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Stability of solitary waves for coupled nonlinear schrödinger equations
Nonlinear
Analysis.
Theory,
Methods
&Applications.
Vol. 26, No. 5. pp. 933-939. 1996 Copyright Q 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X196 $15.00+ .O!l
Pergamon 0362-546X(94)00340-8
STABILITY
OF SOLITARY WAVES FOR COUPLED SCHRijDINGER EQUATIONS
NONLINEAR
MASAHITO OHTA Department of Mathematical Sciences, University of Tokyo, Hongo, Tokyo 113, Japan (Received 25 April
1994; received for publication
17 November 1994)
Key words and phrases: Coupled nonlinear Schrtidinger equations, orbital stability, solitary waves,
concentration-compactness. 1. INTRODUCTION
AND RESULT
In the present paper we consider the stability of solitary waves for the following coupled nonlinear Schrodinger equations
(1.1)
where u and u are complex valued functions of (t, x) E R2, and a is a real constant. The system (1.1) describesthe nonlinear modulations of two monochromatic waves, whose group velocities are almost equal, in optical fibers (see [l, 2]), and it is regarded as a natural extension of the following single nonlinear Schrodinger equation (1.2)
When a > - 1, (1.1) has solitary wave solutions with the form (see [ 1, 21) (u,(t, x), u,(t, x)) = (eicw-c2)c+ic++i~9w(x - 2ct), ei(d-c2)f+icx+i’fpw(x- 2ct)),
(1.3)
where o > 0, c, (Yand P are real constants, and 9,(x) =
20 sech6x. J- a+1
We note that 9, satisfies -g
+ oa, - (a + 1)19,129= 0,
XE R, (1.4w)
i L
9(x) = 9(-x),
x E R
lim 9(x) = 0. x+_+s,
Since (1.1) is invariant under the Galilean transformations W, 4, W, x)) ++(e -ic*f+iqf,
x _ zct), 933
e -ic2r+icxU(f,
x
-
2ct)),
c E
IT?,
934
M. OHTA
and the phase transformations (u(t, x), u(t, x)) - (e’“u(t, x), e’“u(t, x)),
a, P E R
we may consider the case when c = (Y = /I = 0 in (1.3) only. Thus, in what follows, we put (4 U, 4, u, (t, 4) = (eiofpo(xl, e’“‘v, (x)1. We now state our main result. THEOREM1.l. Let a > - 1. Then, for any o > 0, the solitary wave solution (u,(t), v,(t)) of (1.1) is orbitally stable in the following sense: for any E > 0 there exists a 6 > 0 such that if No 9uO)E H’(m) x H’(R) verifies lb0 - V)“llH’ + lb0 - V)Af1 < 6, then the solution (u(t),‘u(i)) of (1.1) with (u(O), u(0)) = (u,, u,J satisfies sup inf (Ilu(t) - eio’ryV7,(INL+ IIu(t) - eip~yyvI,llH~I< E, t E:IKci.6.y EIR where T,,u(x) = v(x - y) and H’(lR) is the usual Sobolev space. Remark 1.2. When a = 1, (1.1) is also invariant under the rotations in (u, v)
(u(t*xl, et, 9) - w, xl, w, 4)
cos 6 sin 6
( _ sin6 cos()>
9
6 E I?.
Due to this additional symmetry, when a = 1, one can also discuss the stability of solitary waves based on the study of a linearized operator (seeGrillakis et al. [3], and Stubbe and VAzquez [4]). However, when a # 1, since (1.1) is not invariant under the rotations in (u, u), the situation is quite different from that of a = 1. Remark 1.3. It is known that for any (u,,, u,J E H’(lR) x H’(R) there exists a unique solution
(u(t), u(t)) of (1.1) with (u(O), u(O)) = ( uO, uO)in C(lR; H’(R) x H’(Q), the conservation law Ilw)llL~ = Il%h2, s40,
and (u(t), u(t)) satisfies (1.5)
llwllL~ = Il%IILb
u(t)) = JY%, uo),
(1.6)
for any t E IR(see, e.g. Chapter 4 in Cazenave [5]), where
Recently, many authors have been studying the problem of stability and instability of solitary waves for the single nonlinear Schrodinger equations like (1.2) (see, e.g. [6-131). In the next section, following Cazenave and Lions [7], we will consider the minimization problem 1(A) = inf(E(u, u) : U, u E H’(R),
Ilu@ = Ilull$ = d),
A > 0,
(1.7A)
Coupled nonlinear SchrBdinger equations
935
and we will show that the set s(A) of minimizers for (1.7A) is stable for any A > 0. Moreover, we will prove that W(o))
w > 0,
= ((e’“r,v, , eiPr,pJ : a, P, y E W,
where A(o) = II~,II~z = 4&/(a + 1). In this paper, we denote the norms of the Lebesgue space Lp(lR) and the Sobolev space ff’(Q by II-lb and II-lh, respectively. We consider L’(R) as a real Hilbert space with the inner product OD (u, ~9~2= Re u(x)u(x) dx. s -m We also regard H’(lR) and H’(fR) x H’(R) as real Hilbert spacessimilarly. 2. PROOF
OF THEOREM
1.1
In what follows, we use the following notations
E(u,u) = ~~;~I:,+ ll$
- ;(lluss + II43 - [rm lz.4m4x)12~~
2
E,(u)
=
/III
- q
2
II&b
L2
au 2 E2W =IIG L2 II- ; Ibll:~9
1(A, p) = inf{E(u, v) : 2.4,u E H’(m), ((2.4@= A, Ilull~2 Zj(13)= inf{Ej(u) : u E H’(R), IIuII:~ = A],
I(A) = &I, A),
=
p),
j = 1,2,
A(w) = -+,
$(A) = ((u, u) E H'(R) x H'(IR) :1(A) = E(u, u), llult:z = llujj;z = A), $jl(A) = (u E H'(IR):Z,(13) = E,(u),
~~24~~2
=
A).
We note that E(u, 4 = W(u),
E(u,O) = E(0, u) = E,(u),
lIdI
= +I.
We split the proof of theorem 1.1 into the following two steps. 2.1. Let a > - 1. Then, for any A > 0, the set s(A) is stable in the following sense: for any E > 0 there exists a 6 > 0 such that if (uO, uO)E H’(R) X H’(R) verifies
PROPOSITION
inf < 6, ~Iluo- dllHl + II%- (0211H~I (co’, lo3ES(V then the solution (u(t), u(f)) of (1.1) with (u(O), v(O)) = (uo, u,) satisfies
M. OHTA
936
PROPOSITION
2.2. Let a > -1. Then, the relation S@(w)) = l(e’“~,~,, , e’“q,)
: a, A y E RI
holds for any w > 0. Theorem 1.1 follows from propositions 2.1 and 2.2 immediately. In order to prove propositions 2.1 and 2.2, we make some preparations. LEMMA 2.3. Let a > -1. Then, the following hold:
(1) m, Pu)2 Zl(4 + Z,(P), 2, v 2 0; (2) Z1(A(w)) = E,(q?,) = -&
03j2 = - f
2 A3(o),
9
0
> 0;
( > (3) 84 = 21,(A), 2 > 0, and (vu, co,) E 6@(W, o > 0; (4) Z(A) < Z(((Y, j?) + Z(A - (Y,1 - p) for any (YE [0, A] and p E [0, A] satisfying 0 < (Y+ p < 2A.
Proof. (1) Since 2 JYWJu(x)[~[u(x)~~dx 5 Ilull: + Ilull~~, we have ECU,u) 2 E,(u) + E,(u), which implies (1). (2) By theorem 8.1.7 in Cazenave [5], we have $h@(oN = le”s,v,
w > 0,
: 8,~ E W,
which implies that Z,@(o)) = E,(cp,). Moreover, from the facts that
we get 4
J%(cJ = -------3(a + 1)
3/2
.
(3) For any o > 0, we have Z@(o)) 5 E(q,, 9,) = 2E,(q,,) = 2Z,(A(o)), which together with (1) implies (3). (4) From (l), we have Z(cY,p) + Z(A - 01,A - p> 2 Z&x) + Z,(A - a) + I#)
+ Z,(lt - p).
Moreover, from (2), we have Z,(A) < II(a) + Z,(A - a) for any 01E (0, A). Thus, from (3), we have Z(A) < Z(cr,8) f Z(A - (;Y,A - j3) for any (YE (0, A) and p E (0, A). In other cases, for example, when (Y = 0 and p > 0, since Z(0, p) = Z,(P) i = -j 2 0,
1 a2 0-4 B3,
if a > 0, if-l
Coupled
nonlinear
SchrBdinger
931
equations
we have Z(0, /?) > Zr(p). Thus, we obtain Z(A) = 21,(A) I Z&3) + ZI(A - p> + Z,(A) < Z(0, /3) + I(& A - p). The other cases can be proved similarly.
n
LEMMA 2.4. Let a > -1 and A > 0. If a sequence ((uj, uj)) c H’Q’R) x H’(lR) verifies Ilujlliz -+ A, llu$2 + A and E(uj, ~j) + Z(A), then there exists a subsequence ](Uj,, Uj,)) of ((Uj’ vj)] and a family 1~~~~1 c IR such that
(ryj,uj'3 ryj, vjp)+
in H’(R) x H’(R)
(9’9 $1
for some (f$, (p2) E s(A). Proof. The proof of lemma 2.4 relies on the concentration compactness principle introduced by Lions [14, 151. We have already proved the key inequality, the strictly subadditive condition, in lemma 2.3 (4). Thus, lemma 2.4 can be proved in the same way as in Section IV of 1151. n Proof of proposition 2.1. We prove by contradiction. If the set s(A) is not stable, then there exist an so > 0 and a sequence ((Q, voj)j c ZZ’(R) x ZZ1(R) such that
inf
iII%j - PIIlffl + II”f)j- P211ff11 + 0,
(2.1)
(d,P2) ES(A)
sup tEiR(s,,J$EfsIh) tll”jCt) -
PIIIH’+ IIvj(t) - u1211H11 2 &03
(2.2)
where (uj(t), Uj(t)) is a solution of (1.1) with (uj(t), vj(t)) = (u,,~, voj). By continuity (2.2), we can take tj E R such that
(co, ~fso tIl”j(‘j) - P1llH’+ llvj(tj) - (p211fflJ = Eg,
in
t
and (2.3)
From (2.1) and the conservation laws (1.5) and (1.6), we have
II”jCtj)llL2 = ll”Ojll~2 + A9 E(“j(tj),
uj>
=
IIvjCtj>II~2 = II”Ojll~2 + Av 9 uOj)
-Q”Oj
+ z(A)-
From (2.4), (2.5) and lemma 2.4, there exist a subsequence and a family (yj,) C IR such that (ry,.uj’(tj*)3
ry,r vj’(tj’))
+ (u7’, V2)
for some (cp’, q2) E S(A). However, since (r-,,$, completes the proof of proposition 2.1. n Next, we give the proof of proposition Ke’“qv,,
e’“qp,)
Thus, in order to prove proposition
(2.4)
((ujr(tjr),
(2.5) vj,(c,))] of
((uj(tj),
uj($))]
in N’(R) x N’(R)
rpY,,p2) E s(A), this contradicts
(2.3). This
2.2. From lemma 2.3 (3), we have
: ~,P,Y E RI C S@(w)>,
w > 0.
2.2, we may show that 0 > 0.
(2.6)
938
M. OHTA
LEMMA 2.5. Let a > -1 and A > 0. If (u, u) E S(A), then ]u(x)j = IV(X)/ holds for any x E R. Proof. Let (u, u) E $&I). Then, it follows from lemma 2.3 that
Z(A) = E(u, u) 2 E,(u) + E,(u) 2 21&I) = Z(A). Thus, we have
which implies that
Hence, we obtain that Iu(x)] = Iu(x)1 for any x E R. n 2.2. As stated above, we may show (2.6). Let (u, II) E S@(o)). Then, there exists a Lagrange multiplier (q , w2) E lR2such that
Proof of proposition
- (alu12 + lu12)u = 0,
XE R,
+ 02u - (lu12 + alu12)u = 0,
x E R.
dx2 + qu
(2.7) z
From lemma 2.5, we have ]u(x)l = Iu(x)] for any x E R, so that from (2.7) we have d2u 2 + WlU - (a + l)lU~2U = 0, dx
XE R,
-- d2u dx2 + 02u - (a + l)]u12u = 0,
x E R.
Again by theorem 8.1.7 in Cazenave [5], we get or, o2 > 0, and there exist real constants (Y,j?,y, and y2 such that ia u = ei8rysy, fp,, . u = e ~y,~,,~ Since ]u(x)l = 1u(x)] f or any x E iR and ]]ul& = ]]u]l~~= n(w), it follows from (2.8) and (2.9) that y, = y2 and o = o, = 02. The proof is completed. n Remark 2.6. When a > 0, the system (1.1) has other solitary wave solutions (e’“‘W,(x), 0) and
(0, eiofly,(x)), where o > 0 and y,(x) =
2w a sech&x.
We can prove that these solitary waves are orbitally stable in the same way as in the proof of theorem 1.1. Acknowledgements-The author would like to express his deep gratitude to Professor Yoshio Tsutsumi for his helpful advice. The author would also like to thank Mr. Ken-Ichi Nakamura and Mr. Tetsu Mizumachi for their fruitful discussions.
Coupled nonlinear
Schrodinger equations
939
REFERENCES 1. NEWBOULT G. K., PARKER D. F. & FAULKNER T. R., Coupled nonlinear Schrodinger equations arising in the study of monomode step-index optical fibers, J. math. Phys. 30, 930-936 (1989). 2. WADATI M., IIZUKA T. & HISAKADO M., A coupled nonlinear Schrodinger equation and optical solitons, J. phys. Sot. Japan 61, 2241-2245 (1992). 3. GRILLAKIS M., SHATAH J. & STRAUSS W. A., Stability theory of solitary waves in the presence of symmetry, II, J. funct. Analysis 94, 308-348 (1990). 4. STUBBE J. & VAZQUEZ L., A nonlinear Schrodinger equation with magnetic field effect: solitary waves with Math. 46, 493-499 (1989). internal rotation, Portugaliae 5. CAZENAVE T., An introduction to nonlinear SchrGdinger equations, Textos de Metodos Matematicos, Vol. 22. Instituto de Matematica-UFRJ, Rio de Janeiro (1989). 6. BERESTYCKI H. & CAZENAVE T., Instabilite des etats stationnaires dans les Cquations de Schrodinger et de Klein-Gordon non lintaires, C. r. Acud. Sci. Paris 293, 489-492 (1981). 7. CAZENAVE T. & LIONS P.-L., Orbital stability of standing waves for some nonlinear Schrodinger equations, COT?lmU?lS math. Phys. 85, 549-561 (1982). 8. GRILLAKIS M., SHATAH J. & STRAUSS W. A., Stability theory of solitary waves in the presence of symmetry, I, J. funct. Analysis 74, 160-197 (1987). 9. ILIEV I. D. & KIRCHEV K. P., Stability and instability of solitary waves for one-dimensional singular Schrodinger equations, Diff. Integral Eqns 6, 685-703 (1993). 10. SHATAH J., Stable standing waves of nonlinear Klein-Gordon eauations. Communs math. Phvs. 91. 313-327 (1983). 11. SHATAH J. & STRAUSS W. A., Instability of nonlinear bound states, Communs math. Phys. 100, 173-190 (1985). 12. WEINSTEIN M. I., Nonlinear Schrodinger equations and sharp interpolation estimates, Communs math. Phys. 87, 567-576 (1983). 13. WEINSTEIN M. I., Lyapunov stability of ground states of nonlinear dispersive evolution equations, Communs pure uppl. Math. 39, 51-68 (1986). 14. LIONS P. L., The concentration-compactness principle in the calculus of variations. The locally compactness, part 1, Ann. Inst. H. Poincare’ Analysis non lindaire 1, 109-145 (1984). 15. LIONS P. L., The concentration-compactness principle in the calculus of variations. The locally compactness, part 2, Ann. Inst. H. PoincarP Analysis non linkaire 1, 223-283 (1984).
Analysis.
Theory,
Methods
&Applications.
Vol. 26, No. 5. pp. 933-939. 1996 Copyright Q 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X196 $15.00+ .O!l
Pergamon 0362-546X(94)00340-8
STABILITY
OF SOLITARY WAVES FOR COUPLED SCHRijDINGER EQUATIONS
NONLINEAR
MASAHITO OHTA Department of Mathematical Sciences, University of Tokyo, Hongo, Tokyo 113, Japan (Received 25 April
1994; received for publication
17 November 1994)
Key words and phrases: Coupled nonlinear Schrtidinger equations, orbital stability, solitary waves,
concentration-compactness. 1. INTRODUCTION
AND RESULT
In the present paper we consider the stability of solitary waves for the following coupled nonlinear Schrodinger equations
(1.1)
where u and u are complex valued functions of (t, x) E R2, and a is a real constant. The system (1.1) describesthe nonlinear modulations of two monochromatic waves, whose group velocities are almost equal, in optical fibers (see [l, 2]), and it is regarded as a natural extension of the following single nonlinear Schrodinger equation (1.2)
When a > - 1, (1.1) has solitary wave solutions with the form (see [ 1, 21) (u,(t, x), u,(t, x)) = (eicw-c2)c+ic++i~9w(x - 2ct), ei(d-c2)f+icx+i’fpw(x- 2ct)),
(1.3)
where o > 0, c, (Yand P are real constants, and 9,(x) =
20 sech6x. J- a+1
We note that 9, satisfies -g
+ oa, - (a + 1)19,129= 0,
XE R, (1.4w)
i L
9(x) = 9(-x),
x E R
lim 9(x) = 0. x+_+s,
Since (1.1) is invariant under the Galilean transformations W, 4, W, x)) ++(e -ic*f+iqf,
x _ zct), 933
e -ic2r+icxU(f,
x
-
2ct)),
c E
IT?,
934
M. OHTA
and the phase transformations (u(t, x), u(t, x)) - (e’“u(t, x), e’“u(t, x)),
a, P E R
we may consider the case when c = (Y = /I = 0 in (1.3) only. Thus, in what follows, we put (4 U, 4, u, (t, 4) = (eiofpo(xl, e’“‘v, (x)1. We now state our main result. THEOREM1.l. Let a > - 1. Then, for any o > 0, the solitary wave solution (u,(t), v,(t)) of (1.1) is orbitally stable in the following sense: for any E > 0 there exists a 6 > 0 such that if No 9uO)E H’(m) x H’(R) verifies lb0 - V)“llH’ + lb0 - V)Af1 < 6, then the solution (u(t),‘u(i)) of (1.1) with (u(O), u(0)) = (u,, u,J satisfies sup inf (Ilu(t) - eio’ryV7,(INL+ IIu(t) - eip~yyvI,llH~I< E, t E:IKci.6.y EIR where T,,u(x) = v(x - y) and H’(lR) is the usual Sobolev space. Remark 1.2. When a = 1, (1.1) is also invariant under the rotations in (u, v)
(u(t*xl, et, 9) - w, xl, w, 4)
cos 6 sin 6
( _ sin6 cos()>
9
6 E I?.
Due to this additional symmetry, when a = 1, one can also discuss the stability of solitary waves based on the study of a linearized operator (seeGrillakis et al. [3], and Stubbe and VAzquez [4]). However, when a # 1, since (1.1) is not invariant under the rotations in (u, u), the situation is quite different from that of a = 1. Remark 1.3. It is known that for any (u,,, u,J E H’(lR) x H’(R) there exists a unique solution
(u(t), u(t)) of (1.1) with (u(O), u(O)) = ( uO, uO)in C(lR; H’(R) x H’(Q), the conservation law Ilw)llL~ = Il%h2, s40,
and (u(t), u(t)) satisfies (1.5)
llwllL~ = Il%IILb
u(t)) = JY%, uo),
(1.6)
for any t E IR(see, e.g. Chapter 4 in Cazenave [5]), where
Recently, many authors have been studying the problem of stability and instability of solitary waves for the single nonlinear Schrodinger equations like (1.2) (see, e.g. [6-131). In the next section, following Cazenave and Lions [7], we will consider the minimization problem 1(A) = inf(E(u, u) : U, u E H’(R),
Ilu@ = Ilull$ = d),
A > 0,
(1.7A)
Coupled nonlinear SchrBdinger equations
935
and we will show that the set s(A) of minimizers for (1.7A) is stable for any A > 0. Moreover, we will prove that W(o))
w > 0,
= ((e’“r,v, , eiPr,pJ : a, P, y E W,
where A(o) = II~,II~z = 4&/(a + 1). In this paper, we denote the norms of the Lebesgue space Lp(lR) and the Sobolev space ff’(Q by II-lb and II-lh, respectively. We consider L’(R) as a real Hilbert space with the inner product OD (u, ~9~2= Re u(x)u(x) dx. s -m We also regard H’(lR) and H’(fR) x H’(R) as real Hilbert spacessimilarly. 2. PROOF
OF THEOREM
1.1
In what follows, we use the following notations
E(u,u) = ~~;~I:,+ ll$
- ;(lluss + II43 - [rm lz.4m4x)12~~
2
E,(u)
=
/III
- q
2
II&b
L2
au 2 E2W =IIG L2 II- ; Ibll:~9
1(A, p) = inf{E(u, v) : 2.4,u E H’(m), ((2.4@= A, Ilull~2 Zj(13)= inf{Ej(u) : u E H’(R), IIuII:~ = A],
I(A) = &I, A),
=
p),
j = 1,2,
A(w) = -+,
$(A) = ((u, u) E H'(R) x H'(IR) :1(A) = E(u, u), llult:z = llujj;z = A), $jl(A) = (u E H'(IR):Z,(13) = E,(u),
~~24~~2
=
A).
We note that E(u, 4 = W(u),
E(u,O) = E(0, u) = E,(u),
lIdI
= +I.
We split the proof of theorem 1.1 into the following two steps. 2.1. Let a > - 1. Then, for any A > 0, the set s(A) is stable in the following sense: for any E > 0 there exists a 6 > 0 such that if (uO, uO)E H’(R) X H’(R) verifies
PROPOSITION
inf < 6, ~Iluo- dllHl + II%- (0211H~I (co’, lo3ES(V then the solution (u(t), u(f)) of (1.1) with (u(O), v(O)) = (uo, u,) satisfies
M. OHTA
936
PROPOSITION
2.2. Let a > -1. Then, the relation S@(w)) = l(e’“~,~,, , e’“q,)
: a, A y E RI
holds for any w > 0. Theorem 1.1 follows from propositions 2.1 and 2.2 immediately. In order to prove propositions 2.1 and 2.2, we make some preparations. LEMMA 2.3. Let a > -1. Then, the following hold:
(1) m, Pu)2 Zl(4 + Z,(P), 2, v 2 0; (2) Z1(A(w)) = E,(q?,) = -&
03j2 = - f
2 A3(o),
9
0
> 0;
( > (3) 84 = 21,(A), 2 > 0, and (vu, co,) E 6@(W, o > 0; (4) Z(A) < Z(((Y, j?) + Z(A - (Y,1 - p) for any (YE [0, A] and p E [0, A] satisfying 0 < (Y+ p < 2A.
Proof. (1) Since 2 JYWJu(x)[~[u(x)~~dx 5 Ilull: + Ilull~~, we have ECU,u) 2 E,(u) + E,(u), which implies (1). (2) By theorem 8.1.7 in Cazenave [5], we have $h@(oN = le”s,v,
w > 0,
: 8,~ E W,
which implies that Z,@(o)) = E,(cp,). Moreover, from the facts that
we get 4
J%(cJ = -------3(a + 1)
3/2
.
(3) For any o > 0, we have Z@(o)) 5 E(q,, 9,) = 2E,(q,,) = 2Z,(A(o)), which together with (1) implies (3). (4) From (l), we have Z(cY,p) + Z(A - 01,A - p> 2 Z&x) + Z,(A - a) + I#)
+ Z,(lt - p).
Moreover, from (2), we have Z,(A) < II(a) + Z,(A - a) for any 01E (0, A). Thus, from (3), we have Z(A) < Z(cr,8) f Z(A - (;Y,A - j3) for any (YE (0, A) and p E (0, A). In other cases, for example, when (Y = 0 and p > 0, since Z(0, p) = Z,(P) i = -j 2 0,
1 a2 0-4 B3,
if a > 0, if-l
Coupled
nonlinear
SchrBdinger
931
equations
we have Z(0, /?) > Zr(p). Thus, we obtain Z(A) = 21,(A) I Z&3) + ZI(A - p> + Z,(A) < Z(0, /3) + I(& A - p). The other cases can be proved similarly.
n
LEMMA 2.4. Let a > -1 and A > 0. If a sequence ((uj, uj)) c H’Q’R) x H’(lR) verifies Ilujlliz -+ A, llu$2 + A and E(uj, ~j) + Z(A), then there exists a subsequence ](Uj,, Uj,)) of ((Uj’ vj)] and a family 1~~~~1 c IR such that
(ryj,uj'3 ryj, vjp)+
in H’(R) x H’(R)
(9’9 $1
for some (f$, (p2) E s(A). Proof. The proof of lemma 2.4 relies on the concentration compactness principle introduced by Lions [14, 151. We have already proved the key inequality, the strictly subadditive condition, in lemma 2.3 (4). Thus, lemma 2.4 can be proved in the same way as in Section IV of 1151. n Proof of proposition 2.1. We prove by contradiction. If the set s(A) is not stable, then there exist an so > 0 and a sequence ((Q, voj)j c ZZ’(R) x ZZ1(R) such that
inf
iII%j - PIIlffl + II”f)j- P211ff11 + 0,
(2.1)
(d,P2) ES(A)
sup tEiR(s,,J$EfsIh) tll”jCt) -
PIIIH’+ IIvj(t) - u1211H11 2 &03
(2.2)
where (uj(t), Uj(t)) is a solution of (1.1) with (uj(t), vj(t)) = (u,,~, voj). By continuity (2.2), we can take tj E R such that
(co, ~fso tIl”j(‘j) - P1llH’+ llvj(tj) - (p211fflJ = Eg,
in
t
and (2.3)
From (2.1) and the conservation laws (1.5) and (1.6), we have
II”jCtj)llL2 = ll”Ojll~2 + A9 E(“j(tj),
uj
=
IIvjCtj>II~2 = II”Ojll~2 + Av 9 uOj)
-Q”Oj
+ z(A)-
From (2.4), (2.5) and lemma 2.4, there exist a subsequence and a family (yj,) C IR such that (ry,.uj’(tj*)3
ry,r vj’(tj’))
+ (u7’, V2)
for some (cp’, q2) E S(A). However, since (r-,,$, completes the proof of proposition 2.1. n Next, we give the proof of proposition Ke’“qv,,
e’“qp,)
Thus, in order to prove proposition
(2.4)
((ujr(tjr),
(2.5) vj,(c,))] of
((uj(tj),
uj($))]
in N’(R) x N’(R)
rpY,,p2) E s(A), this contradicts
(2.3). This
2.2. From lemma 2.3 (3), we have
: ~,P,Y E RI C S@(w)>,
w > 0.
2.2, we may show that 0 > 0.
(2.6)
938
M. OHTA
LEMMA 2.5. Let a > -1 and A > 0. If (u, u) E S(A), then ]u(x)j = IV(X)/ holds for any x E R. Proof. Let (u, u) E $&I). Then, it follows from lemma 2.3 that
Z(A) = E(u, u) 2 E,(u) + E,(u) 2 21&I) = Z(A). Thus, we have
which implies that
Hence, we obtain that Iu(x)] = Iu(x)1 for any x E R. n 2.2. As stated above, we may show (2.6). Let (u, II) E S@(o)). Then, there exists a Lagrange multiplier (q , w2) E lR2such that
Proof of proposition
- (alu12 + lu12)u = 0,
XE R,
+ 02u - (lu12 + alu12)u = 0,
x E R.
dx2 + qu
(2.7) z
From lemma 2.5, we have ]u(x)l = Iu(x)] for any x E R, so that from (2.7) we have d2u 2 + WlU - (a + l)lU~2U = 0, dx
XE R,
-- d2u dx2 + 02u - (a + l)]u12u = 0,
x E R.
Again by theorem 8.1.7 in Cazenave [5], we get or, o2 > 0, and there exist real constants (Y,j?,y, and y2 such that ia u = ei8rysy, fp,, . u = e ~y,~,,~ Since ]u(x)l = 1u(x)] f or any x E iR and ]]ul& = ]]u]l~~= n(w), it follows from (2.8) and (2.9) that y, = y2 and o = o, = 02. The proof is completed. n Remark 2.6. When a > 0, the system (1.1) has other solitary wave solutions (e’“‘W,(x), 0) and
(0, eiofly,(x)), where o > 0 and y,(x) =
2w a sech&x.
We can prove that these solitary waves are orbitally stable in the same way as in the proof of theorem 1.1. Acknowledgements-The author would like to express his deep gratitude to Professor Yoshio Tsutsumi for his helpful advice. The author would also like to thank Mr. Ken-Ichi Nakamura and Mr. Tetsu Mizumachi for their fruitful discussions.
Coupled nonlinear
Schrodinger equations
939
REFERENCES 1. NEWBOULT G. K., PARKER D. F. & FAULKNER T. R., Coupled nonlinear Schrodinger equations arising in the study of monomode step-index optical fibers, J. math. Phys. 30, 930-936 (1989). 2. WADATI M., IIZUKA T. & HISAKADO M., A coupled nonlinear Schrodinger equation and optical solitons, J. phys. Sot. Japan 61, 2241-2245 (1992). 3. GRILLAKIS M., SHATAH J. & STRAUSS W. A., Stability theory of solitary waves in the presence of symmetry, II, J. funct. Analysis 94, 308-348 (1990). 4. STUBBE J. & VAZQUEZ L., A nonlinear Schrodinger equation with magnetic field effect: solitary waves with Math. 46, 493-499 (1989). internal rotation, Portugaliae 5. CAZENAVE T., An introduction to nonlinear SchrGdinger equations, Textos de Metodos Matematicos, Vol. 22. Instituto de Matematica-UFRJ, Rio de Janeiro (1989). 6. BERESTYCKI H. & CAZENAVE T., Instabilite des etats stationnaires dans les Cquations de Schrodinger et de Klein-Gordon non lintaires, C. r. Acud. Sci. Paris 293, 489-492 (1981). 7. CAZENAVE T. & LIONS P.-L., Orbital stability of standing waves for some nonlinear Schrodinger equations, COT?lmU?lS math. Phys. 85, 549-561 (1982). 8. GRILLAKIS M., SHATAH J. & STRAUSS W. A., Stability theory of solitary waves in the presence of symmetry, I, J. funct. Analysis 74, 160-197 (1987). 9. ILIEV I. D. & KIRCHEV K. P., Stability and instability of solitary waves for one-dimensional singular Schrodinger equations, Diff. Integral Eqns 6, 685-703 (1993). 10. SHATAH J., Stable standing waves of nonlinear Klein-Gordon eauations. Communs math. Phvs. 91. 313-327 (1983). 11. SHATAH J. & STRAUSS W. A., Instability of nonlinear bound states, Communs math. Phys. 100, 173-190 (1985). 12. WEINSTEIN M. I., Nonlinear Schrodinger equations and sharp interpolation estimates, Communs math. Phys. 87, 567-576 (1983). 13. WEINSTEIN M. I., Lyapunov stability of ground states of nonlinear dispersive evolution equations, Communs pure uppl. Math. 39, 51-68 (1986). 14. LIONS P. L., The concentration-compactness principle in the calculus of variations. The locally compactness, part 1, Ann. Inst. H. Poincare’ Analysis non lindaire 1, 109-145 (1984). 15. LIONS P. L., The concentration-compactness principle in the calculus of variations. The locally compactness, part 2, Ann. Inst. H. PoincarP Analysis non linkaire 1, 223-283 (1984).