new approach to periodic solutions of integrable equations and

NEW APPROACH TO PERIODIC SOLUTIONS OF INTEGRABLE EQUATIONS A N D N O N L I N E A R THEORY OF M O D U L A T I O N A L INSTABILITY

A.M. KAMCHATNOV

a'b

alnstitute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow Region, 142092, Russic bTRINITI, Troitsk, Moscow Region, 142092, Russia

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Physics Reports 286 (1997) 199-270

New approach to periodic solutions of integrable equations and nonlinear theory of modulational instability A . M . K a m c h a t n o v a'b, ~ aInstitute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow Region, 142092, Russia b TRINITI, Troitsk, Moscow Region, 142092, Russia Received August 1996; editor: D.L. Mills Contents

1. Introduction 2. Periodic solutions of nonlinear integrable equations 2.1. Basic principles of the finite-band integration method 2.2. Nonlinear Schrrdinger equation 2.3. AB-system 2.4. Self-induced transparency 2.5. Heisenberg model 2.6. Uniaxial ferromagnet (easy axis case) 2.7. Stimulated Raman scattering 2.8. Derivative nonlinear Schrrdinger equation 3. Whitham modulation theory 3.1. Averaging method 3.2. Whitham equations for NLS, DNLS, Heisenberg model and uniaxial ferromagnet cases 3.3. Whitham equations for the AB-system 3.4. Whitham equations for SIT case 3.5. Whitham equations for stimulated Raman scattering case

202 203 203 206 213 215 218 221 229 235 242 242

243 245 246

3.6. Phase velocity of nonlinear waves and Whitham equations 4. Nonlinear theory of modulational instability and formation of solitons on the sharp front 4.1. Asymptotic evolution of an initially local disturbance in modulationally unstable systems described by the NLS equation 4.2. Domain formation in one-dimensional uniaxial ferromagnet 4.3. Formation of solitons on the sharp front of a pulse in the AB-system 4.4. Formation of solitons on the sharp front of the SIT pulse 4.5. Formation of solitons on the sharp front of the SRS pulse 5. Conclusion References

247 249

249 256 257 259 263 268 268

247

Abstract

A new method of finding the periodic solutions for the equations integrable within the framework of the AKNS scheme is reviewed. The approach is a modification of the known finite-band integration method, based on the re-parametrization of the solution with the use of algebraic resolvent of the polynomial defining the solution in the finite-band integration method. This approach permits one to obtain periodic solutions in an effective form necessary for applications. The periodic

1E-mail: [email protected]. 0370-1573/97/$32.00 © 1997 Elsevier Science B.V. All rights reserved

PH S 0 3 7 0 - 1 5 7 3 ( 9 6 ) 0 0 0 4 9 - X

A.M. Kamchatnov/Physics Reports 286 (1997) 199-270

201

solutions are found for such systems as the nonlinear Schrrdinger equation, the derivative nonlinear Schrrdinger equation, the Heisenberg model, the uniaxial ferromagnet, the AB system, and self-induced transparency and stimulated Raman scattering equations. The modulation Whitham theory describing the slow modulation of periodic waves is expressed in a form convenient for applications. The Whitham equations are obtained for all abovementioned cases. The technique developed is applied to the nonlinear theory of modulational instability describing the transformation of a local disturbance expanding into a nonuniform region presented as a modulated periodic wave whose evolution is governed by the Whitham equations. This theory explains the formation of solitons on the sharp front of a long pulse.

PACS." 03.40.Kf; 42.65.Sf; 47.35.+i; 75.30.Ds Keywords: Solitons; Periodic solutions; Whitham equations; Modulational instability

A.M. Kamchatnov/ Physies Reports 286 (1997) 199-270

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1. Introduction

It is well known that the notion of solitons penetrates into all contemporary nonlinear physics. To observe this, it is sufficient to look through several monographs or review articles [1-10]. But this search shows also that the concept of a single soliton (or rarified gas of solitons) is not sufficient for adequate description of some real situations. Simple examples of those are high-temperature quasi-one-dimensional systems [9, 11] and the process of solitons formation from initially intensive and long pulse [4, 12]. In such situations, it is necessary to use another type of solutions of the nonlinear equation under consideration - the periodic ones which may be considered as a 'dense lattice' of solitons, if the distance between solitons is of the same order as soliton's width. In mathematical literature the problem of finding the periodic solutions of integrable equations has been considered for a long time. Firstly, it was solved for the case of Korteweg-de Vries (KdV) equation ut -

6uux + u~x = 0

in the remarkable articles [13-18] with the use of nontrivial methods of algebraic geometry and functional analysis. It was found that (multi-) periodic solution of KdV equation is determined by a hyperelliptic curve l2 =

(1)

and can be expressed in terms of the corresponding theta-functions. Later this method was generalized on the sine-Gordon equation, Utt -- Uxx ~- s i n u ,

and nonlinear Schr6dinger (NLS) equation, iut + u~ &

2

lUl2 U ~--- 0

in [19--23] (see also the review article [24] about this 'finite-band method'). While in all these cases the formulas for periodic solutions look very similar, in the sine-Gordon and (focusing) NLS equations cases the additional problem of extraction of the 'real' solutions arises (see, e.g., [25-28]). The periodic solutions obtained by this method have rather complicated form which prevents their applications to real physical situations. In Section 2 of this report we present a simple modification of the finite-band integration method which solves this 'effectivization' problem in the simplest and the most important for applications one-phase case (the solution depends only on one phase W = x - Vt, not speaking about the exponential factor depending on 'pseudophase'). This method will permit us to find periodic solutions for a number of important equations. The solutions are parametrized by the zeros of the algebraic resolvent of the polynomial P(,~) in (1) without any additional constraints arising in the usual finite-band method. The advantage of such a presentation becomes especially important in discussions of the modulated periodic waves. In real physical situations the periodic wave is modulated, that is the parameters defining it depends on space and time coordinates. If such a dependence is slow (the parameters change little in one wavelength and one period), then the evolution equations for these parameters can be averaged over

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203

fast oscillations of the wave, and this averaging yields the Whitham modulation equations [29, 30, 4]. Whitham obtained his modulation equations for KdV periodic solution ('cnoidal wave') without use of the inverse scattering transform (IST) method. Later it was found that the Whitham method is formulated most naturally in terms of this method [31]. That permitted one to obtain the Whitham equations goveming the modulations of sine-Gordon [32] and NLS [33] wavetrains. Section 3 is devoted to simple and convenient method of obtaining the Whitham equations which can be applied to a number of physically important equations. The Whitham equations govem the parameters 2i (the polynomial P(2) zeros). Two different cases should be distinguished: (a) the parameters 2i are real, and (b) the parameters 2i are complex. The first one corresponds to the modulationally stable waves. The solution of Whitham equations leads to the description of evolution of long and intensive pulses decaying ultimately into separate solitons [12, 34-42]. The other case corresponds to the modulational instability of the periodic wave. In Section 4 the technique developed is applied to the nonlinear theory of modulational instability. It has been known for a long time that the plane wave with constant amplitude is unstable with respect to small perturbations in many physical systems [43-45], and, in particular, when such a wave is described by the NLS equation (see, e.g., [46]). The computer simulations [1], [47,48] showed that the local perturbation transforms into an expanding nonuniform oscillatory region. Its central part has maximal modulation and looks like a number of solitons while weakly modulated edges propagate with some group velocity along the unmodulated wave. The whole region can be described as modulated nonlinear periodic wave which evolution is govemed by the corresponding Whitham equations [49-52]. Analogous problems arise in the theory of formation of solitons on the sharp front of a long pulse. In Section 3 we shall consider such an approach to this kind of problems. We think that the examples presented in this report show wide,applicability of the methods under consideration.

2. Periodic solutions of nonlinear integrable equations 2.1. Basic principles o f the finite-band integration method

A great number of physically important equations can be integrated by the inverse scattering transform method in framework of the AKNS scheme (see, e.g., [53,5,8]), and we confine ourselves to this scheme. It is based on possibility of presenting the equation describing the evolution of the wave uk(x,t), which propagates along the x-axis, as compatibility condition of two linear systems: aq,,lax

=

+ G4,

,

-

,

(2) a

21ax =

(3) a~z/at = c~1 - A~2 .

204

A.M. Kamchatnov/Physies Reports 286 (1997) 199-270

Requiring ~ 2 ~ , 2 / O x ~ t = ~2~tl,2/~t~X , we obtain the equations ~F/~t - ~A/~x + C G - B H = O,

(4)

~G/~t - ~B/~x + 2 ( B F - A G ) = O, OH/~t - OC/~x + 2 ( A H - C F ) = O .

The coefficients in (2),(3) depend on functions u k ( x , t ) and on the arbitrary spectral parameter 2. They are chosen so that Eqs. (4) coincide with the evolution equations under consideration when one takes into account the arbitrariness of 2. Considering (2) as a scattering problem of waves ~kl,~2 on some potential defined by the coefficients F , G , H , we find that the evolution of the 'scattering data' due to (4) (i.e., actually, due to the evolution equation under consideration) is very simple (see, e.g., [53, 5, 8]). This permits one to 'linearize', in principle, the problem of evolution of the initial pulse or to find some special classes of particular solutions, such as multi-soliton solutions which correspond to nonreflective potentials [4-8]. The finite-band integration method uses different technique briefly discussed here. For the first time it was developed for the KdV equation in [13-18]. In KdV equation case the operator on the right-hand side of (2) is such that one of the components (~bl,~k2) can be excluded and this operator reduces to the second-order differential operator L coinciding with the quantum-mechanical Schrrdinger operator for a particle moving in the potential u(x, t). If evolution of u(x, t) is governed by the KdV equation, then the spectrum of L does not depend on t and in the case of periodic potential u ( x ) it consists of some number of bands separated by lacunae. The corresponding Bloch function of the quantum particle moving in the periodic potential is a single-valued function of the spectral parameter 2 defined on the two-sheet Riemann surface, obtained by joining together two complex planes of spectral parameter 2 with cuts along lacunae. The evolution of u ( x , t ) is determined by the movement of the points of the auxiliary spectrum, where the Bloch function has zeros, along this Riemann surface. In the KdV case the operator L is self-adjoint and its spectrum lies on the real axis, so that lacunae are the segments of this axis. One can show that each point of auxiliary spectrum lies inside 'its' lacuna and moves along cycle around this lacuna. But if the operator in (2) is not self-adjoint, the spectrum can be complex and the term 'lacuna' no longer has a clear meaning. Therefore, the loci of the auxiliary spectrum points are not prescribed beforehand, and their determination is not a simple problem. Just this difficulty gives rise to the above-mentioned ineffectiveness of the finite-band method in the case of equations corresponding to not self-adjoint operator (2) in the AKNS scheme. Let us proceed to a more detailed formulation of the method. The systems (2), (3) have two basic solutions, 0P1,~92) and (~01, ~02) , which can be used to build a vector with the spherical components, f = -(i/2)(~9~q~2 + ff2qh),

g = ~blq~,,

h = -~b2~p2,

(5)

satisfying the following linear systems: ~ f /~x = - i l l 9 + iGh , ~9/~x=2iGf+2Fg, ~h/~x = - 2 i H f

- 2Fh,

(6)

A.M. Kamchatnov / Physics Reports 286 (1997) 199-270

205

and ~ f / ~ t = - i C 9 + iBh ,

(7)

O9/~t = 2 i B f + 2 A g , Oh/Ot = - 2 i C f - 2 A h .

During the evolution according to (6) and (7), the length of the vector with components (5) is preserved, that is the quantity f 2 _ oh = P(2)

(8)

is independent of x and t and, hence, is the function of 2. Periodic solutions are distinguished by the condition that P(2) be a polynomial in 2. We can consider this condition as an ansatz which will be justified by the results obtained in what follows. The simplest nontrivial one-phase periodic solution corresponds usually to the fourth degree polynomial, 4

P(2) = 1--[ (2 - 2i) = A4 -

s 1 A3 ~- $2 22 - s 3 A q- $4 ,

(9)

i=1

where At are the zeros of the polynomial. They are just the parameters determining the periodic solution and comprise the above-mentioned constant specmnn of the 'Bloch wave' problem. As we shall find in each particular case, the systems (6) and (7) have the polynomial solution f = J~A2 - f 2 + f z ,

9(A) = ~b(2)(A - # ) ,

h(2) = - @ 2 ) ( 2 - # * ) ,

(10)

with depending on uk(x,t) coefficients. The dependence of the so-called 'auxiliary spectrum point' (the zero of 9-function in complex A-plane) #(x, t) on x and t is determined by simple equations [17, 23], which can be derived by substitution of (10) into the second equations of systems (6), (7) and putting A = #: ~# ~x -

2iG(#)~.. ~

Jr#),

~#_ 8t

2iB(#) dp(#~ f ( # ) "

(11)

One can find an analogous equations for functions #*(x, t). Now let us take into account the relation which (8) gives at A = #, f(#) = ~-~,

(12)

and the fact that the coefficients of f ( # ) in (11) prove to be, as we shall see below in each particular case, constant, so that ~# ~-~ = const, i ~ - ~ ,

~# ~t =

const,i ~

(13)

that is # depends only on the phase W = x - Yt,

d # / d W = const- ix/fi-(#),

(14)

where the phase velocity equals to V = B ( # ) / G ( # ) = C(#* ) / H ( # * ) = constant.

(15)

206


In case of fourth degree polynomial P(h) (see (9)) the solution of Eq. (14) can be easily expressed in terms of elliptic functions, and, when we know #(x, t), it is not difficult to find the solution of the equation under consideration. Just this approach, generalized on the higher degree polynomials P(h), is the foundation of the finite-band integration method [4,24]. But long ago [19,20] it was noted that initial conditions #(0, 0) for Eq. (14) and uk(0, 0) have to be chosen so that condition (8) fulfils. This condition can be fulfilled without any difficulties in the case of self-adjoint operators (2), when hi lie on the real axis and # moves around its segment between two zeros of the polynomial P(h). But in the case of not self-adjoint operators (2) and complex spectrum hi the problem of finding the locus of #(x, t) may be rather complicated. As we noted above, just this problem makes the solution found in this way ineffective. The situation becomes even worse when we consider nonuniform problem with varying parameters hi so that condition (8) is now local and changes with evolution of hi. It is clear that one can obtain an effective form of the solution only if the constraint (8) is satisfied automatically. That prompts the method of the solution of the problem. Indeed, the identity (8) must be fulfilled during the evolution of the periodic wave and each its state can be taken as initial condition. Therefore, this identity (8) determines the curve on the complex h-plane along which the parameter ~t moves during the evolution according to Eq. (14). On introducing the coordinate v along this curve, so that p = #(v), and on passing to the motion equations for v, we become sure that (8) fulfils automatically. This approach was suggested in [54] for NLS and DNLS equations and has shown its applicability to a number of other equations. 2.2. Nonlinear Schr6dinger equation

Nonlinear Schr6dinger (NLS) equation is one of the most important integrable equations. It has universal nature and describes the evolution of wave envelopes in media with dispersion and weak nonlinearity [1,5, 6]. Discovery of the integrability of the NLS equation by the IST method promoted to a large extent the understanding of the generality of this method and its subsequent fast development. The periodic solutions of the NLS equation has been studied in [20] by the standard finite-band method (see also [21, 55, 56]). We shall follow here to the approach of Ref. [54]. Let us consider the 'difficult' from the point of view of the finite-band method the focusing NLS equation, iut + u~ + 2 [ul2u = 0.

(16)

Eqs. (4) coincide with (16) if one takes as their coefficients the following expressions: F=-i2, G=iu, H=iu*, A = i l u l 2 - 2 i 2 2, B = - u x + 2 i h u ,

C=u~+2ihu*.

(17)

Then the solution of the systems (6) and (7) has the form f = h2

-

-

f h -]- f 2 ,

g = iu(2 -- # ) ,

h = iu*(2 -- #*),

(18)

provided the coefficients of these polynomials satisfy the equations ~fi/~x = ~ fi/~t = 0 , ~fz/~X = -ilu[e(/z - / ~ * ) ,

(19) ~f2/Ot = -U*Ul~ - uxu*#* ,

(20)

A.M. Kamchatnov/PhysicsReports 286 (1997) 199-270 Ux = 2iu(/~ - f ) ,

ut = 2ilu[ 2 + 2lUg + 4iuf2,

u,# + U~x = - 2 i u f 2 ,

ut# + u#t = 2ilul2u/~ + 2f2Ux

207

(21)

(22)

as well as their complex conjugate. Besides that we must satisfy condition (8), which after inserting of (18) takes the form (22 - f 2 + f2) 2 + [u12(2 - / ~ ) ( 2 - #*) = •4

__ S1,~3 ~_ $2/~2 __ $3,~ " + $4 .

(23)

Comparing the coefficients of 2k on both sides of (23) leads to the conservation laws

2 f = sl,

f z + 2f2 + ]u[2 = s2,

2 f f2 + lu[2(/~ +/.t*) = s3,

(24)

f22 + lul2## * = s4.

(25)

Eqs. (24) give

f = &/2,

f2 = 1(s2 - ( s ] / 4 ) - lu12),

(26)

that is f is constant in accordance with (19) and (20) can be reduced to the equations

~lu]2/Sx = 2ilul2(/.t - #*),

~[ul=/Ot-- 2~1 lulZ(~ - #*) = sl(alul2/Ox).

(27)

Eqs. (21) yield the important relation between u and #:

8u/~x = 2iu(/~

-

Sl/2),

~u/Ot = 2i(s2 -- S]/4)U + Sl ~U/~X,

(28)

and Eqs. (22) give the evolution equations for/~:

~#/~x = - 2 i f ( u ) = - 2 i ~ ,

~#/~t = - 2 i & f ( / ~ ) = sl~#/~x.

(29)

Recall that they can be obtained by means of substitution of g from (18) into the second equations of (6) and (7) and putting 2 = # (see (13)). As follows from (27) and (29), lu[2 and/~ depend only on the phase

W = x - Vt,

(30)

where the nonlinear phase velocity V of the periodic solution of the NLS equation is expressed in terms of zeros 2i, i = 1,2, 3,4, of the polynomial P(2): 4 V = -$1 = 7E ~ii=1

(31)

The dependence of the field intensity lul 2 and of ~ on the phase W is defined by the equations

dlu]2/dW = 2i[ul2(/, - / ~ * ) ,

dl(dW = - 2 i Pv/-PT~.

(32)

From (28) we have

u(x, t) = exp[2i(s2 - s~/4)t] ~(W),

(33)

provided

d'~/dW = 2i(/.t - Sl/2)~.

(34)

208


Now let us recall that #(W) must move with change of W along some curve on the complex plane to satisfy condition (8), that is we have to take into account Eqs. (25). We see that the intensity v = ]u[2 can serve as a suitable parameter along this curve, so that Eqs. (25) take the form # # * = - ( 1 / 4 v ) [ v 2 - 2 v ( p + s 2 / 8 ) + s q + p2 _ 4 r ] ,

# + #* = s / 2 - q / v ,

(35)

where we have introduced the standard in algebra of polynomials notations 32

s = sl,

p = s2 - ~s 1,

r

I S 2 ( S 2 __ 3 S ~ )

=

s 4 -~-

1

12

q = i s l ( s 2 - ~sl) - s3

(36)

l~SlS3 •

Thus we see that # and #* are the solutions of the quadratic equation whose coefficients are given by expressions (35). It is remarkable that its discriminant is equal to ~ l ( v ) / v 2, where ~(v)

= v 3 - 2 p v 2 + (p2 _ 4 r ) v + q2

(37)

is the cubic resolvent of the polynomial P(2) (see, e.g., [57]). Thus, we have #, #* -- (s/4) - [q -t- ~ ] / 2 v .

(38)

As is known [57], the zeros vl, v2, v3 of the cubic resolvent ~ ( v ) are related to the zeros 2,-,i = 1,2,3,4, of the polynomial P(2) by the formulae •1 z

1 --~(21

V2 •

1 --~(/~1 -- 22 -- 23 "-~ 2 4 ) 2,

V3 z

1 --~(21

-- 22 -q- 23 -- 2 4 ) 2 ,

-~ 22

-

-

23

-

-

(39)

/~4) 2 •

The quantity v = [u[ z is real by definition and can oscillate between two positive zeros (39) of the resolvent ~(v). Therefore, the zeros 2i of the polynomial P(2) have to consist of two complex conjugate pairs 21 = ~ + iy,

22 =/~ + i6,

23 =

~ - iT,

24 =

/~ - i5,

(40)

and formulae (39) yield vl = - ( ~ - f l ) 2 ,

v2=(7--6) 2 ,

v 3 = ( 7 + 3 ) 2.

(41)

This means that v oscillates in the interval v2 _< v < v3, where the resolvent ~ ( v ) is negative, and for the trajectory of # (its locus) we have #(v)=(s/4)-[q+i~]/2v,

v2 0 the resolvent's zeros are ordered according to vt < - 1 < v2 < v3 < 1 < v4 and $3 oscillates in the interval -1

(194)

7 2 ,


235

/3~/72 -- G2 -~- ]7V//32 + 0 .2

(b)

vl=

0.2

/3v/]72 _ 0.2 _ > / / 3 2 "1)4 ~

,

+ 0.2

v2=v3=0,

for - / 3 2 < 0.2 < ]72,

0" 2

? -]72 (C)

1)1 = V4 =

--

0"2

,

1)2 =

--•3

=

--

(208)

~ / 3 2 "~- O'2 0.2

__/32 for

0.2
_/32 the zeros v2 and v3 coincide with each other what leads to the wave with constant amplitude. We shall write here the solution for the case (a) 0.2 > 72: $3 = -(1/0.)V/0.2 - ]72 ,

R3

A v / - ~ _ ]72 , 2aX/(de/4) _q_]72

( 20.v - ]72 )

S+ = (]7/0.) exp ~ 1 v/(A2/4) + y2 ~ R+

=

,

]7 exp {. / 20.V/-~- _ ]72 ) 1 { . 20.~/(A2/4) + ]72 \ v/(A2/4) + ]72 /

(209)

However, at 0.2 < -/32 the other two zeros Vl and v4 coincide, and we have special periodic solution _/32 x/Q- + ]72 cos(2x//32 + ]72W) _ V/-ff_/32

$3=

J

V/"ffq-]72-vlff-/32c°s(2v/f12q-]72W)

'

j

=

_0.2

>

f12 .

The same behavior holds for ~ ¢ 0, as we can see from Fig. 6, where the dependence o f resolvent's zeros on 0.2 is shown in the case o f the parameters values ~ = ]7 = 1, /3 = 2, a = 0. These curves can be considered as deformations o f the curves in Fig. 5, when we pass from 6 = 2 to 6 = 0. Again the zeros v2, v3 coincide for 0.2 > - 4 (/32 = 4), and Vl, v4 for 0.2 < - 4 . In the former region the periodic solution goes to the wave with constant amplitude and in the latter region it goes to the special periodic wave with m = 0. It is important that in both cases the wave is expressed in terms of complex spectrum 2i, what leads to its modulational instability. The periodic solution for the two-photon propagation case [68] can be found in the same w a y [71]. It should be noted only, that due to different normalization condition - S 21 - S ~2 +S~2 = 1 (instead o f (179)), the variable IS31 > l and oscillates between another pair of resolvent's zeros vl and v4, which leads to slight modification o f the formulae obtained in SRS case. By now we have considered two classes o f resolvents: Ferrari resolvent (37) and the resolvent (135) with its particular case (111), corresponding to the limit J -+ 0 (the fourth zero v4 oc 1/J goes to infinity). N o w we shall turn to one more example with a different resolvent.

2.8. Derivative nonlinear Schrrdinger equation The derivative nonlinear Schrrdinger ( D N L S ) equation

iu~ + Uxx -4- 2i(]ul2U)x = 0

(210)

plays important role in the physics o f ultrashort optical pulses, wave propagation in magnetized plasma, etc. Its integrability by the IST method was established by Kaup and Newell [72] with derivation o f the corresponding soliton solutions. Here we shall find its periodic solution (see [54, 73]).

A.M. Kamchatnov/ Physics Reports 286 (1997) 199-270

236

I

-1o

I

I

1o

2o

VI=V4

Fig. 6. T h e s a m e c u r v e s as in Fig. 5 b u t for the s p e c t r u m v a l u e s "~1 = 1 + i, 23 = 1 - i, ,~2 = 24 = 2. T h e s e c u r v e s c a n be c o n s i d e r e d as d e f o r m a t i o n s o f t h o s e in Fig. 5 as w e go f r o m 6 = 2 to 6 = 0.

We shall discuss the DNLS equation (210) with a minus sign before the last term. This sign can be easily inverted by means of simple substitutions. The DNLS equation can be expressed as a compatibility condition (4), if we take [74] F = -222 ,

G = 22u,

A = -(8i24 + 4ilu[222), C = 823u * + ( - 2 i u * +

H = 22u* , B = 823u + (2iUx + 41u12u)2,

(211)

4lul2u*)2.

Now, as one can find from (6) and (7), the polynomial (8) contains only the even degrees o f 2, and nontrivial periodic solutions correspond to the sixth and eighth degree polynomials P(2). It will be clear from the following that solutions corresponding to the sixth degree of P ( 2 ) are particular cases of solutions corresponding to the eighth degree of P(2). Therefore, we assume that P ( 2 ) is equal to 4

P ( 2 ) = 1--[ (22 -- ) 2 ) = 28 __ Sl)~6 q_ $2/~4 __ S3j~2 -4- S 4 ,

(212)

i=1

where +2i are the zeros of the polynomial. Then (6) and (7) lead to the expressions f = 24 - s122 + f2 ,

9 = u2( 22 - P) ,

h = u'2(22 - #*)

(213)

and u satisfies the equations

~u/~x =

-4iu(fl - #),

au/Ot=

8iu[Zf2 - ( f l - #)(2f~ + ]ul2)],

(214)


237

where the quantities f l , f 2 , [u[2,P,# * are connected by the following constraint, which is a result of

(8): (24 -- f,22 + f2) 2 - lul2

2(2 2 -

2

-

-- P ( 2 ) .

(215)

The dependence of # on x and t can be obtained from (6) and (7) if one puts 22 = # and takes into account that f ( # V 2 ) = ~ .

~l~/~x = 4i~/r~-/~1/2),

81~/~t = 8i(2f~ + [ u l 2 ) ~ ) .

(216)

As in preceding cases, we see that constraint (215) can be considered as an equation for the locus o f / ~ in the complex 2 plane, and, as in the NLS equation case, the variable v = lul 2 is the natural coordinate along this locus. Comparing the coefficients of 2 t on both sides of (215), we have 2fl + v = Sl ,

f l 2 + 2f2 + v(# + #*) = s2 ,

2 f l f 2 + v~#* = s3 ,

(217)

f 2 = s4 .

One can obtain from this system the expression for

I~(v) = (1/8v)[4s2 + 8 x / ~ - (v

-

$1) 2 :~ i x / Z ~ v ) ] ,

(218)

where ~ ( v ) is a fourth-degree polynomial in v: ~(y)

= y4 __ 4SlY3 + (6s 2 _ 8s2 + 48x/~)v 2

- (4s~ - 16s,s2 + 64s3 + 32sl x/&)v + (-s~ + 4s2 -I- 8x/~) 2 .

(219)

The root in these formulae is considered to be v'& = 21222324. The resolvent (219) zeros are related to the zeros of P ( 2 ) by simple symmetric formulae: the zeros V1 = (21 -{- 22 + 23 -- 24) 2 ,

V2 = (21 -~ 22 -- 23 -~ 24) 2 ,

Y3 = (21 -- 22 + 23 "q- 2 4 ) 2 ,

1)4 ~--_( - - 2 1 ~- 22 "]- 23 ~- 24)2

(220)

correspond to the upper sign in (219), and the zeros Yl = (21 "~- 22 ~- 23 -~- 24) 2 ,

Y2 = (21 ~- 22 -- 23 -- 24) 2 ,

Y3 = (21 - - 22 -'~ 23 -- 24) 2 ,

Y4 ~" ( - - 2 1 "Jr- 22 -~- 23 - - 24) 2

(221)

correspond to the lower sign. This can be proved by a simple check of the Virte formulae. The passage mentioned above to the sixth-degree polynomial P ( 2 ) can be accomplished by removing one of the zeros 2i. As follows from (216) and first formula (217), the variable # depends only on the phase

W = x + 2sit,

dl~/dW = 4 - 4 i ~ / ~ 1/2) .

From (214) and (217) we find that

dlu[2/dW = 4ilul2(/.t - # * ) ,

(222)

238


and then (218) gives

dv/dW = v / - J l ( v ) .

(223)

This equation can be easily resolved by means of elliptic functions. If v is known, then u(x, t) can be obtained from (214). With help of (217) we get

~u ~u/~t = 1 6 i v ~ u + 2sl ~--~ so that

u(x, t) = e x p ( 1 6 i x / ~ t ) ~ ( W ) ,

(224)

where tT(x,t) should satisfy the equation

d~/dW

=

1 4i ( - 1 s 1 ~- ~v -~- #)/~.

(225)

It is clear that the zeros 2i should be numbers such that v oscillates between two positive values. The polynomial ~ ( v ) has four zeros vg, which are given by (220) or (221) depending on the choice of sign in (219). If only two v; are real and positive, then let us enumerate 2~ so that these vi are vl, v2, and Vl > v2. If all the vi are real and positive, then let us enumerate 2i so that vl _> v2 >_ v3 >_ v4. Thus, as it is clear from (223), the variable v can oscillate in the intervals vl > v > v2 or v3 > v > v4, where ~ ( v ) v > v2. (iii) All four 2i are real and

21 ~ 22 >__23 >_ 24.

(229)

Both (220) and (221) yield the real and positive vi corresponding to different periodic solutions for which the variable v oscillates in the intervals Vl > v > v2 or v3 > v > v4. (iv) If two 2i are complex conjugate and two others are real 21 = ~ + ifl,

)~2 = ~ - ifl,

23 = 7,

24 = 3 ,

(230)

239


then ( 2 2 0 ) yields vl = (2~ + 7 - 6) 2

V2 = (2~ - 7 + 6 ) 2

v3 = (y + 6 + 2ifl) 2 ,

v4 = (7 + 6 - 2ifl) z

(231)

and ( 2 2 1 ) leads to the same values o f vi with different sign before 6. N o w w e shall turn to finding the periodic solutions. Let us discuss at first the case w h e n the variable v oscillates in the interval v~ _> v > v2 and v3, v4 are also real. W e shall c h o o s e initial conditions so that v = v~ at W = 0. T h e n ( 2 2 3 ) leads to the solution v =

71(V2 - - V4) -~ (71 - - V2)V4 sn2(~/(vl - v3)(v2 - v 4 ) W / Z , m )

- v3)(v2 - 7 4 ) W / 2 , m )

V2 - - 74 -~- (71 - - 7 2 ) s n 2 ( x / ( v l

,

(232)

where m =

(71 - - 72)(~) 3 - - 7 4 )

(233)

(71 - - 73)(112 - - 7 4 )

N o w w e introduce the zeros o f the Weierstrass cubic b y m e a n s o f the expressions el = 1 1 2 ( v l - v3)(v2 - v4) - (vl - v2)(v3 - v 4 ) ] , e2 = ~[2(Vl - v2)(v3 - v4) - (Vl - v3)(vz - 7 4 ) ] , e3 = - - 1 [ ( 7 1

(234)

- - 7 2 ) ( 7 3 - - 7 4 ) ~- (71 - - 7 3 ) ( 7 2 - - 7 4 ) ] .

T h e n ( 2 3 2 ) c a n be expressed in the f o r m ~(rV) - ~(p) v = Vl gd(W) - ga(x) '

(235)

w h e r e the parameters x and p are defined b y the equations 1

f0(/C) = e3 -- ~(Vl -- v2)(71 -- 1~3),

~O(p) = e3

--

1(74/71)(71

--

1)2)(71

--

73)-

(236)

A f t e r substitution o f ( 2 3 5 ) into (225), one can integrate the equation b y m e a n s o f ( 5 1 ) and get the expression for the periodic solution o f the D N L S equation u(x, t ) = exp

[( i

v/_ff X

3v q-

--S1 + ~

K

271

W + p rv o-(p)o'2(W + to)

1

/ ) ,

vl > v > vz .

(237)

T h e case w h e n v oscillates in the interval 73 > v > v4 can be considered in the s a m e way. Initial conditions are c h o s e n so that v = v4 at W = 0. F o r v w e get the expression (rv) -

ga(K) '

v = v4ga(W)_

(238)

w h e r e x and p are n o w defined b y 1

f0(/ Vs of the small modulation wave. The solution found corresponds to the step-like initial data 1E(~,0)]=27

for ¢ > 0 ,

IE( ,0)I=0

for ~ < 0 .

(351)

and is stable at both its edges - in the soliton region and in the region of small modulation (where 12 > 27, as one can see from (348). The velocity v = ¢/~ (v > 0) is connected with the velocity Vphys in physical variables x and t (see (85)) by the relation Vphys= v/(1 + v). We see that Vphys < 1 (dimensionless speed of light equals to unity) and again solitons move slower than the perturbation of the uniform region. Thus, the solitons are created at the back region of the long light pulse propagating through the resonant medium.

4.5. Formation of solitons on the sharp front of the S R S pulse The Whitham equations for the SRS periodic wave coincide with those for the SIT wave in the sharp-line limit after replacement A -* - A / 2 (see Sections 3.4-3.6). Therefore their solution can be obtained from Eqs. (339), (343) by means of the same replacement. But this time the connection of the Riemann invariants 2i with the resolvent's zeros is given by Eqs. (144), (145) with reversed sign of fl and J = 0 -2 (see Figs. 5 and 6). The limit 6 = 0 yields the unmodulated wave for 0-2 > _ f 1 2 , i.e. our periodic wave matches with the unmodulated region only if arising values of fl satisfy this inequality. In Fig. 19 the plots of moving Riemann invariants 22 = fl + i6 and 24 = f l - i6 are shown (see Eqs. (342), (343) with A ~ -A/2). We see that the pair of complex Riemann invariants arises at 22--24 =fl(0), where /~(0) is given by (345) (with A --~ -A/2). It is important that the plane wave solution (see Eq. (209)) does not depend on /~ and therefore matches with our modulated periodic wave at the small modulation edge (m = 0, 6 = 0). The dependence of vl, v2, v3 on x is shown in Fig. 20. At the plane wave boundary (m = 0) we have v2 = v3 and at the soliton edge ( m = 1) we have vl =v2. The dependence of $3 (see Eq. (195)) on ¢ at two moments of 'time' r is shown in Fig. 21. These plots demonstrate the process of soliton formation on the front of the SRS pulse [70]. The two-photon propagation case [71] needs only a slight modification of the SRS case, because now fl is positive and $3 =E1E~ +E2E~ oscillates between the resolvent's zeros vl and v4. The final results for the oscillatory region are shown in Fig. 22.


264

IE(~)I 2 "~ = 0 . 5

120

i00

80

60

J

40

20

(a)

i

i

i

|

i

20

40

60

80

i00

IE()I

A

i20

A

x = 1.0

iO0

80

t

60

40

20

(b)

i

i

i

50

100

150

i

200

Fig. 18. The dependence of light intensity IE(~)I 2 on space coordinate ~ at (a) t = 0 . 5 and (b) t = 1.0 in the case of initially step-like pulse described by the SIT equations (c~ = 7 = 1, A = 0.4).

A.M. Kamchatnov / Physics Reports 286 (1997) 199-2 70

265

0,5

i

I

1.25

1.5

i

-0.5

-i

Fig. 19. The loci of the Riemann invariants 22 and 24 in the case of self-similar solution of the Whitham equations for SRS system ()q = 1 + i, 22 = 1 - i). The difference between this plot and Fig. 14 is due to the nonzero value of the parameter A = 0.4.

i

zo

i

,

o

40

,

,

,

60

,

•

80

-0.2

-0.4

-0.6

-0.8

-1 Vl -1.2 Fig. 20. The dependence of resolvent's zeros •1, ~)2,1J3 on the self-similar variable ~ = ~/'r of the Whitham equations for the SRS system.


266

S3

•

-

-

,

.

.

.

.

.

25

.

.

.

.

,

.

50

,

.

.

,

75

.

.

.

.

.

100

.

.

.

.

,

1Z5

.

.

.

.

150

-0,2

-0.4

-0.6

-o::I (a)

$3

-

-

,

50

-

-

,

i00

-0.2

-

-

,

150

-

,

.

,

|

200

.

.

.

.

I

,

,

,

,

250

I

,

,

,

,

300

x

=

4.0

-0.4

-o. 6

-0.8

-1

(b)

J

L

Fig. 21. The dependence of the variable S 3 (difference of two waves intensities, see equation (176)) on the space variable at (a) v = 2 and (b) r = 4 for initially step-like pulse in the SRS system.

.4. M. Kamchatnov / Physics Reports 286 (1997) 199-270

1.6

267

S3

f

1.5

1,4

F

x = 8.0

1.3

1,2

1.1

/

(a)

, ,500

2,

, 600

700

1.6 $3 1.5 1.4

800

x = 16.0

1.3

f

1,2

1,1

J (b)

. 1000

1200

1400

1600

Fig. 22. The dependence o f the variable $3 (the total intensity o f two w a v e s ) on the space variable ~ at (a) z = 8 and ( b ) z = 16 for initially step-like pulse (ct = ), = 1, A = 4 . 0 ) .

268

A.M. Kamehatnov / Physics Reports 286 (1997) 199-2 70

5. Conclusion We suppose that the presented examples in this report demonstrate the effectivity of the developed methods in the framework of the AKNS scheme. One may hope that this approach can be generalized on different classes of integrable equations which will permit one to investigate some other physical systems. We should also note that the Gurevich-Pitaevskii-type problem with the step-like initial pulse is only very particular case described by the self-similar solution of the Whitham equations. It would be interesting to generalize such an approach on more arbitrary form of initial pulses. Numerical simulation (see [1,48]) shows that probably two-phase periodic modulated solutions may be necessary for the description of arising nonuniform region, which poses the problem of the corresponding generalization of the above methods. We hope to see fast progress in this interesting field of theoretical and mathematical physics.

Acknowledgements I am very grateful to M.J. Ablowitz, V.M. Agranovich, V.R. Chechetkin, A.L. Chernyakov, G.A. El, E.V. Ferapontov, F. Ginovart, A.V. Gurevich, V.P Kotlyarov, A.L. Krylov, V.R. Kudashev, A.C. Newell, V.G. Nosov, M.V. Pavlov, H. Steudel, V.F. Tuganov, A.A. Vedenov for valuable discussions.

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