Now, taking the real part
where L U
i U
n n
U
n,
+ Fn
n, xx
[(LnUn)xVn
of
(L n
([U112
lln)V n’ x
iNm [2)Un +
(2.3)
]’
k U
n n
and
making the use of (1.2), we get
m
m
Z (V
(I/2i)
1
m
n
z
+ </)
U -U
n x
C
n
m
n
n
(1/2) Z
V
x n t
n
(lUll)4 x +
n=l
= [U)t
</)(<
n
2
k"
(lUn 2 )xxx +
2) x
m
2
(/)
n=l
Z
n =I
(IUn,
x
2) x
<lUll ) x
n
n[
Now, making the use of the assumption (1.3) on
A(x)
to
arctan
,
kn,
is from the assumption on k
(ax), where a
multlplyiug (2.4) by
integrating in x from
n’
using the technique of integrntion by part and makin the use of (2,2) and
(2.3),
we get
m
0
and 0
ir z
12
-r n-l (fUn
+
lun ,x
2
+
lUn 14)
dx dt <-
(2.5)
Let r > 0 and B be soth such that B(x)
1 for lx
r, B(x)
0 for lx
2r
B
i. Multiplying (2.1) by B and integrating In x from -2r to
2r, we get
2r
2
2r n(lunl
)t dx
b
/2r
(lUnlZ
-2r
+
f2r_2r BUn2
dx
iun
x
12) dx
for some positive constant b.
Let 0 <
(t-
tl)
1 < t, then
fr_r [Un[2
"ft I f2r
-2r B[Un[
t
Let t
I
r
fr
t
inn [2dx
Hence, by (2.5)
,ACOWLEDGEMENT.
2-10138.
2
I,
dx _< (t
dxds
+
ten
(b + 1)
t
1
[Un [2(x
tl
t
I
(s-
2r
2
2r([Un[
t)dx
f2r
_2r B([Un[2
tl)
+ ]U
0 as t
nx
2)
t dx[ds
dxds
Q.E.D.
This work was supported by George Mason University Research Grant
LOCAL ENERGY DECAY FOR WAVES IN A NONUNIFORM MEDIUM
623
REFERENCES
i.
CIIEN, II.II. and LIU, C.S., Solitons in Nonlinear Media, Pls. Rev. Lett. 37 (1976),
693-697.
2.
CIIEN, I!.ii. and LIU, C.S., Sollton Generation at Resonance and Density Modificatlon in Laser-lrradated Plasmas, _Pl__s. Rev. Lett. 39 (1977), 1147-1151.
3.
CHEN, !I.II. and LIU, C.S.,
Irradiated
Langm1=|r So]Itons and
Resonance
LaserExperiments,
Absorpt|on ]n
PJasmas, in Plasma Pl=sics, Nonlinear Theory
and
4.
(Ed. Hans Wilhelmsson) 211-221, P.le1=m Pub., 1977.
NEWELL, A.C., Nonlinear Tunne]llng, J. Math. Pl__s_. 19 (1978), 1126-]133.
5.
GUPTA M.R., SOM, B.K.
6.
GUPTA, M.R., Exact Inverse Scattering Solution of a Non-llnear Evolution Equation in a Non-uniform Medium, Pl__._s. Lett. 72A ([979), 420-422.
7.
GUPTA, M.R. and RAY, J., Extension of
and DASCUPTA, B., Exact Solution of Damped Nonlinear
Schr6dlnger Equation for a Parabolic Density Profile, Phi. Lett. 69A (1978),
172-174.
inverse Scattering Method to Nonlinear
Evolutlon Equation in Nonuniform Medium, J. Math. phys
2183.
22 (1981), 2180-
8.
BERKIIOER, A.L. and ZAKIIAROV, V.E., Self Excitation of Waves with Different Polarizations in Nonlinear Media, Soviet Ply__s_. JETP 31 (1910), 486-490.
9.
ELPIIICK, C., The Quantum Spectral Transform Method [or the One- and Two-Component
Nonlinear Schrdlnger Model, J.
A 16 (1983), 4013-4024.
i0.
II.
.
KAISER, H.-C., Nlerungslsungen fr en System Gekoppelter Nlchtllnearer
Schrdlnger-Glelclungen, Math. Nachr’. I01 (1981), 213-293.
LIN, J.E., Asymptotic Behavior in Time of tle Solutions of Three Nonlinear Partial Differential Equations, J. Differential Equations 29 (1978), 467-413.