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The cubic nonlinear Schrödinger equation in a fluid-filled elastic tube
6 January 1997
PHYSICS
LETTERS
A
Physics Letters A 224(1997) 154-158
EISEWER
The cubic nonlinear Schredinger equation in a fluid-filled elastic tube Wen-shan Duan a*b,Ben-ren Wang a, Rong-jue Wei a a Institute of Acoustics and Stare Key Lab of Modem Acoustics, Nanjing University, Nanjing 210093, China b Northwest Normal University. Lunzhou 730070, China
Received 31 May 1996; revised manuscript received 3 October 1996; accepted for publication 14 October 1996 Communicated by A.R. Bishop
Abstract Using the reductive perturbation method, the carrier wave modulation propagating in a fluid-filled elastic tube is investigated. It is showed that such a process can be described by the nonlinear Schriidinger equation. Keywords: Soliton; Nonlinear wave; Nonlinear Schrijdinger equation; Fluid dynamics
1. Introduction The concept of solitons, especially the nonlinear Schrodinger equation, has now become ubiquitous in modern nonlinear science and indeed can be found in various branches of physics [ l-71. In optics, the optical pulses propagating inside single-mode fibers in the presence of both dispersion and nonlinearity can be expressed by the nonlinear Schriidinger equation (NLSE) [ 1,2] . In a nonlinear lattice it has been shown [3,4] that single-mode excitations of a chain with nearest-neighbor interactions with cubic or quartic nonlinearity are governed by the nonlinear Schrodinger equation (NLSE). The NLSE can also be obtained directly from the fundamental equations for deep water waves [ $61. A derivation of the NLSE consistent to third-order for the propagation of an acoustic wave in a duct, including a wavemaker-like forcing, was given by Aranha, Yue and Mei [7]. A steady soliton solutions was observed experimentally by Wu [ 81. An analysis of this nonpropagating soli-
03759601/97/$17.00
ton was made extensively and the NLSE was obtained by many authors [g-13], where they took account of the additional effects of damping, surface tension and the parametrical drive. Sloshing waves in a rectangular channel in the vicinity of the second cutoff frequency can also be described by the NLSE [ 141. The research work on solitons may be classified by two kinds of physically interesting waves, the slowly varying wave and carrier wave modulation, particularly in a nonlinear lattice and in a nonlinear water wave. The dynamical behaviour of these two kinds of solitons can be described by the KdV equation and the nonlinear Schriidinger equation (FUSE), respectively. Recently some authors [ 15-181 studied the KdV equation in the similar system of a fluid-filled elastic tube. In this paper the NLSE will be first obtained for this fluid-filled elastic tube to describe the weakly nonlinear carrier wave modulation.
Copyright 0 1997 Published by Elsevier Science B.V. All rights reserved.
PII SO375-9601(96)00796-7
W.-s. Duan et al./Physics
2. Equation
of motion
We shall consider an incompressible fluid that is confined within an infinitely long circular cylinder. The wall of the cylinder is composed of an elastic tube. A localized pressure increase in the fluid (only axial variations are considered ) causes a radially symmetric expansion of the elastic tube in the region of the pressure increase. The equations governing the fluid motion are those of conservation of mass and momentum given by
Letters A 224 (1997) 154-158
155
Schrodinger equation might be expected to arise in a the description of a nonlinear wave in a fluid-filled elastic tube, we look for a solution in the form of Fourier series, A’
=
Feng
1 +
A(“.1)(5,7)e”(k+-w’).
(7)
(1) (2) where u is the velocity of the fluid, A (x, t) is the crosssectional area of the cylinder, pa is the constant density of the fluid and p( X, t) is the fluid pressure. A third equation relating the area of the tube to the pressure must also be obtained. Since the area is seen to play a role analogous to that of the density in a compressible fluid, the equation that we seek plays the role of an equation of state and it can be given as [ 15-181
a*( A - Ao) -%A-Ao) pwa2
+
atz
,
(3)
where pw is the density of the tube wall, h and E are the thickness and the Young’s modulus of the tube respectively, r( x, t) is the radius of the tube and a is its equilibrium radius, A0 = mz*. The quantity ( E/p,a2)‘/2 is the angular frequency for radial vibration of the tube. Introducing dimensionless quantities through the definitions A = ra2A’, p = pop’, t = Tt’, x = Lx’, u = ( L/T) u’, where po = Eh/2a,T2 = pwa2/E and L* = pWah/2po, we have the dimensionless equations
The expansion parameter E is assumed to be much less than unity and the coefficients A(“,‘), p(“,‘), u(“*~) are assumed to vary slowly over a wavelength +k-’ and also during one cycle wo-‘, we will show that A(‘-” satisfies a cubic Schrodinger equation. For At ‘3’) to be real we must have A(‘~‘) = AC’,-‘)*. It will’be found that we need retain only terms with n = l,, 2,3 and I= 1,2 to obtain this result. We expect A”,‘) to represent a disturbance that is much longer than a wavelength moving at isome velocity V, this may be accomplished by assuming that A(‘*‘) is a function of a variable 6 = E(X - vt).
In addition, slow temporal variations in this profile may be introduced by allowing a higher-order timedependence variable 7 = e*t.
(11)
Substitution of Eqs. (7)-(9) into E!qs. (4)-(6) and separation of terms according to the first three powers of E yields at O(e) -iloA(‘.”
(4)
u(l.I)
+ ilku”*‘) = 0,
p”.”
(12)
= kp’W
(13)
’
0
(5)
(10)
= (1 _ &,*)41*0,
(14)
at O( E2> _ilwA(*,‘) We now look for a solution in the form of a slowly modulated wavetrain. To understand how the cubic
_ VA:l.” + u:‘*‘) + ilku(2.l) (Wu(lJ-d
+ilkxA 4
= 0,
(15)
156
W.-s. Duan et al./Physics
_ilwuC23’) _ h”vl)
f
+ c
ikpu
(L94JlJ-9)
we have (-V2
9 + pl’.‘)
+ ilkp(**‘)
Letters A 224 (1997) 154-158
= 0,
(16)
~(~3’) = (1 _ 12~*),4(*vf) + 2VilwA:‘*‘),
(17)
+ l)A$lvo)
= 0. As long as V2# 1
we see that A(*s”) has no 5 dependence. Since a term A(‘*‘)( T) will not be of interest for propagation problems, we set A(‘*O) = ~(~9’) = p(‘*O) = 0. For I= 1 we find that
-iwA(2S1) + ikuc2T1)+ (w/k - V)A:lS1) = 0,
at O( E3>
(25)
_iwu(2*1) + ik( 1 _ u~)A(~,‘) + [ 1 - ti2 - Vw(2k+
l/k)]A;l’l’
= 0.
(26)
For I= 2 we obtain + ilkA(2.9)U(1,1-9)]
_ilwu(3*1)
_
h(2v” 6
= 0,
+
(18) (1,q)
~(~9~) 7 fC”
ilku (1,9)u(2*[-9)
c
+
= 0,
(27)
uf
_&2.2)
9 +
+A@.*) + k&2’ + ,(A”.‘)j2
(lJ-9)
+ k( 1 _4&)A(2*2) + !2!k!~A(1v1)12= 0,
p:‘~~) + ilkp(3.‘) = 0,
(28)
9
(19)
To obtain nonzero where jA(1*‘)12 = A(‘,‘)A(‘,-‘). A(‘,‘), we have from Eqs. (25) and (26)
(20)
‘=&l:k?)2
p(3v’) = (1 _ ~‘LW~)A(~,I)_ 2iloAi’v’f + V2Agi”’ + 2ViZoAy”‘.
We now separate terms according to powers of 1. For terms of first order in E we find that no information is obtained for I= 0. For 1 = 1 and 1= -1 we obtain the dispersion relation for the linear theory, namely 2
k2
(21)
to =jqs
A(‘,‘) = $(I,I) w
(22)
’
(23) ’
(131) = (1 _ W2)A(l.l)+
(24)
With w determined by Eq. (21) we can see that A”.“’ = p”.” = n (lJ) =0 for 1 < -1 and 1 > 1. From the second order equation we find that for 1= 0 _VA;‘.O’ + ujl*a) = 0, Considering that P
A’2.2’ = &/A”s”]~,
(30)
u(2.2) = w - 1 _ 1 IA(1.‘)12. k ( 2k2 >
(31)
(1.0) = A(l.0) ,
_VAp.a) + @“’ + (A(t,t)u(l.-t)
+ A(‘.-‘)u(l.‘))~
= 0,
kp(l.l’ w
P
and from Eqs. (27) and (28)
In the third order equation we have at I = 0
and
u(l,l) =
(29)
_$l,o)
_$2.0’ P
(32) + p;2JJ) + (UKl)uU~-l))f
= 0,
(2.0) = At&o)
From Eqs. (32) -( 34) can be obtained
(33) (34)
that
A’2.0’ =&($+2V;)lA”*“li+F,(~), (35)
+p;l’o’ = 0. u(*.o) =&(~+$V)lA”.“12+F2(7).
(36)
157
W.-s. Duan et al./ Physics Letters A 224 (1997) 154-158
Similarly
in the third order equation
_iWA(3,‘) _ VA:“’ + ik(A”.‘)U’2.0’ + A’2.0)U(1.1)
_i,,(3.‘) + ik(u P
at E = 1 we have
(47)
+ A(1,-1)U(2.2)
+ A’2,2’u”,-9
= 0,
_ ~“,I’ %!
+ pi”‘)
(l,1)u(2.0)
+ uw)u(2,29
(37)
+ ikp(3.‘) + *:1*1) = 0,
_ 2iwAJ’,‘)
If we want to obtain the solution of Eqs. (37)-( 39), the variables of A(3*1),~(3*‘), P(~*‘) will disappear from these equations. The coefficient of Ai2*” is also zero. Then we obtain from Eqs. (37)-(39) + F(T)]
= 0, (40)
where
•t H=2o(k’+
4 (l+k2)2
H
I
l),
(48)
2E
Z=vsech(r+kT-00) x exp[ -ik[
(39)
+ iA”*“[GIA”*1’12
7=--T.
As is well known, the solution of Eq. (46) is (38)
+ V2Akjg” + 2ViwAk2.‘).
+iEAkk,‘)
transformations,
+ uk2s1) + ikU(3,‘) + A$‘,‘)
(3.l) = (1 _ ,2)A(3,‘)
HA”.‘) 7
by using the following
- $i(v - k2)T - icro].
(49)
The solitary wave solution represented by Bq. (49) has four parameters. They are 7, which repreBents the amplitude and the pulse width of the solitary,wave, k, which represents its speed and two pammete& which represent the phase constant, 00 and (TO.We mote here that the pulse height v is inversely proportional to the pulse width v-‘, and that the constant k, whiph represents the speed of the pulse transmission, is :independent of the pulse height v. This is different ifrom the KdV soliton, where the speed of the soliton is proportional to the pulse height [ 16-181.
Acknowledgment ’
(41) (42)
This research is supported by the Foundlations of Natural Science, Natural Nonlinear Science ‘of China and the Foundation of Science of Jiang-Su Province.
3k2
E=(1 +k2)2’ F(T)
= kw
2F2(7)
(43)
.
+ ;F,(,)
(44)
> If F( 7) = 0, Eq. (40) becomes SchrSdinger equation iHA:‘,‘)
_ “A::.”
the cubic nonlinear
_ GA(‘,‘)IA(‘v’)(2
= 0.
(45)
Here E/H is proportional to d2w/dk2, because d2w/dk2 = -3k2/w( l+k2)3 and E/H = 3k2/2w( 1+ k2)3. G is negative if k < 2, then Eq. (45) will have a dark soliton. On the other hand, G is positive if k > 2. In this case there is a bright soliton. If G > 0 we have from Eq. (45) i& + iZ[t
+ 2/Z12 = 0
(46)
References [ 1 ] G.P Agrawal, Nonlinear fiber optics (Academic Press, New York, 1989). [ 2 1 A. Hasegawa, Optical solitons in fibers (Springer. Berlin, 1989). [3] K. Yoshimum and S. Watanabe. J. Phys. Sot. Japan 60 (1991) 82. [4] V.V. Konotop, Phys. Rev. E 53 ( 1996) 2843. [S] H. Hashimoto and H. Ono, J. Phys. Sot. Japan 33 ( 1972) 605. [6] CC. Mei, Dynamics of water waves (Academic Press. Beijing, 1984) [in Chinese]. [7] J.A. Aranha. D.K.P. Yue and C.C. Mei. J. Fluid’ Mech. 121 (1982) 465. [S] J.R. Wu et al.. Phys. Rev. L.ett. S2 (1984) 1421. [9] J.W. Miles, J. Fluid Mech. 148 (1984) 451. [IO] A. Larraza and S Putterman, J. Fluid Mech. 148 { 1984) 443. [ II] B.R. Wang and R.J. Wei, Acta Phys. Sinica 35 ( 1986) 1547 [in Chinese 1.
158
W.-s. Duan et d/Physics
1121 R.J. Wei et al.. Chin. Phys. Lett. 3 (1986) 213. [ 131 J.R. Yan and Y.P. Mei, Europhys. Lett. 23 ( 1993) 335. [ 141 E. Kit, L. Shemer and T. Miloh, J. Fluid Mech. 181 ( 1987) 265. [ 151 G.L. Lamb, Elements of sol&on theory (Wiley, New York, 1982).
Letters A 224 (1997) 154-158
[ 161 S. Yomosa, I. Phys. Sot. Japan 56 (1987) 506. [ 171 J.F. Paquerot and M. Remoissenet. Phys. Lett. A 194 ( 1994) 77.
[ 181 W.S. Duan, B.R. Wang and R.J. Wei, J. Phys. Sot. Japan 65 ( 1996) 945.
PHYSICS
LETTERS
A
Physics Letters A 224(1997) 154-158
EISEWER
The cubic nonlinear Schredinger equation in a fluid-filled elastic tube Wen-shan Duan a*b,Ben-ren Wang a, Rong-jue Wei a a Institute of Acoustics and Stare Key Lab of Modem Acoustics, Nanjing University, Nanjing 210093, China b Northwest Normal University. Lunzhou 730070, China
Received 31 May 1996; revised manuscript received 3 October 1996; accepted for publication 14 October 1996 Communicated by A.R. Bishop
Abstract Using the reductive perturbation method, the carrier wave modulation propagating in a fluid-filled elastic tube is investigated. It is showed that such a process can be described by the nonlinear Schriidinger equation. Keywords: Soliton; Nonlinear wave; Nonlinear Schrijdinger equation; Fluid dynamics
1. Introduction The concept of solitons, especially the nonlinear Schrodinger equation, has now become ubiquitous in modern nonlinear science and indeed can be found in various branches of physics [ l-71. In optics, the optical pulses propagating inside single-mode fibers in the presence of both dispersion and nonlinearity can be expressed by the nonlinear Schriidinger equation (NLSE) [ 1,2] . In a nonlinear lattice it has been shown [3,4] that single-mode excitations of a chain with nearest-neighbor interactions with cubic or quartic nonlinearity are governed by the nonlinear Schrodinger equation (NLSE). The NLSE can also be obtained directly from the fundamental equations for deep water waves [ $61. A derivation of the NLSE consistent to third-order for the propagation of an acoustic wave in a duct, including a wavemaker-like forcing, was given by Aranha, Yue and Mei [7]. A steady soliton solutions was observed experimentally by Wu [ 81. An analysis of this nonpropagating soli-
03759601/97/$17.00
ton was made extensively and the NLSE was obtained by many authors [g-13], where they took account of the additional effects of damping, surface tension and the parametrical drive. Sloshing waves in a rectangular channel in the vicinity of the second cutoff frequency can also be described by the NLSE [ 141. The research work on solitons may be classified by two kinds of physically interesting waves, the slowly varying wave and carrier wave modulation, particularly in a nonlinear lattice and in a nonlinear water wave. The dynamical behaviour of these two kinds of solitons can be described by the KdV equation and the nonlinear Schriidinger equation (FUSE), respectively. Recently some authors [ 15-181 studied the KdV equation in the similar system of a fluid-filled elastic tube. In this paper the NLSE will be first obtained for this fluid-filled elastic tube to describe the weakly nonlinear carrier wave modulation.
Copyright 0 1997 Published by Elsevier Science B.V. All rights reserved.
PII SO375-9601(96)00796-7
W.-s. Duan et al./Physics
2. Equation
of motion
We shall consider an incompressible fluid that is confined within an infinitely long circular cylinder. The wall of the cylinder is composed of an elastic tube. A localized pressure increase in the fluid (only axial variations are considered ) causes a radially symmetric expansion of the elastic tube in the region of the pressure increase. The equations governing the fluid motion are those of conservation of mass and momentum given by
Letters A 224 (1997) 154-158
155
Schrodinger equation might be expected to arise in a the description of a nonlinear wave in a fluid-filled elastic tube, we look for a solution in the form of Fourier series, A’
=
Feng
1 +
A(“.1)(5,7)e”(k+-w’).
(7)
(1) (2) where u is the velocity of the fluid, A (x, t) is the crosssectional area of the cylinder, pa is the constant density of the fluid and p( X, t) is the fluid pressure. A third equation relating the area of the tube to the pressure must also be obtained. Since the area is seen to play a role analogous to that of the density in a compressible fluid, the equation that we seek plays the role of an equation of state and it can be given as [ 15-181
a*( A - Ao) -%A-Ao) pwa2
+
atz
,
(3)
where pw is the density of the tube wall, h and E are the thickness and the Young’s modulus of the tube respectively, r( x, t) is the radius of the tube and a is its equilibrium radius, A0 = mz*. The quantity ( E/p,a2)‘/2 is the angular frequency for radial vibration of the tube. Introducing dimensionless quantities through the definitions A = ra2A’, p = pop’, t = Tt’, x = Lx’, u = ( L/T) u’, where po = Eh/2a,T2 = pwa2/E and L* = pWah/2po, we have the dimensionless equations
The expansion parameter E is assumed to be much less than unity and the coefficients A(“,‘), p(“,‘), u(“*~) are assumed to vary slowly over a wavelength +k-’ and also during one cycle wo-‘, we will show that A(‘-” satisfies a cubic Schrodinger equation. For At ‘3’) to be real we must have A(‘~‘) = AC’,-‘)*. It will’be found that we need retain only terms with n = l,, 2,3 and I= 1,2 to obtain this result. We expect A”,‘) to represent a disturbance that is much longer than a wavelength moving at isome velocity V, this may be accomplished by assuming that A(‘*‘) is a function of a variable 6 = E(X - vt).
In addition, slow temporal variations in this profile may be introduced by allowing a higher-order timedependence variable 7 = e*t.
(11)
Substitution of Eqs. (7)-(9) into E!qs. (4)-(6) and separation of terms according to the first three powers of E yields at O(e) -iloA(‘.”
(4)
u(l.I)
+ ilku”*‘) = 0,
p”.”
(12)
= kp’W
(13)
’
0
(5)
(10)
= (1 _ &,*)41*0,
(14)
at O( E2> _ilwA(*,‘) We now look for a solution in the form of a slowly modulated wavetrain. To understand how the cubic
_ VA:l.” + u:‘*‘) + ilku(2.l) (Wu(lJ-d
+ilkxA 4
= 0,
(15)
156
W.-s. Duan et al./Physics
_ilwuC23’) _ h”vl)
f
+ c
ikpu
(L94JlJ-9)
we have (-V2
9 + pl’.‘)
+ ilkp(**‘)
Letters A 224 (1997) 154-158
= 0,
(16)
~(~3’) = (1 _ 12~*),4(*vf) + 2VilwA:‘*‘),
(17)
+ l)A$lvo)
= 0. As long as V2# 1
we see that A(*s”) has no 5 dependence. Since a term A(‘*‘)( T) will not be of interest for propagation problems, we set A(‘*O) = ~(~9’) = p(‘*O) = 0. For I= 1 we find that
-iwA(2S1) + ikuc2T1)+ (w/k - V)A:lS1) = 0,
at O( E3>
(25)
_iwu(2*1) + ik( 1 _ u~)A(~,‘) + [ 1 - ti2 - Vw(2k+
l/k)]A;l’l’
= 0.
(26)
For I= 2 we obtain + ilkA(2.9)U(1,1-9)]
_ilwu(3*1)
_
h(2v” 6
= 0,
+
(18) (1,q)
~(~9~) 7 fC”
ilku (1,9)u(2*[-9)
c
+
= 0,
(27)
uf
_&2.2)
9 +
+A@.*) + k&2’ + ,(A”.‘)j2
(lJ-9)
+ k( 1 _4&)A(2*2) + !2!k!~A(1v1)12= 0,
p:‘~~) + ilkp(3.‘) = 0,
(28)
9
(19)
To obtain nonzero where jA(1*‘)12 = A(‘,‘)A(‘,-‘). A(‘,‘), we have from Eqs. (25) and (26)
(20)
‘=&l:k?)2
p(3v’) = (1 _ ~‘LW~)A(~,I)_ 2iloAi’v’f + V2Agi”’ + 2ViZoAy”‘.
We now separate terms according to powers of 1. For terms of first order in E we find that no information is obtained for I= 0. For 1 = 1 and 1= -1 we obtain the dispersion relation for the linear theory, namely 2
k2
(21)
to =jqs
A(‘,‘) = $(I,I) w
(22)
’
(23) ’
(131) = (1 _ W2)A(l.l)+
(24)
With w determined by Eq. (21) we can see that A”.“’ = p”.” = n (lJ) =0 for 1 < -1 and 1 > 1. From the second order equation we find that for 1= 0 _VA;‘.O’ + ujl*a) = 0, Considering that P
A’2.2’ = &/A”s”]~,
(30)
u(2.2) = w - 1 _ 1 IA(1.‘)12. k ( 2k2 >
(31)
(1.0) = A(l.0) ,
_VAp.a) + @“’ + (A(t,t)u(l.-t)
+ A(‘.-‘)u(l.‘))~
= 0,
kp(l.l’ w
P
and from Eqs. (27) and (28)
In the third order equation we have at I = 0
and
u(l,l) =
(29)
_$l,o)
_$2.0’ P
(32) + p;2JJ) + (UKl)uU~-l))f
= 0,
(2.0) = At&o)
From Eqs. (32) -( 34) can be obtained
(33) (34)
that
A’2.0’ =&($+2V;)lA”*“li+F,(~), (35)
+p;l’o’ = 0. u(*.o) =&(~+$V)lA”.“12+F2(7).
(36)
157
W.-s. Duan et al./ Physics Letters A 224 (1997) 154-158
Similarly
in the third order equation
_iWA(3,‘) _ VA:“’ + ik(A”.‘)U’2.0’ + A’2.0)U(1.1)
_i,,(3.‘) + ik(u P
at E = 1 we have
(47)
+ A(1,-1)U(2.2)
+ A’2,2’u”,-9
= 0,
_ ~“,I’ %!
+ pi”‘)
(l,1)u(2.0)
+ uw)u(2,29
(37)
+ ikp(3.‘) + *:1*1) = 0,
_ 2iwAJ’,‘)
If we want to obtain the solution of Eqs. (37)-( 39), the variables of A(3*1),~(3*‘), P(~*‘) will disappear from these equations. The coefficient of Ai2*” is also zero. Then we obtain from Eqs. (37)-(39) + F(T)]
= 0, (40)
where
•t H=2o(k’+
4 (l+k2)2
H
I
l),
(48)
2E
Z=vsech(r+kT-00) x exp[ -ik[
(39)
+ iA”*“[GIA”*1’12
7=--T.
As is well known, the solution of Eq. (46) is (38)
+ V2Akjg” + 2ViwAk2.‘).
+iEAkk,‘)
transformations,
+ uk2s1) + ikU(3,‘) + A$‘,‘)
(3.l) = (1 _ ,2)A(3,‘)
HA”.‘) 7
by using the following
- $i(v - k2)T - icro].
(49)
The solitary wave solution represented by Bq. (49) has four parameters. They are 7, which repreBents the amplitude and the pulse width of the solitary,wave, k, which represents its speed and two pammete& which represent the phase constant, 00 and (TO.We mote here that the pulse height v is inversely proportional to the pulse width v-‘, and that the constant k, whiph represents the speed of the pulse transmission, is :independent of the pulse height v. This is different ifrom the KdV soliton, where the speed of the soliton is proportional to the pulse height [ 16-181.
Acknowledgment ’
(41) (42)
This research is supported by the Foundlations of Natural Science, Natural Nonlinear Science ‘of China and the Foundation of Science of Jiang-Su Province.
3k2
E=(1 +k2)2’ F(T)
= kw
2F2(7)
(43)
.
+ ;F,(,)
(44)
> If F( 7) = 0, Eq. (40) becomes SchrSdinger equation iHA:‘,‘)
_ “A::.”
the cubic nonlinear
_ GA(‘,‘)IA(‘v’)(2
= 0.
(45)
Here E/H is proportional to d2w/dk2, because d2w/dk2 = -3k2/w( l+k2)3 and E/H = 3k2/2w( 1+ k2)3. G is negative if k < 2, then Eq. (45) will have a dark soliton. On the other hand, G is positive if k > 2. In this case there is a bright soliton. If G > 0 we have from Eq. (45) i& + iZ[t
+ 2/Z12 = 0
(46)
References [ 1 ] G.P Agrawal, Nonlinear fiber optics (Academic Press, New York, 1989). [ 2 1 A. Hasegawa, Optical solitons in fibers (Springer. Berlin, 1989). [3] K. Yoshimum and S. Watanabe. J. Phys. Sot. Japan 60 (1991) 82. [4] V.V. Konotop, Phys. Rev. E 53 ( 1996) 2843. [S] H. Hashimoto and H. Ono, J. Phys. Sot. Japan 33 ( 1972) 605. [6] CC. Mei, Dynamics of water waves (Academic Press. Beijing, 1984) [in Chinese]. [7] J.A. Aranha. D.K.P. Yue and C.C. Mei. J. Fluid’ Mech. 121 (1982) 465. [S] J.R. Wu et al.. Phys. Rev. L.ett. S2 (1984) 1421. [9] J.W. Miles, J. Fluid Mech. 148 (1984) 451. [IO] A. Larraza and S Putterman, J. Fluid Mech. 148 { 1984) 443. [ II] B.R. Wang and R.J. Wei, Acta Phys. Sinica 35 ( 1986) 1547 [in Chinese 1.
158
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