- Email: [email protected]
Orbital stability of solitary waves of the long wave-short wave resonance equations
GUO,
et d.:Orbitd
Stability
of . . .
37
Orbital Stability of Solitary Waves of The Long Wave-Short Wave Resonance Equations 1 Boliig GUO & Lin CHEN (Institute of Applied Physics-and Computational Mathematics, P. 0. Box 8009, Beijing 100088. Chiia ) Abstract: This paper concerns the orbital stability for solitary waves of the long waveshort wave resonance equations. By using a different method from [15], applying the abstract results of Grillakis et al. [8][9] and detailed spectral analysis, we obtain the necessary and sufhcient condition for the stability of the solitary waves. Key words: Solitary wave, stability, Long wave-short wave resonance equations. AMS subject classifications : 35Q55; 35B35.
Introduction Long wave-short wave resonance is a special case of three wave resonances which can occur when the second order nonlinearity arises in the process. Benney [l] presents a general theory for deriving nonlinear partial differential equations which permit both long and short wave solutions. One of the equations proposcd by him is of the form
where a, X, ,LL,v E R with Xpv ;Ic 0 and n real function and e complex function. e is the envelope of the short wave, while n is the amplitude of long wave. It is easy to see that by rescaliig of 2, t. one can chang (1) into
where a E R. The system of (1) arises in the study of surface waves with both gravity and capillary in action (see [2,3]) and also in plasma physics (see [4]). When a = 0, Ma [17] found that (2) can be rewritten in Lax’s formulation and solved its Cauchy problem for them by the inverse scattering method, and Ph. Laurencot [15] confirmed that the solitary-wave solution of (2) is stable. The local and global existence of inital value problem for (2) was considered in [5-71. In this paper, we consider the stability of solitary waves of (2). By applying the abstract theory of Grillakis et al. [S-9] and detailed spectral analysis, we obtain the orbital stability of solitary waves. For the other types of equations, such as nonlinear Schr&linger equation, KdV equation and BO equation, the orbital stability of solitary waves were considered in [lO-141. This paper is organized as follows: in section 1, we state the results of the existence of solitary waves, in section 2, we state the assumptions and the stability results. ‘The article =a.~received
OIL Apr.
1,1996
38
in Nonliaca~
Conummicationw
1.
The existence
Science
b Numcricd
of solitary
Simulation
Vd.1.
NoLs(Jul.
1~96)
waves
Consider the following system
(3) Let E(t, 2) = e -iwteig(r-st)-
Gl,w
(x
-
vt),
n(t, 2) = %/,w(x - vt)
(4)
(5)
be the solitary waves of (3), where w, q, u are real numbers, &,,a and n,.,Tware real functions. then we have
nw.,(x) = fisech*(
(6)
9x)
Q = %. Finally, we have Theorem
1 . For any real constanta CO
then (6).
2.
e&t
solitarg
Main
w, v, o satisfying
=4w+v2<0,
v>O
waves of (8) in the fom
and
1-av>O,
(7)
with n,., , i,.. , q, v, w and a satisfying
of (d)-(5),
results
Let u’ = (r, n). The function space in which we shall work is X = H&,+,,(R) with real inner product (cl,
tii)
= Be
pna
+ ElZ2
+ erZ2t)d=
x I&,,(R),
(8)
I
The dual space of X ia X’ I:X-rX’defrnedby
= H&,,,,(R)
<
IUi,U<
x L&,,(R),
>=
(Ui,
there is a natural
Ui),
isomorphism
(9)
where < a, +) denotes the pairing between X and X’. < f, ii >= Re By (9)-(W), it is obvious I=
(fin + f2Z)dz.
(10)
GUO. et al.:Orbital
Let Ti, T2 be one-parameter
Stability
of..
.
39
groups of unitary operator on X defined by
Tl(sl)u’(.)
= ii(. - 31)
T2(s2)ii(.) = (e(++
for u’(s) E X,
sr E R,
, n(a)) for C(a) E X,
(11)
s2 E R.
(12)
Obviously
,
-i Tao) =i 1. 0
It follows from Theorem 1 and (3) that there exist (G.~(x), n,,,(x)) of (3). with L~.L~ defined by
solitary
waves
Tl(ut)Tz(wt)
(13)
Let
GAv(x) = e’*‘2,,,(x). (15) In this and the following sections, we shall consider the orbital stability of solitary waves T~(~t)T~(wt)O~,,(z) of (3). Note that equation (3) is invariant under TX(.) and Tz(.), we define the orbital stability as follows: Deibition: The solitary wave T~(at)Tz(wt)&,+,( t ) is orbitally stable if for all E > 0 there exists 6 > 0 with the following property. If IlG - *(lx < 6 and u’(t) is a solution of (3) in some interval [O, to) with u’(O) L i;o, then C(t) can be continued to a solution in O
inf Ilii(t) - T~(s~)T~(s~)~~~x < e.
II<:;!& 8lERsaER
(16)
Otherwise Tl(vt)T2(wt)3,,,(z) is called orbitally unstable. So long as w , v are fixed we write E, i. n for eWw,o, i,,. , n,., . Define (17) It is easy to verify that E(C) is invaraint the flow of (3). Namely E(TI(~I)T~(s~)u~ and for any t E R, u(t) ia a flow of (3)
under Tl and T2, and formally
= E(G),
for
any
sr,as E R,
conserved under
(18)
40
Cdcatiopr
in Nonlinear
Science
k N
E(qt))
+
Simulation
1
Vol.1.
= E(ii(0)).
with a skew-symmetrically
mm)
(19)
Note that equation (3) can be written as the following Hamiltonian g
No.s(Jui.
system
= JE’(u’)
(20)
linear operator J defined by
J=(--;
2g)
iw
(
and S’(C) is the F’rechet derivative of E(C). As in [S-9], we define
B1 =
7
such that T;(O) = JB1,
such that q(O) = JB2, Ol(u3
1 < B1ii,u’ >= -; 5
=
Qz(u’)
J
n’dz + $Im I,( c,E)dz,
J
1 2=;
=
(22)
R
/aladz.
(23)
R
As in [8-g], by (13)-( 19), ‘we can prove that 91(Z(s1)T2(82)C)
=
QI($
= Q2(%
Qi(WdTz(32)~)
(24)
for any 81,82 E R. And for any t E R, C(t) is a flow of (3)
QdW
=
QdW)),
Qz(W)
= Q2(G(O)).
(25)
Furthermore E’(@,,,)
- vQ:(%,w)
- wQ:(@w,v) = 0,
where E’, Qi, Qi are the Fbechet derivatives of E, 81 ad
E’(+
(
-c=z+;,;a’e’2r)
92,
with
,
(26)
GUO,
et d.:Orbitd
Stability
of..
41
.
Define an operator from X to X’ H wtn = E”(Ou.,) with 3 = ($1,
$2)
- uQ’l’(RAts) - 4’:@&dJ,
(27)
*:
E X7 and
Observe that H,,, is self-adjoint in the sense that H:,a = El,,,. This means that I-lH,*, is a bounded self-adjoint operator on X. The ‘spectrum’ of HW,,, consists of the real numbers X such that H,,, - XI is not invertible. We claim that X = 0 belongs to the spectrum of H w,v * By (18) (24)(26) and (27), it is easy to prove that Hw,,T’,(OPw..b)
= 0.
(29)
Hw.JW)%,.,(z) = o.
(30)
Let 2 = {k1T:(0)@,&)
+ knT:(O)‘&.,jz)
!‘kl. k2 E R}.
t31j
By (29) and (30), 2 is contained in the kernel of HvI.,. Assumption 1. (Spectral decomposition of H,.,) The space X is decomposed as a direct sum X=N+Z+P
f32)
where Z is defined above, N is a finite-dimensional < H,+,.,ti. ii><
0
subspace such that
for
Of ii~ N
(33)
ti>_> 6lliiIIg
for u’~ P
(34)
and P is a closed subspace such that < H,,,ii,
with some constant 15> 0 independent of ii. We define d(w, u) : R x R 4 R by (35) and define ,Y’(w, v) to be the Hessian of function d. It is a symmetric bilinear form. Before stating the theorem for the initial-value problem for (3) (see [7]), we define the function spaces XT by X;
= {u : [0, TJ x R + Clu E C([O, Tj; H3’?, %, us, u,, E L’=(R; L2[0, TJ)}
Theorem 2 . For any eQ E H312(R) and no E HI, then eksts a unique function [0, T] x R -+ C sat&lying (S), and such that r E .X$ and n E L2 for all T > 0.
E:
Let us now state our main results about stability of solitary wave T~(vt)T2(wt)+,,,(z). Theorem 3 . Under the condition of Theorem 1. the soliiary waves Ti(ut) Tz(Wt) cP,,D.(z) of (3) with the ezpression (13)-(15) ape orbitally stable if and only ifw < -.$f$$$ .
References
42
Cdcationn
in Nonlinear
!kimce
& Numorkd
Simulation
Vol.1,
No.B(Jd.
lQQ6)
[l] D. J. Benny, Stud. appl. Math. 55, 93 (1976). [2] V. D. Djordjevic and L. G. Redekopp, J. Fluid Mech. 78. 703-714 (1977). [3] D. J. Benny, Stud. appl. Math. 56, 81-94 (1977). [4] L. Vega, Proc. Am. math. Sot. 102, 874878 (1988). [5] Guo BoIing,J. Math. Res..Exposition.No.l, 69-76 (1987). [S] Guo Bohng and Pan xiude, Chin. Phys. Lett.,7(6), 241 (1990). [7] M. Tautsumi and S. Hatano, Nonlinear AnaIysis.TMA. Vol 22. No.2, 155-171 (1994). [B] M. Grillakis, J. Shatah aud W. Strauss, J. Funct. Anal, 74. 160-197 (1987). [9] M. GriUakis, J. Shatah, and W. St&m, J.Funct. Anal. 94. 308-348 (1990). [lo] Guo BoIing and Wu Yaping, preprint. [ll] J. Shatah and W. Strauss, Comm. Math. Phys., 100. 173-190 (1985). [12] J. L. Bona, P. E. Souganidia and W. A. Strau3a, Proc. R. Sot. Lond. A 411. 395-412 (1987). i13] J. P. Albert, J. L. Bona and D. B. Henry, Physica D. 24. 343-366 (1987). [14] J. P. Albert and J. L. Bona, IMA J. Appl. Math. 46. 1-19 (1991). [15] PH. Laurencot, Nonlinear Analysis. TMA. Voi 24. No 4. 509527 (1995). 1161M. Reed and B. Simon,Fourier analysis. Self-Adjointness. 1975. [17] Ma Yan-Chow, Stud. appl. Math. 50. 201-221 (1978).
Numerical Eqation ’
Simulation
of Weakly
Damped
force KdV
Lixin TIAN & NaiIong GUO (Department of Mathmatics and Physics,Jiangsu University of Science and technology, Zhcrijiang 212013, Jiangsu, PXChina.) Abstract: In thie paper we set approximate inertial manifold of weakIy damped forced KdV equation and get the numerical simulation . The result using the method is as same as the numerical simuiation. Key Words: appraxima te inertial manifold, weakly dampled forced KdV equation, se&adjoint operator
Introduction One of the mcmt important’and the dynamical anaIy& of s-time ‘Thepaperrrsrreai*sdonJltno.
interesting subjects in the field of nonlinear science is system. The etudy of this subject is developing along
13,lQQe
et d.:Orbitd
Stability
of . . .
37
Orbital Stability of Solitary Waves of The Long Wave-Short Wave Resonance Equations 1 Boliig GUO & Lin CHEN (Institute of Applied Physics-and Computational Mathematics, P. 0. Box 8009, Beijing 100088. Chiia ) Abstract: This paper concerns the orbital stability for solitary waves of the long waveshort wave resonance equations. By using a different method from [15], applying the abstract results of Grillakis et al. [8][9] and detailed spectral analysis, we obtain the necessary and sufhcient condition for the stability of the solitary waves. Key words: Solitary wave, stability, Long wave-short wave resonance equations. AMS subject classifications : 35Q55; 35B35.
Introduction Long wave-short wave resonance is a special case of three wave resonances which can occur when the second order nonlinearity arises in the process. Benney [l] presents a general theory for deriving nonlinear partial differential equations which permit both long and short wave solutions. One of the equations proposcd by him is of the form
where a, X, ,LL,v E R with Xpv ;Ic 0 and n real function and e complex function. e is the envelope of the short wave, while n is the amplitude of long wave. It is easy to see that by rescaliig of 2, t. one can chang (1) into
where a E R. The system of (1) arises in the study of surface waves with both gravity and capillary in action (see [2,3]) and also in plasma physics (see [4]). When a = 0, Ma [17] found that (2) can be rewritten in Lax’s formulation and solved its Cauchy problem for them by the inverse scattering method, and Ph. Laurencot [15] confirmed that the solitary-wave solution of (2) is stable. The local and global existence of inital value problem for (2) was considered in [5-71. In this paper, we consider the stability of solitary waves of (2). By applying the abstract theory of Grillakis et al. [S-9] and detailed spectral analysis, we obtain the orbital stability of solitary waves. For the other types of equations, such as nonlinear Schr&linger equation, KdV equation and BO equation, the orbital stability of solitary waves were considered in [lO-141. This paper is organized as follows: in section 1, we state the results of the existence of solitary waves, in section 2, we state the assumptions and the stability results. ‘The article =a.~received
OIL Apr.
1,1996
38
in Nonliaca~
Conummicationw
1.
The existence
Science
b Numcricd
of solitary
Simulation
Vd.1.
NoLs(Jul.
1~96)
waves
Consider the following system
(3) Let E(t, 2) = e -iwteig(r-st)-
Gl,w
(x
-
vt),
n(t, 2) = %/,w(x - vt)
(4)
(5)
be the solitary waves of (3), where w, q, u are real numbers, &,,a and n,.,Tware real functions. then we have
nw.,(x) = fisech*(
(6)
9x)
Q = %. Finally, we have Theorem
1 . For any real constanta CO
then (6).
2.
e&t
solitarg
Main
w, v, o satisfying
=4w+v2<0,
v>O
waves of (8) in the fom
and
1-av>O,
(7)
with n,., , i,.. , q, v, w and a satisfying
of (d)-(5),
results
Let u’ = (r, n). The function space in which we shall work is X = H&,+,,(R) with real inner product (cl,
tii)
= Be
pna
+ ElZ2
+ erZ2t)d=
x I&,,(R),
(8)
I
The dual space of X ia X’ I:X-rX’defrnedby
= H&,,,,(R)
<
IUi,U<
x L&,,(R),
>=
(Ui,
there is a natural
Ui),
isomorphism
(9)
where < a, +) denotes the pairing between X and X’. < f, ii >= Re By (9)-(W), it is obvious I=
(fin + f2Z)dz.
(10)
GUO. et al.:Orbital
Let Ti, T2 be one-parameter
Stability
of..
.
39
groups of unitary operator on X defined by
Tl(sl)u’(.)
= ii(. - 31)
T2(s2)ii(.) = (e(++
for u’(s) E X,
sr E R,
, n(a)) for C(a) E X,
(11)
s2 E R.
(12)
Obviously
,
-i Tao) =i 1. 0
It follows from Theorem 1 and (3) that there exist (G.~(x), n,,,(x)) of (3). with L~.L~ defined by
solitary
waves
Tl(ut)Tz(wt)
(13)
Let
GAv(x) = e’*‘2,,,(x). (15) In this and the following sections, we shall consider the orbital stability of solitary waves T~(~t)T~(wt)O~,,(z) of (3). Note that equation (3) is invariant under TX(.) and Tz(.), we define the orbital stability as follows: Deibition: The solitary wave T~(at)Tz(wt)&,+,( t ) is orbitally stable if for all E > 0 there exists 6 > 0 with the following property. If IlG - *(lx < 6 and u’(t) is a solution of (3) in some interval [O, to) with u’(O) L i;o, then C(t) can be continued to a solution in O
inf Ilii(t) - T~(s~)T~(s~)~~~x < e.
II<:;!& 8lERsaER
(16)
Otherwise Tl(vt)T2(wt)3,,,(z) is called orbitally unstable. So long as w , v are fixed we write E, i. n for eWw,o, i,,. , n,., . Define (17) It is easy to verify that E(C) is invaraint the flow of (3). Namely E(TI(~I)T~(s~)u~ and for any t E R, u(t) ia a flow of (3)
under Tl and T2, and formally
= E(G),
for
any
sr,as E R,
conserved under
(18)
40
Cdcatiopr
in Nonlinear
Science
k N
E(qt))
+
Simulation
1
Vol.1.
= E(ii(0)).
with a skew-symmetrically
mm)
(19)
Note that equation (3) can be written as the following Hamiltonian g
No.s(Jui.
system
= JE’(u’)
(20)
linear operator J defined by
J=(--;
2g)
iw
(
and S’(C) is the F’rechet derivative of E(C). As in [S-9], we define
B1 =
7
such that T;(O) = JB1,
such that q(O) = JB2, Ol(u3
1 < B1ii,u’ >= -; 5
=
Qz(u’)
J
n’dz + $Im I,( c,E)dz,
J
1 2
=
(22)
R
/aladz.
(23)
R
As in [8-g], by (13)-( 19), ‘we can prove that 91(Z(s1)T2(82)C)
=
QI($
= Q2(%
Qi(WdTz(32)~)
(24)
for any 81,82 E R. And for any t E R, C(t) is a flow of (3)
QdW
=
QdW)),
Qz(W)
= Q2(G(O)).
(25)
Furthermore E’(@,,,)
- vQ:(%,w)
- wQ:(@w,v) = 0,
where E’, Qi, Qi are the Fbechet derivatives of E, 81 ad
E’(+
(
-c=z+;,;a’e’2r)
92,
with
,
(26)
GUO,
et d.:Orbitd
Stability
of..
41
.
Define an operator from X to X’ H wtn = E”(Ou.,) with 3 = ($1,
$2)
- uQ’l’(RAts) - 4’:@&dJ,
(27)
*:
E X7 and
Observe that H,,, is self-adjoint in the sense that H:,a = El,,,. This means that I-lH,*, is a bounded self-adjoint operator on X. The ‘spectrum’ of HW,,, consists of the real numbers X such that H,,, - XI is not invertible. We claim that X = 0 belongs to the spectrum of H w,v * By (18) (24)(26) and (27), it is easy to prove that Hw,,T’,(OPw..b)
= 0.
(29)
Hw.JW)%,.,(z) = o.
(30)
Let 2 = {k1T:(0)@,&)
+ knT:(O)‘&.,jz)
!‘kl. k2 E R}.
t31j
By (29) and (30), 2 is contained in the kernel of HvI.,. Assumption 1. (Spectral decomposition of H,.,) The space X is decomposed as a direct sum X=N+Z+P
f32)
where Z is defined above, N is a finite-dimensional < H,+,.,ti. ii><
0
subspace such that
for
Of ii~ N
(33)
ti>_> 6lliiIIg
for u’~ P
(34)
and P is a closed subspace such that < H,,,ii,
with some constant 15> 0 independent of ii. We define d(w, u) : R x R 4 R by (35) and define ,Y’(w, v) to be the Hessian of function d. It is a symmetric bilinear form. Before stating the theorem for the initial-value problem for (3) (see [7]), we define the function spaces XT by X;
= {u : [0, TJ x R + Clu E C([O, Tj; H3’?, %, us, u,, E L’=(R; L2[0, TJ)}
Theorem 2 . For any eQ E H312(R) and no E HI, then eksts a unique function [0, T] x R -+ C sat&lying (S), and such that r E .X$ and n E L2 for all T > 0.
E:
Let us now state our main results about stability of solitary wave T~(vt)T2(wt)+,,,(z). Theorem 3 . Under the condition of Theorem 1. the soliiary waves Ti(ut) Tz(Wt) cP,,D.(z) of (3) with the ezpression (13)-(15) ape orbitally stable if and only ifw < -.$f$$$ .
References
42
Cdcationn
in Nonlinear
!kimce
& Numorkd
Simulation
Vol.1,
No.B(Jd.
lQQ6)
[l] D. J. Benny, Stud. appl. Math. 55, 93 (1976). [2] V. D. Djordjevic and L. G. Redekopp, J. Fluid Mech. 78. 703-714 (1977). [3] D. J. Benny, Stud. appl. Math. 56, 81-94 (1977). [4] L. Vega, Proc. Am. math. Sot. 102, 874878 (1988). [5] Guo BoIing,J. Math. Res..Exposition.No.l, 69-76 (1987). [S] Guo Bohng and Pan xiude, Chin. Phys. Lett.,7(6), 241 (1990). [7] M. Tautsumi and S. Hatano, Nonlinear AnaIysis.TMA. Vol 22. No.2, 155-171 (1994). [B] M. Grillakis, J. Shatah aud W. Strauss, J. Funct. Anal, 74. 160-197 (1987). [9] M. GriUakis, J. Shatah, and W. St&m, J.Funct. Anal. 94. 308-348 (1990). [lo] Guo BoIing and Wu Yaping, preprint. [ll] J. Shatah and W. Strauss, Comm. Math. Phys., 100. 173-190 (1985). [12] J. L. Bona, P. E. Souganidia and W. A. Strau3a, Proc. R. Sot. Lond. A 411. 395-412 (1987). i13] J. P. Albert, J. L. Bona and D. B. Henry, Physica D. 24. 343-366 (1987). [14] J. P. Albert and J. L. Bona, IMA J. Appl. Math. 46. 1-19 (1991). [15] PH. Laurencot, Nonlinear Analysis. TMA. Voi 24. No 4. 509527 (1995). 1161M. Reed and B. Simon,Fourier analysis. Self-Adjointness. 1975. [17] Ma Yan-Chow, Stud. appl. Math. 50. 201-221 (1978).
Numerical Eqation ’
Simulation
of Weakly
Damped
force KdV
Lixin TIAN & NaiIong GUO (Department of Mathmatics and Physics,Jiangsu University of Science and technology, Zhcrijiang 212013, Jiangsu, PXChina.) Abstract: In thie paper we set approximate inertial manifold of weakIy damped forced KdV equation and get the numerical simulation . The result using the method is as same as the numerical simuiation. Key Words: appraxima te inertial manifold, weakly dampled forced KdV equation, se&adjoint operator
Introduction One of the mcmt important’and the dynamical anaIy& of s-time ‘Thepaperrrsrreai*sdonJltno.
interesting subjects in the field of nonlinear science is system. The etudy of this subject is developing along
13,lQQe