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One-dimensional modulation instability of wave packets in media with resonant dispersion
PHYSICA ELSEVIER
Physica A 241 (1997) 128-132
One-dimensional modulation instability of wave packets in media with resonant dispersion S.A. Rybak*, O.M. Zozulya N.N.Andreev Acoustic Institute, Shvernika Ul. 4, Moscow 117036, Russia
Abstract Modulation instability of small-amplitude quasimonochromatic wave packets in media containing compact oscillators is analysed in the one-dimensional case. Conservative media with quadratic nonlinearity and linear nondamped oscillators are considered. Frequency range of unstable wave-packet propagation is found. It is shown that instability results from resonant longwave-short-wave interaction. P A C S : 42.25.Bs; 05.60.+w; 43.25.Ed Keywords: Waves; Oscillators; Instability; Modulation
1. Introduction Instability o f quasimonochromatic waves propagating in nonlinear dispersive media is an extensively-studied subject ascending to pioneering works on modulation instability o f wavetrains on water surface in hydrodynamics (often referred to as Benjamin-Feir instability) and stimulated Mandel'shtamm-Brillouin scattering of light in nonlinear optics. The nature o f this instability is universal for a great variety of wave phenomena and can be easily comprehended on the basis o f three-wave (or four-wave) interaction. To outline it briefly let us, e.g., consider propagation of a quasimonochromatic wave packet (with central wave vector k and bandwidth 6 ~ k ) in a medium with quadratic nonlinearity. In course o f their nonlinear interaction the main harmonic with kl = k and a small sideband harmonic of the wave packet with k2 = k + Ak (where A k > 6 ) engender the harmonics with k3 = 2k + Ak and k4 = Ak. The interaction between either of the generated harmonics with k3,4 and the main harmonic with kl can result in the increase of the sideband with k2 (which depends on the nonlinear coupling among * Corresponding author. Fax: (7-095) 2088055; e-mail: [email protected]. 0378-4371/97/$15.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved PH S0378-4371 (97)00071-X
S.A. Rybak, O.M. Zozulya/Physica A 241 (1997) 128-132
129
the triad participants). If the net interaction in the triads {kl,k2,k3} and {kl,k2,k4} gives rise to the sideband growth (at the expense of the energy of the main harmonic), then the wave packet is unstable towards slow modulation of its envelope or carrier frequency. It is worth noting that nonlinear coupling in a triad can be resonant at some frequencies leading to the total energy interchange among the triad participants. For the triads {kl,k2,k3} and {kl,k2,k4} the resonance conditions are: k l + k2 = k 3 ,
CO(kl ) + c o ( k 2 ) : c o ( k 3 ) ,
k l - k2 = k4,
CO(kl ) - CO(k2 ) : CO(k4 ) ,
respectively. The first condition reduces to 2CO(k)~ CO(2k) and can be met only in the absence (or for very artificial forms) of dispersion relation. The second reduces to VCO(k) ,-~ CO(Ak)/Ak,
(1)
i.e., the condition of group velocity synchronism between short and long waves, which is satisfied in a number of wave models [1]. In this paper we analyze modulation instability of small-amplitude quasimonochromatic wave packets in media containing linear identical oscillators and specified by nonlinearity of second order [2]. The proposed model is pertinent to many physical applications, e.g., acoustic waves in bubbly liquids and solids, polaritons in dielectric crystals, etc. We show that the consequent resonant dispersion relation results in the resonant long-wave-short-wave interaction, i.e., the equality (1) between the group velocities of short and long waves is met at some frequency CO, and thus this interaction play the dominant role in the stability of wavepacket propagation. For the sake of simplicity all the analysis is confined to nondissipative propagation and one dimension.
2. Stability analysis Let us consider the one-dimensional wave equation: Ut, -- C2 blxx = (SU 2 -- O'(P)tt ,
(2)
where the "thermodynamic" variable ~p is incorporated to describe the dynamics of oscillators coupled via the wave field u: qht + co20~o= co2u ;
(3)
here e is the coefficient of quadratic nonlinearity ( e > 0 ) , COo the eigenfrequency of oscillators, a the dispersion coefficient, being proportional to the concentration of oscillators. We assume, as is generally the case, that the phase speed c(co) of linear waves satisfies c(oo)>~c(O), i.e., a > 0 . To define the evolution of a quasimonochromatic wave packet we apply the multiple scaling technique as follows. Introducing slow variables X = px, T = #t, )(2 = 1~2x,where/~ ~ 1, we seek the solution to (2) and (3) in the form: U ~- ,tL(Uo -{- ,LtUl -~- ,Lt2U2 -~- " " • ),
q~ = ~((#0 "l- ~.lq~l "at-]A2 ~02 -}- ' ' ") ,
(4)
130
S.A. Rybak, O.M. Zozulya/Physica A 241 (1997) 128-132
where
uo = ae iO + (*),
tp0 = A e i0 + (*) ;
u l = b + d e 2 i ° +(*),
q } l = B + C e i ° +Dei° + ( * ) ; . . .
(5)
0 = k x - cot is the wave phase; k, 0) are the central wave number and frequency of the packet, respectively. The amplitudes of harmonics a, A, b, B, C, d, D are assumed to be slowly varying functions having the form a = a ( X , T, X2), so that their derivatives with respect to x and t can be written as 0
{3
{3
0
/~2 0
Taking this into account and substituting (4) and (5) into the system (2) and (3) we equate the terms of the same order of e i0 and/t. For the first order of kt and the main harmonic e i0 we find the well-known resonant dispersion relation for linear waves:
0-0)2 0)2 0)2 _ c2k 2 _ 0)2 _ 0)2 "
(6)
The second order of/~ results in the equation of geometrical optics for e i°, and corresponds to the wave propagation with permanent envelope: aTq-0)tax = 0 ,
where0),=
(Co~)
(0)2_0)2)2
(0)2-7 bT0)v7- 0)4
(7)
is the group speed of short waves. In this order of p we also find the amplitude of the second harmonic e2iO: d = ca2 (0)2 _ 0)2)(40)2 _ 0)2) 300)2092
(8)
The third order of/~ gives rise to the equation of quasioptic approximation for e i°, which can be reduced to
• 80)2
.0)t/
ax: + t~c2 (ab + a ' d ) - t20),axx = 0,
(9)
as a result of the o / ' substitution:
0)t,
2 2 0)(0)2 _ 0)2 )2 [(0)2 _ 0)2)(0)2 -H 30) 2 ) - 30-{04] 0-0)0C0 [(0)2 _ 0)2)2 + 0-0)4]3
(lo)
To close (9) we need to determine the long-wave amplitude b and thus to consider terms with e ° that ensue from the fourth order of/~. We find 2e (a*a)rr, brr - c~bxx-- 1+0-
where c l = x / l +co 0-
= 0 ) ' o,=0
(11)
is the group speed of long waves. The system of two equations (9) and (11) describe the case of three-wave interaction, where the two waves are close to each other and
S.A. Rybak, O.M. ZozulyalPhysica A 241 (1997) 12&132
131
both represented by ae i0, and the third wave represented by b. The system of equations of the form (9) and (11) was for the first time derived to describe the interactions between Langmuir and ion-acoustic waves in plasma [3]. From (7) it follows that a = a ( X - 09'T,X2), which enables us to determine b: (2~)
b=
~
a*a
1_(cd09,)2.
(12)
This expression suggests the feasibility of resonant long-wave-short-wave interaction. To verify whether the resonance condition 09'= ct is satisfied at some frequencies we rewrite (12) in the explicit form as
{2ga*a~
(092-092)3(092--(1 + ¢r)092) b = \ 0.°92 j (0)2 _ 09~)2(092 _ 3o)20) _ 0.092(0/4 30920092+ 3094)" _
Assuming a ~ 1, i.e., far from eigenfrequency 090 the dispersion is small, we find the presence of this resonance at 09, ~ v~090(1 + a/8). To formulate the stability criterion it is sufficient to note that (ab + a*d)~a*a 2, so that (9) is nothing but nonlinear Shrrdinger equation. Its solutions are unstable (towards disturbances with small wave numbers) when the terms a*a2 and axx have the same swing:
09,(aba,a2 + a'd) < 0 .
(13)
From (10) it follows that the group speed 09' has only one point of extremum, namely minimum, which lies in the interval of nontransparency (090,090x/1 + cr). Excluding this interval from the analysis we find that instability occurs in the range 1.45 ~
< - - < v/3, 090
(14)
where the corrections of the order a are omitted. This inequality demonstrates that the long-wave-short-wave interaction play the key role in the stability of wave packet propagation in media with resonant dispersion. Let us now consider the limit cases of high and low frequencies. In the low-frequency limit (i.e., 0994090) we can rewrite (3) as: q) = U -- 0902 Utt .
Its substitution in (2) results in the equation of Boussinesq type: (1 -~- ~7)Utt -- C2 Uxx = ~ ( U 2 ) t t q- 0.0902Utn, ,
From (14) it follows that wave packets are stable in this limit, which is consistent with one-dimensional Boussinesq model. In the high-frequency limit (i.e., 09>>090) the term 092~o in (3) is negligible so that the wave equation reduces to: u,, -
co Uxx =
2).
-
0.090 u.
132
S.A. Rybak, O.M. Zozulya/Physica A 241 (1997) 128-132
Equation of this type arises in a variety of applications, e.g., it describes the high frequency limit for internal waves in a rotating ocean (where ~o0 is the frequency of rotation) [4], acoustic waves in fluctuating or turbulent media (where ~o0 = co~L, and L is the correlation length) [5], strain waves in curved rods [6], etc. From (14) it follows that wave packets are stable in such media; however, we found no results in literature on this subject.
3. Conclusions One-dimensional modulation instability of quasimonochromatic wave packets of small amplitude has been investigated for media with resonant dispersion relation and nonlinearity of second order. It is shown that nonlinear self-interaction of the wave packet can result in resonant generation of long waves, if the carrier frequency is tuned to some frequency ~o,, corresponding to the condition of group velocity synchronism. Nearness to this resonance underlies the modulation instability, which in case of exact tuning degenerates into decay instability in three-wave system. The development of decay instability is out of the scope of the analysis since the technique is valid only if the amplitudes of the generated waves are small compared to the amplitude of the wave packet. However, in the absense of dissipation it is plausible to suggest the total energy interchange among participants of the three-wave interactions.
References [1] D.J. Benney, Stud. Appl. Math. 56 (1977) 81. [2] S.A. Rybak and Yu.I. Skrynnikov, in: Nonlinear World, IV Int. Workshop, Kiev, 1989, Proc. (World Scientific, Singapore, 1989) p. 664. [3] V.E. Zakharov, Sov. JETP 62 (1972) 1745. [4] L.A. Ostrovsky, Sov. Oceanology 18 (1978) 120. [5] E.S. Benilov and E.N. Pelinovsky, Sov. JETP 67 (1988) 98. [6] S.A. Rybak and Yu.I. Skrynnikov, J. de Physique IV 4 (1994) 805.
Physica A 241 (1997) 128-132
One-dimensional modulation instability of wave packets in media with resonant dispersion S.A. Rybak*, O.M. Zozulya N.N.Andreev Acoustic Institute, Shvernika Ul. 4, Moscow 117036, Russia
Abstract Modulation instability of small-amplitude quasimonochromatic wave packets in media containing compact oscillators is analysed in the one-dimensional case. Conservative media with quadratic nonlinearity and linear nondamped oscillators are considered. Frequency range of unstable wave-packet propagation is found. It is shown that instability results from resonant longwave-short-wave interaction. P A C S : 42.25.Bs; 05.60.+w; 43.25.Ed Keywords: Waves; Oscillators; Instability; Modulation
1. Introduction Instability o f quasimonochromatic waves propagating in nonlinear dispersive media is an extensively-studied subject ascending to pioneering works on modulation instability o f wavetrains on water surface in hydrodynamics (often referred to as Benjamin-Feir instability) and stimulated Mandel'shtamm-Brillouin scattering of light in nonlinear optics. The nature o f this instability is universal for a great variety of wave phenomena and can be easily comprehended on the basis o f three-wave (or four-wave) interaction. To outline it briefly let us, e.g., consider propagation of a quasimonochromatic wave packet (with central wave vector k and bandwidth 6 ~ k ) in a medium with quadratic nonlinearity. In course o f their nonlinear interaction the main harmonic with kl = k and a small sideband harmonic of the wave packet with k2 = k + Ak (where A k > 6 ) engender the harmonics with k3 = 2k + Ak and k4 = Ak. The interaction between either of the generated harmonics with k3,4 and the main harmonic with kl can result in the increase of the sideband with k2 (which depends on the nonlinear coupling among * Corresponding author. Fax: (7-095) 2088055; e-mail: [email protected]. 0378-4371/97/$15.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved PH S0378-4371 (97)00071-X
S.A. Rybak, O.M. Zozulya/Physica A 241 (1997) 128-132
129
the triad participants). If the net interaction in the triads {kl,k2,k3} and {kl,k2,k4} gives rise to the sideband growth (at the expense of the energy of the main harmonic), then the wave packet is unstable towards slow modulation of its envelope or carrier frequency. It is worth noting that nonlinear coupling in a triad can be resonant at some frequencies leading to the total energy interchange among the triad participants. For the triads {kl,k2,k3} and {kl,k2,k4} the resonance conditions are: k l + k2 = k 3 ,
CO(kl ) + c o ( k 2 ) : c o ( k 3 ) ,
k l - k2 = k4,
CO(kl ) - CO(k2 ) : CO(k4 ) ,
respectively. The first condition reduces to 2CO(k)~ CO(2k) and can be met only in the absence (or for very artificial forms) of dispersion relation. The second reduces to VCO(k) ,-~ CO(Ak)/Ak,
(1)
i.e., the condition of group velocity synchronism between short and long waves, which is satisfied in a number of wave models [1]. In this paper we analyze modulation instability of small-amplitude quasimonochromatic wave packets in media containing linear identical oscillators and specified by nonlinearity of second order [2]. The proposed model is pertinent to many physical applications, e.g., acoustic waves in bubbly liquids and solids, polaritons in dielectric crystals, etc. We show that the consequent resonant dispersion relation results in the resonant long-wave-short-wave interaction, i.e., the equality (1) between the group velocities of short and long waves is met at some frequency CO, and thus this interaction play the dominant role in the stability of wavepacket propagation. For the sake of simplicity all the analysis is confined to nondissipative propagation and one dimension.
2. Stability analysis Let us consider the one-dimensional wave equation: Ut, -- C2 blxx = (SU 2 -- O'(P)tt ,
(2)
where the "thermodynamic" variable ~p is incorporated to describe the dynamics of oscillators coupled via the wave field u: qht + co20~o= co2u ;
(3)
here e is the coefficient of quadratic nonlinearity ( e > 0 ) , COo the eigenfrequency of oscillators, a the dispersion coefficient, being proportional to the concentration of oscillators. We assume, as is generally the case, that the phase speed c(co) of linear waves satisfies c(oo)>~c(O), i.e., a > 0 . To define the evolution of a quasimonochromatic wave packet we apply the multiple scaling technique as follows. Introducing slow variables X = px, T = #t, )(2 = 1~2x,where/~ ~ 1, we seek the solution to (2) and (3) in the form: U ~- ,tL(Uo -{- ,LtUl -~- ,Lt2U2 -~- " " • ),
q~ = ~((#0 "l- ~.lq~l "at-]A2 ~02 -}- ' ' ") ,
(4)
130
S.A. Rybak, O.M. Zozulya/Physica A 241 (1997) 128-132
where
uo = ae iO + (*),
tp0 = A e i0 + (*) ;
u l = b + d e 2 i ° +(*),
q } l = B + C e i ° +Dei° + ( * ) ; . . .
(5)
0 = k x - cot is the wave phase; k, 0) are the central wave number and frequency of the packet, respectively. The amplitudes of harmonics a, A, b, B, C, d, D are assumed to be slowly varying functions having the form a = a ( X , T, X2), so that their derivatives with respect to x and t can be written as 0
{3
{3
0
/~2 0
Taking this into account and substituting (4) and (5) into the system (2) and (3) we equate the terms of the same order of e i0 and/t. For the first order of kt and the main harmonic e i0 we find the well-known resonant dispersion relation for linear waves:
0-0)2 0)2 0)2 _ c2k 2 _ 0)2 _ 0)2 "
(6)
The second order of/~ results in the equation of geometrical optics for e i°, and corresponds to the wave propagation with permanent envelope: aTq-0)tax = 0 ,
where0),=
(Co~)
(0)2_0)2)2
(0)2-7 bT0)v7- 0)4
(7)
is the group speed of short waves. In this order of p we also find the amplitude of the second harmonic e2iO: d = ca2 (0)2 _ 0)2)(40)2 _ 0)2) 300)2092
(8)
The third order of/~ gives rise to the equation of quasioptic approximation for e i°, which can be reduced to
• 80)2
.0)t/
ax: + t~c2 (ab + a ' d ) - t20),axx = 0,
(9)
as a result of the o / ' substitution:
0)t,
2 2 0)(0)2 _ 0)2 )2 [(0)2 _ 0)2)(0)2 -H 30) 2 ) - 30-{04] 0-0)0C0 [(0)2 _ 0)2)2 + 0-0)4]3
(lo)
To close (9) we need to determine the long-wave amplitude b and thus to consider terms with e ° that ensue from the fourth order of/~. We find 2e (a*a)rr, brr - c~bxx-- 1+0-
where c l = x / l +co 0-
= 0 ) ' o,=0
(11)
is the group speed of long waves. The system of two equations (9) and (11) describe the case of three-wave interaction, where the two waves are close to each other and
S.A. Rybak, O.M. ZozulyalPhysica A 241 (1997) 12&132
131
both represented by ae i0, and the third wave represented by b. The system of equations of the form (9) and (11) was for the first time derived to describe the interactions between Langmuir and ion-acoustic waves in plasma [3]. From (7) it follows that a = a ( X - 09'T,X2), which enables us to determine b: (2~)
b=
~
a*a
1_(cd09,)2.
(12)
This expression suggests the feasibility of resonant long-wave-short-wave interaction. To verify whether the resonance condition 09'= ct is satisfied at some frequencies we rewrite (12) in the explicit form as
{2ga*a~
(092-092)3(092--(1 + ¢r)092) b = \ 0.°92 j (0)2 _ 09~)2(092 _ 3o)20) _ 0.092(0/4 30920092+ 3094)" _
Assuming a ~ 1, i.e., far from eigenfrequency 090 the dispersion is small, we find the presence of this resonance at 09, ~ v~090(1 + a/8). To formulate the stability criterion it is sufficient to note that (ab + a*d)~a*a 2, so that (9) is nothing but nonlinear Shrrdinger equation. Its solutions are unstable (towards disturbances with small wave numbers) when the terms a*a2 and axx have the same swing:
09,(aba,a2 + a'd) < 0 .
(13)
From (10) it follows that the group speed 09' has only one point of extremum, namely minimum, which lies in the interval of nontransparency (090,090x/1 + cr). Excluding this interval from the analysis we find that instability occurs in the range 1.45 ~
< - - < v/3, 090
(14)
where the corrections of the order a are omitted. This inequality demonstrates that the long-wave-short-wave interaction play the key role in the stability of wave packet propagation in media with resonant dispersion. Let us now consider the limit cases of high and low frequencies. In the low-frequency limit (i.e., 0994090) we can rewrite (3) as: q) = U -- 0902 Utt .
Its substitution in (2) results in the equation of Boussinesq type: (1 -~- ~7)Utt -- C2 Uxx = ~ ( U 2 ) t t q- 0.0902Utn, ,
From (14) it follows that wave packets are stable in this limit, which is consistent with one-dimensional Boussinesq model. In the high-frequency limit (i.e., 09>>090) the term 092~o in (3) is negligible so that the wave equation reduces to: u,, -
co Uxx =
2).
-
0.090 u.
132
S.A. Rybak, O.M. Zozulya/Physica A 241 (1997) 128-132
Equation of this type arises in a variety of applications, e.g., it describes the high frequency limit for internal waves in a rotating ocean (where ~o0 is the frequency of rotation) [4], acoustic waves in fluctuating or turbulent media (where ~o0 = co~L, and L is the correlation length) [5], strain waves in curved rods [6], etc. From (14) it follows that wave packets are stable in such media; however, we found no results in literature on this subject.
3. Conclusions One-dimensional modulation instability of quasimonochromatic wave packets of small amplitude has been investigated for media with resonant dispersion relation and nonlinearity of second order. It is shown that nonlinear self-interaction of the wave packet can result in resonant generation of long waves, if the carrier frequency is tuned to some frequency ~o,, corresponding to the condition of group velocity synchronism. Nearness to this resonance underlies the modulation instability, which in case of exact tuning degenerates into decay instability in three-wave system. The development of decay instability is out of the scope of the analysis since the technique is valid only if the amplitudes of the generated waves are small compared to the amplitude of the wave packet. However, in the absense of dissipation it is plausible to suggest the total energy interchange among participants of the three-wave interactions.
References [1] D.J. Benney, Stud. Appl. Math. 56 (1977) 81. [2] S.A. Rybak and Yu.I. Skrynnikov, in: Nonlinear World, IV Int. Workshop, Kiev, 1989, Proc. (World Scientific, Singapore, 1989) p. 664. [3] V.E. Zakharov, Sov. JETP 62 (1972) 1745. [4] L.A. Ostrovsky, Sov. Oceanology 18 (1978) 120. [5] E.S. Benilov and E.N. Pelinovsky, Sov. JETP 67 (1988) 98. [6] S.A. Rybak and Yu.I. Skrynnikov, J. de Physique IV 4 (1994) 805.