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Example of shock waves in unstable media: the focusing nonlinear Schrödinger equation
25 July 1994
PHYSICS
LETTERS
A
Physics Letters A 190 (1994) 255-258
ELSEVIER
Example of shock waves in unstable media: the focusing nonlinear Schrbdinger equation Ramil’ F. Bikbaev, Vadim R. Kudashev Institute OfMathematics, Chernyshevskii 112, Ufa 450000, Russian Federation Received 16 March 1994; revised manuscript received 25 March 1994; accepted for publication 19 May 1994 Communicated by A.R. Bishop
Abstract Dissipationless shock waves in modulational unstable one-dimensional media are investigated for the simplest example of the integrable focusing nonlinear Schriidinger (NS) equation. Our approach is based on the construction of a special exact solution of the Whitham-NS system, which “partially saturates” the modulational instability.
1. Introduction
Jfm=2a, =a,
It is well known that the focusing NS equation, v(x,O)=O,
iu,+u,+21uj2u=0,
(1)
may be rewritten as a hydrodynamic type system
x
x20, (4’ )
It is natural to call the solution v(x, t), v(x, t) ] or u(x, t) of the problem (l)-(4) the shock wave solution.
f;+2cfv)X=O
9
Vt+2w,-2f,=t(2fulf-fl/f2)x,
(2)
where
U=f"Z exp(igL v=h.
(3)
We consider ( 1 )- (3 ) with the following step-like initial condition,
Jfrn=2, =l,
x
v(x,O)=O, equivalent to Elsevier Science B.V. SSDIO375-9601(94)00402-B
(4)
The main purpose of this Letter is to describe the qualitative behaviour of the shock wave at t s 1. We note that this problem is much more difficult than the analogous problem for stable models [ 1,2 1. The typical and simplest shock problem in the modulational stable model was first studied by Gurevich and Pitaevskii [ 1 ] as an example for the Korteweg-de Vries (KdV) equation. In Ref. [ I] the modulational theory developed by Whitham [ 3 ] was used. It is important that the Whitham system for the KdV equation as well as for other stable models [ 31 is hyperbolic, which means that Whitham characteristic speeds are everywhere real: Im( Si) = 0. However for unstable models these speeds are generally speaking complex [ 31: Im(&) $0 (elliptic case).
R.F. Bikbaev, V.R. Kudashev /Physics Letters A 190 (1994) 255-258
256
This complicates very much any analytical investigation of unstable models due to exponential growth of the linear perturbations of the (locally) constant background. In particular there are obvious obstacles in studying of shock problems in unstable systems, which are closely related to the classical problem of the modulational instability of the monochromatic wave in ( 1) (see, for example, Ref. [ 4 ] ). In this Letter we demonstrate how to overcome these difficulties on the concrete example ( 1 )- (4). We shall show that a famous strategy [I ] may be modified and successfully applied to ( 1)- (4 ) . Our approach is based on the concept of “partially saturating modulational instability solutions” of elliptic Whitham-NS equations proposed in Ref. [ 5 ] for the description of the modulational instability in ( 1). A special subclass of Whitham-NS system solutions with vanishing imaginary parts Im (S,) = 0 of some (not all! ) characteristic speeds (see ( 12 ), ( 17 ) ) play the key role in our construction.
AAl2 1_iu1232,A31
s=u+
>
s,=u+ 1 -P&4/~,,
s,=s:,
s,=sl;,
(9)
pan,
where ,l ii= 2 i - Aj, K, E are the complete elliptic integrals of the first and the second kind, respectively, m=l2J43l~32~,4.
3. Solution of the Whitham-NS
system and longtime behaviour of the NS-hydrodynamic shock wave
We construct an approximate solution of the shock problem ( 1)-( 4) for t B 1 as follows. In the “external” region on the (x, t) plane: (X-CO,s-x’(t)), the solution is a zero-phase one,
Jfrn=2,
x
,
z)o(x,t) =o . 2. One-phase NS equation solution and its Whitham-NS
modulations
It is well known that the one-phase solution of ( 1), (2) has the form J;(e) =f3 + ti -f,)dn’
fi-fi “‘fX 3 eFx-ut)
(6)
wherefiafafiaO>j& A=,/xaO, dn is the Jacobi elliptic function. The elliptic spectral curve corresponding to (5 ), (6) has branching points Ai, i=l,2,3,4;1,=;1:,&=Ifsuchthat (cf.Ref. [6]) n,~a-iy=$U-t~-?ii(~+~), n,~8-is=tU+fJ-f3-ti(~-~).
(7)
The Whitham-NS equations for ( 1)) (2)) ( 5 ) - ( 7 ) can be presented in the diagonal form (cf. Ref. [ 6 ] )
2
+s,(n)
2
=o
)
i= 1,2,
3,4,
In the oscillation region 0
(t) the “internal” solution (5 ), ( 6 )
(8), (9). In this the Whitham-NS (11)
Im(S3) =O , t Re(S,) -x=g(p, (5)
U&9)=+u-~&).
(10)
A1Econst,
m} ,
1-e;
’
6) ,
(12)
where g(/?, 6) is an arbitrary smooth function of its variables, which is determined from the initial conditions. Let us consider the simplest case g=O (cf. Refs. [ 1,5,7] ). From the initial condition (4) we get y=l,
(Y=O,
Ai=-i.
(13)
Proposition. The system ( 12)) ( 13) with g= 0 is compatible and has a unique solution for 62 0, /.I> 0 in the “internal” region O
x=x+ --XI ) O
/I= I/,/%7x’/48t,
At the boundaryx=x+
J2zx’/2fi
t.
(14)
(t) the solution cf,, y ) from
R.F. Bikbaev, KR. Kudashev /PhysicsL.ettersA 190 (1994) 255-258
( 5 ), (6 ) is continuously glued with 1 from (4). If x/ t+ + 0 the points (A,, &) closely come to the points (A,, A,). In this limit our solution cf;, y) from (5), (6) degenerates into the stationary soliton (breather),correspondingtoL2=&=i,A1=A3=-i. The behaviour of 2,=2,(x/t) on the complex plane 1 is shown in Fig. 1. The shock wave solution amplitude I u(x, t) 1=,/fm for t>> 1 is given in Fig. 2. Remark 1. Due to the Galilean invariance of ( 1)
257
The corresponding changes in formulae ( 1 1)- ( 14) are as follows: x++4bt+x+, IZ,+b+l,, A,+b+lZ,. Remark 2. The mirror symmetry x+ -x allows to solve the shock problem symmetric to (4)
Jfm=l,
XGO,
=2, V(X,0)
x>o,
=o .
(16)
the above analysis may be easily extended to the case of the shock wave with more general initial conditions,
We do not go into the details, but just note that the corresponding solution of the Whitham-NS system (8) has the form
Jfrn=L
x
i13=const,
x30,
and the picture of the solution u (x, t ) at t B 1 is symmetric with respect to the x-t -x transformation to the picture shown in Fig. 2.
=l, v(x, 0) =2b.
(15)
Im(S,)=O,
tRe(S,)-x=0,
(17)
Remark 3. The problems (4), (15), (16) are of course the simplest examples of the shock problems in the unstable model ( 1 ), (2). As we have shown above in these simplest cases the traditional ideas of Refs. [ 1,2 ] can be applied (after proper modification). For more general initial data it is necessary to invent a new technique. For example, how to solve the following shock problem,
U(x,O)=e’Z’, CO,
xv*)
x30?
We stress that in this paper unstable shock waves are considered in the framework of the GurevichPitaevskii [ 1] strategy, i.e. on the heuristical level. For example the problem of smooth gluing of the solution in the “external” and “internal” region (including the gluing of the solution phase) cannot be considered by this method. For example it may appear that the Whitham system solution described in our paper (see Fig. 1) corresponds to the 1-O shock problem (see the preceding formula). Such an effect is connected with the phase shift unambiguity in the Whitham theory. The problem of a more correct choice of boundary conditions will be the subject of our future investigation.
Fig. 1. The behaviour of&on the complex plane A.
I0
X-CO,
-I
Fig. 2. The picture of the shock wave amplitude 1u(x, t) 1 for tw 1.
Remark 4. Note that the above shock wave process ( 1)- ( 14) has the same typical time t x 1 as the ordinary modulational instability ( 1Im Si 1N 1 at t = 0).
258
R.F. Bikbaev, V.R. Kudashev /Physics
The work of V.R.K. was supported, in part, by “But-RUTEX”, and by an INTAS grant.
References [ 1] A.V. Gurevich and L.P. Pitaevskii, Sov. Phys. JETP 38 (1974) 291. [2] A.V. Gurevich and A.L. Krylov, Zh. Eksp. Teor. Fiz. 92 (1987) 1684;
Letters A 190 (1994) 255-258
R.F. Bikbaev, Teor. Mat. Fiz. 86 (1989) 474; Zap. Nauch. Sem. POMI 199 (1992) 25. [3] G.B. Whitham, Proc. Roy. Sot. A 283 (1965) 238; Linear and nonlinear waves (Wiley, New York, 1974). [4] B.B. Kadomtsev, Collective phenomena in plasmas (Nauka, Moscow, 1976) [in Russian]. [ 51R.F. Bikbaev and V.R. Kudashev, submitted to Pis’ma Zh. Eksp. Teor. Fiz. ( 1994). [6] M.V.Pavlov, Teor. Mat. Fiz. 71 (1987) 351. [7] A.M. Kamchatnov, Phys. Lett. A 162 (1992) 389.
PHYSICS
LETTERS
A
Physics Letters A 190 (1994) 255-258
ELSEVIER
Example of shock waves in unstable media: the focusing nonlinear Schrbdinger equation Ramil’ F. Bikbaev, Vadim R. Kudashev Institute OfMathematics, Chernyshevskii 112, Ufa 450000, Russian Federation Received 16 March 1994; revised manuscript received 25 March 1994; accepted for publication 19 May 1994 Communicated by A.R. Bishop
Abstract Dissipationless shock waves in modulational unstable one-dimensional media are investigated for the simplest example of the integrable focusing nonlinear Schriidinger (NS) equation. Our approach is based on the construction of a special exact solution of the Whitham-NS system, which “partially saturates” the modulational instability.
1. Introduction
Jfm=2a, =a,
It is well known that the focusing NS equation, v(x,O)=O,
iu,+u,+21uj2u=0,
(1)
may be rewritten as a hydrodynamic type system
x
x20, (4’ )
It is natural to call the solution v(x, t), v(x, t) ] or u(x, t) of the problem (l)-(4) the shock wave solution.
f;+2cfv)X=O
9
Vt+2w,-2f,=t(2fulf-fl/f2)x,
(2)
where
U=f"Z exp(igL v=h.
(3)
We consider ( 1 )- (3 ) with the following step-like initial condition,
Jfrn=2, =l,
x
v(x,O)=O, equivalent to Elsevier Science B.V. SSDIO375-9601(94)00402-B
(4)
The main purpose of this Letter is to describe the qualitative behaviour of the shock wave at t s 1. We note that this problem is much more difficult than the analogous problem for stable models [ 1,2 1. The typical and simplest shock problem in the modulational stable model was first studied by Gurevich and Pitaevskii [ 1 ] as an example for the Korteweg-de Vries (KdV) equation. In Ref. [ I] the modulational theory developed by Whitham [ 3 ] was used. It is important that the Whitham system for the KdV equation as well as for other stable models [ 31 is hyperbolic, which means that Whitham characteristic speeds are everywhere real: Im( Si) = 0. However for unstable models these speeds are generally speaking complex [ 31: Im(&) $0 (elliptic case).
R.F. Bikbaev, V.R. Kudashev /Physics Letters A 190 (1994) 255-258
256
This complicates very much any analytical investigation of unstable models due to exponential growth of the linear perturbations of the (locally) constant background. In particular there are obvious obstacles in studying of shock problems in unstable systems, which are closely related to the classical problem of the modulational instability of the monochromatic wave in ( 1) (see, for example, Ref. [ 4 ] ). In this Letter we demonstrate how to overcome these difficulties on the concrete example ( 1 )- (4). We shall show that a famous strategy [I ] may be modified and successfully applied to ( 1)- (4 ) . Our approach is based on the concept of “partially saturating modulational instability solutions” of elliptic Whitham-NS equations proposed in Ref. [ 5 ] for the description of the modulational instability in ( 1). A special subclass of Whitham-NS system solutions with vanishing imaginary parts Im (S,) = 0 of some (not all! ) characteristic speeds (see ( 12 ), ( 17 ) ) play the key role in our construction.
AAl2 1_iu1232,A31
s=u+
>
s,=u+ 1 -P&4/~,,
s,=s:,
s,=sl;,
(9)
pan,
where ,l ii= 2 i - Aj, K, E are the complete elliptic integrals of the first and the second kind, respectively, m=l2J43l~32~,4.
3. Solution of the Whitham-NS
system and longtime behaviour of the NS-hydrodynamic shock wave
We construct an approximate solution of the shock problem ( 1)-( 4) for t B 1 as follows. In the “external” region on the (x, t) plane: (X-CO,s-x’(t)), the solution is a zero-phase one,
Jfrn=2,
x
,
z)o(x,t) =o . 2. One-phase NS equation solution and its Whitham-NS
modulations
It is well known that the one-phase solution of ( 1), (2) has the form J;(e) =f3 + ti -f,)dn’
fi-fi “‘fX 3 eFx-ut)
(6)
wherefiafafiaO>j& A=,/xaO, dn is the Jacobi elliptic function. The elliptic spectral curve corresponding to (5 ), (6) has branching points Ai, i=l,2,3,4;1,=;1:,&=Ifsuchthat (cf.Ref. [6]) n,~a-iy=$U-t~-?ii(~+~), n,~8-is=tU+fJ-f3-ti(~-~).
(7)
The Whitham-NS equations for ( 1)) (2)) ( 5 ) - ( 7 ) can be presented in the diagonal form (cf. Ref. [ 6 ] )
2
+s,(n)
2
=o
)
i= 1,2,
3,4,
In the oscillation region 0
(t) the “internal” solution (5 ), ( 6 )
(8), (9). In this the Whitham-NS (11)
Im(S3) =O , t Re(S,) -x=g(p, (5)
U&9)=+u-~&).
(10)
A1Econst,
m} ,
1-e;
’
6) ,
(12)
where g(/?, 6) is an arbitrary smooth function of its variables, which is determined from the initial conditions. Let us consider the simplest case g=O (cf. Refs. [ 1,5,7] ). From the initial condition (4) we get y=l,
(Y=O,
Ai=-i.
(13)
Proposition. The system ( 12)) ( 13) with g= 0 is compatible and has a unique solution for 62 0, /.I> 0 in the “internal” region O
x=x+ --XI ) O
/I= I/,/%7x’/48t,
At the boundaryx=x+
J2zx’/2fi
t.
(14)
(t) the solution cf,, y ) from
R.F. Bikbaev, KR. Kudashev /PhysicsL.ettersA 190 (1994) 255-258
( 5 ), (6 ) is continuously glued with 1 from (4). If x/ t+ + 0 the points (A,, &) closely come to the points (A,, A,). In this limit our solution cf;, y) from (5), (6) degenerates into the stationary soliton (breather),correspondingtoL2=&=i,A1=A3=-i. The behaviour of 2,=2,(x/t) on the complex plane 1 is shown in Fig. 1. The shock wave solution amplitude I u(x, t) 1=,/fm for t>> 1 is given in Fig. 2. Remark 1. Due to the Galilean invariance of ( 1)
257
The corresponding changes in formulae ( 1 1)- ( 14) are as follows: x++4bt+x+, IZ,+b+l,, A,+b+lZ,. Remark 2. The mirror symmetry x+ -x allows to solve the shock problem symmetric to (4)
Jfm=l,
XGO,
=2, V(X,0)
x>o,
=o .
(16)
the above analysis may be easily extended to the case of the shock wave with more general initial conditions,
We do not go into the details, but just note that the corresponding solution of the Whitham-NS system (8) has the form
Jfrn=L
x
i13=const,
x30,
and the picture of the solution u (x, t ) at t B 1 is symmetric with respect to the x-t -x transformation to the picture shown in Fig. 2.
=l, v(x, 0) =2b.
(15)
Im(S,)=O,
tRe(S,)-x=0,
(17)
Remark 3. The problems (4), (15), (16) are of course the simplest examples of the shock problems in the unstable model ( 1 ), (2). As we have shown above in these simplest cases the traditional ideas of Refs. [ 1,2 ] can be applied (after proper modification). For more general initial data it is necessary to invent a new technique. For example, how to solve the following shock problem,
U(x,O)=e’Z’, CO,
xv*)
x30?
We stress that in this paper unstable shock waves are considered in the framework of the GurevichPitaevskii [ 1] strategy, i.e. on the heuristical level. For example the problem of smooth gluing of the solution in the “external” and “internal” region (including the gluing of the solution phase) cannot be considered by this method. For example it may appear that the Whitham system solution described in our paper (see Fig. 1) corresponds to the 1-O shock problem (see the preceding formula). Such an effect is connected with the phase shift unambiguity in the Whitham theory. The problem of a more correct choice of boundary conditions will be the subject of our future investigation.
Fig. 1. The behaviour of&on the complex plane A.
I0
X-CO,
-I
Fig. 2. The picture of the shock wave amplitude 1u(x, t) 1 for tw 1.
Remark 4. Note that the above shock wave process ( 1)- ( 14) has the same typical time t x 1 as the ordinary modulational instability ( 1Im Si 1N 1 at t = 0).
258
R.F. Bikbaev, V.R. Kudashev /Physics
The work of V.R.K. was supported, in part, by “But-RUTEX”, and by an INTAS grant.
References [ 1] A.V. Gurevich and L.P. Pitaevskii, Sov. Phys. JETP 38 (1974) 291. [2] A.V. Gurevich and A.L. Krylov, Zh. Eksp. Teor. Fiz. 92 (1987) 1684;
Letters A 190 (1994) 255-258
R.F. Bikbaev, Teor. Mat. Fiz. 86 (1989) 474; Zap. Nauch. Sem. POMI 199 (1992) 25. [3] G.B. Whitham, Proc. Roy. Sot. A 283 (1965) 238; Linear and nonlinear waves (Wiley, New York, 1974). [4] B.B. Kadomtsev, Collective phenomena in plasmas (Nauka, Moscow, 1976) [in Russian]. [ 51R.F. Bikbaev and V.R. Kudashev, submitted to Pis’ma Zh. Eksp. Teor. Fiz. ( 1994). [6] M.V.Pavlov, Teor. Mat. Fiz. 71 (1987) 351. [7] A.M. Kamchatnov, Phys. Lett. A 162 (1992) 389.