Soliton perturbations and the random Kepler problem

F.Kh. Abdullaev,

Physical-Technical Institute, 700084, Tashkent-84, G. Mavlyanov Str. 2-B,

Uzbekistan

J. C. Bronski, 1 G. Papanicolaou

Department of Mathematics, Stanford University, Stanford CA 94305, USA

PACS numbers: 42.65.-k, 42.50. Ar,42.81.Dp

We consider the inuence of randomly varying parameters on the propaga-tion of solitons for the one dimensional nonlinear Schrodinger equation.This models, for example, optical soliton propagation in a �ber whoseproperties vary with distance along the �ber. By using an averaged La-grangian approach we obtain a system of stochastic modulation equationsfor the evolution of the soliton parameters, which takes the form of a ran-domly perturbed Kepler problem. We use the action-angle formulationof the Kepler problem to calculate the statistics of the escape time. Themean escape time for the Kepler problem corresponds, in the optical con-text, to the expected distance until the soliton disintegrates.

1 Introduction

The propagation of solitons in a disordered medium has been extensivelystudied[1{9,15] but there are many important problems that require furtheranalysis. One particular application of interest is optical pulse or soliton prop-agation in a disordered medium. The analysis of this problem is importantfor understanding the behavior of optical communications systems when theoptical properties of the �ber vary with distance. The soliton dynamics in thecase where the �ber properties change adiabatically has been considered byAbdullaev[1], Elgin [10] and Kodama [11]. Kath and Ueda[12,13] have con-sidered the case where the birefringence of the �ber varies randomly. Finally

1 Current Address: University of Illinois Department of Mathematics,1409 W. Green St. Urbana IL, 61801

Preprint submitted to Elsevier Preprint 3 April 1999

Kutz and Wai[15] have considered the e�ect of noise on periodic pulses ina dispersion managed �ber. The survey article of Gredeskul and Kivshar[14]contains a general review.

In this paper we consider the randomly perturbed one-dimensional focusingnonlinear Schrodinger (NLS) equation,

iUt +�

2Uxx + �jU j2U = f(U; t; x) (1)

with the initial condition U(x; 0) given approximately by a soliton. Some per-turbations of particular interest are

Æ Fluctuating quadratic potential term

f(U; x; t) = K(t)x2U(t; x):

Æ Fluctuating nonlinearity

f(U; x; t) = �(t)jU j2U:

Æ Fluctuating dispersion

f(U; x; t) =�

2

(t)Uxx:

Here (t) is a Gaussian, white-noise process, with noise level �2.

In applications to optical �bers the variable which we have called t is actuallydistance along the �ber and the variable we denote by x is actually a time,so that NLS models time dispersion and nonlinearity. Thus the perturbationsare due to random inhomogeneities along the length of the �ber. Throughoutthis paper we will use the variables as we have de�ned in Eqn (1) above, so itshould be understood that when we calculate the mean time for an event tooccur we are actually calculating the mean distance along the �ber until theevent occurs.

In this paper we analyze the e�ect of random inhomogeneities on soliton prop-agation. In particular, we calculate explicit formulas for the mean time untilthe soliton disintegrates because of the weak random inhomogeneities. Wealso calculate the standard deviation of the escape time for the case of thequadratic potential, and give formulas for the moments in the general case.

We begin by using the well-known averaged Lagrangian 2method [16,17,7].We consider approximate solutions that look like a soliton whose parameters,

2 Instead of the averaged Lagrangian method we could use soliton perturbationtheory [19,18] or the conservation laws of NLS[18]

2

Perturbation Mean Exit Time (J0 = 0) Mean Exit Time (J0 >> 1)

�1(t)x2U

2(47�60 ln(2))15�2 �

�21

2563�8�21J

60

�2(t)Uxx / ��22 815�2�22J30�2(t)jU j2U �34

�5 ln(5)�p5 ln(

p5+1p5�1)� 8 ln(2)

���22

8�21�22J0

Table 1The dimensionless mean escape time for an unchirped soliton. The dimensionlesssmall parameters �1 and �2 are given by (3). The initial action J0 is de�ned by (25).

the amplitude, width, phase and chirp, vary with t. In the standard way wesubstitute this trial solution into the Lagrangian and compute the variation ofthe resulting averaged Lagrangian with respect to the soliton parameters to�nd a closed system of ODE's. These stochastic modulation equations can bereduced to a randomly perturbed Kepler problem. This is important becausethe Kepler problem, a Hamiltonian system with two degrees of freedom, is in-tegrable and there exists a well known transformation to action-angle variables.In these variables the Fokker-Planck equation for the evolution of the proba-bility density of the soliton parameters has a simple form, allowing detailedanalysis similar to the one carried out by Papanicolaou, Knapp and White[6]using elliptic functions. A linearized analysis, for oscillations near the bottomof the Kepler potential, has also been considered by Abdullaev [7] for the onedimensional NLS, and by Fibich and Papanicolaou[8] in connection with beamcollapse in two-dimensional critical NLS.

In the action angle variables the Fokker-Planck equation for the probabilitydistribution of soliton parameters becomes a one dimensional di�usion equa-tion with a di�usivity which grows algebraically for large values of the actionvariable. Solution of the one dimensional di�usion equation provides explicitformulas for quantities such as the mean time to reach in�nity. This mean exittime corresponds, in the optical setting, to the mean distance a soliton willtravel along a �ber before breaking up. Thus we are able to explicitly com-pute the expected lifetime of a soliton under various random perturbationsas a function of the the initial conditions and the noise level. The results aresummarized in Table 2.

In this Table all quantities are dimensionless. The dimensionless unit of timeis

�

�2N4; (2)

whereN2 is the L2 norm of the initial pulse U(x; 0), � is the dispersion constantand � is the coeÆcient of the nonlinearity. This choice of scaling is chosen

3

so that the period of the unperturbed problem is one. In this scaling thedimensionless parameters �1 and �2, which are assumed to be small, are givenby

�1 =�K�5=2

�3N6; �2 =

��N2

�1=2: (3)

We note that the mean exit time behaves like ��21;2 which means that it is longerfor smaller noise level �, as one would expect. The dependence of the meanexit time on the initial L2 norm N2 is, however, di�erent for the di�erentperturbations of NLS. For the quadratic potential perturbation the larger theN2 the longer the mean exit time, as can be seen from the form of �1 in (3). Forperturbations of the dispersion and nonlinearity the dimensionless strength ofthe perturbation is �2, which is proportional to N

2. Thus, the mean exit timeis shorter for larger N2.

In the case of the quadratic perturbation it is clear that a stronger soliton, onewith a larger N2, will take longer to disintegrate. In the case of perturbationsin the dispersion or the nonlinearity the situation is more complicated becausea stronger soliton will make the perturbations stronger as well.

The main drawback of the averaged Lagrangian approach that we follow hereis that it fails to include the e�ect of dispersive radiation, and in particular thee�ect of radiation damping. It is possible to include the e�ects of radiation inan averaged Lagrangian approach, as in the work of Kath and Smyth [20], butwe do not do so here. We do, however, want to emphasize that near the bottomof the potential well, for strong solitons, the e�ect of dispersive radiation isexpected to be negligible. It is only near the separatrix, for low, wide solitons,that the e�ects of dispersive radiation are expected to be important.

Our analysis makes no use of the exact integrability of the cubic NLS equationso it is in principle applicable to any dispersive, nonlinear wave equation. Gen-erally, however, the stochastic modulation equations are diÆcult to analyzeexcept when there exists a relatively simple transformation to action-anglevariables, as in the perturbed Kepler problem. Of course, the exit time anal-ysis of the random Kepler problem is of independent interest, independent ofsoliton perturbation theory.

2 Dimensionless variables and scaling

We begin with the perturbed NLS equation

iUt +�

2Uxx + �jU j2U = (t)f(U; x; t); (4)

4

where (t) is the mean-zero Gaussian white noise process satisfying

<(t)(t0)>= 2�2Æ(t� t0): (5)

and f(U; x; t) is the perturbation, which we assume is such that the perturbedequations have a Lagrangian formulation, though this is not strictly necessary.The associated Lagrangian is given by

L =i

2(UtU

� � U�t U)��

2jUxj2 + �

2jU j4 � (t)F (U; x; t): (6)

Some perturbations of particular interest include

Æ Random quadratic potential

f(U; x; t)=Kx2U(x; t);

F (U; x; t)=Kx2jU j2:

Æ Random nonlinearity

f(U; x; t)=�jU j2U;F (U; x; t)=

�

2jU j4:

Æ Random dispersion

f(U; x; t)=Uxx;

F (U; x; t)=�jUxj2:

It is convenient to nondimensionalize the problem, since this makes clear therole of the small parameter in the problem. We introduce the L2 norm of theinitial pulse, given by

2N2 =ZjU j2dx:

The parameters in NLS have the dimensions

[�] =L2

T; [�] =

1

A2T; [N2] = A2L; [K] =

1

L2T;

where A denotes the units of U , and L, T are the units of length and timerespectively. The last of these only appears in the quadratic potential per-turbation. From the �rst three dimensional quantities we can construct the

5

nonlinear length, time and distance scales. If we make the following rescaling

t0 =

N4�2

�

!t; x0 =

N2�

�

!x; U 0 =

s�

�N4

!U

then the problem is nondimensionalized on the natural scales of the unper-turbed problem. Note that in these variables the initial data has an L2 normof two: Z

jU 0j2dx0 = 2:

After this rescaling the dimensionless perturbed NLS is

iU 0t0 +1

2U 0x0x0 + jU 0j2U 0 = �i(t0)f(U 0; x0; t0)

where �i = �1 for the quadratic potential perturbation and �i = �2 for theperturbation of the dispersion and the nonlinearity. Here �1 and �2 are thedimensionless parameters (3). The noise (t0) is now normalized to have unitlevel:

<(t01)(t02)>= 2Æ(t

01 � t02):

We will drop the primes and work with dimensionless variables consistently inthe sequel. The physical quantities can, of course, be recovered by multiplyingby the appropriate scale factors in (2).

3 The modulation equations

In the averaged Lagrangian approach we assume that the solution is given bya soliton-like solution with parameters which vary in time. In particular wemake the ansatz

U(t; x) = A(t)sech

x

a(t)

!exp

hi( (t) + �(t)x2)

i: (7)

Since the L2 norm of U is conserved by the perturbation, the amplitude Aand the width a are related as follows

2 =

1Z�1

jU j2dx = 2A2a: (8)

6

To �nd the evolution equations for the parameters we substitute the ansatzinto the Lagrangian L and integrate out the spatial dependence. This givesthe averaged Lagrangian

L[a; �; ] =1Z

�1L[U ] dx (9)

=�2 t � �2

6�ta

2 � 13a2

� 2�2

3�2a2 +

4

3a:

We then obtain the Euler-Lagrange equations for the functional L

ÆZL[a; �; �t; t]dt = 0; (10)

which give the stochastic modulation equations

at � 2a�= �i(t)f1(a; �); (11)a�t � 2

�2a3+ 2a�2 +

2

�2a2= �i(t)f2(a; �):

Here f1 and f2 arise from the perturbations and have the following forms:

Æ Fluctuating quadratic potential

f1(a; �) = 0 ; f2(a; �) = �a2: (12)

Æ Fluctuating nonlinear term

f1(a; �) = 0 ; f2(a; �) = � 2�2a2

: (13)

Æ Fluctuating group velocity dispersion

f1(a; �) = 2a� ; f2(a; �) =2

�2a3� 2a�2: (14)

Note that in the �rst two cases 3 f1 is identically zero and f2 is a function onlyof the width a and not of the chirp �: In the �rst two problems the phase �can be eliminated from the modulation equations and the resulting equationfor a is the randomly perturbed Kepler problem

att = �U 0(a) + �(t)f2(a; at2a

); (15)

3 In fact this holds for any perturbation which is a function of U alone, and not ofthe derivatives of U

7

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20

1/t**2 - 1/t

Fig. 1. The Kepler Potential

where U(a) is the Kepler potential

U(a) =2

�2a2� 4�2a

: (16)

The analysis of the case where the perturbation is in the group velocity dis-persion is slightly more complicated and will be considered in a later section.

The Hamiltonian for the unperturbed Kepler problem is

H0 =(at)

2

2+ U(a): (17)

The Hamiltonian for the perturbed Kepler problem (15) is

H � H0 + �(t)V � (at)2

2+ U(a) + �(t)a2; (18)

for the uctuating quadratic potential (12), and

H � H0 + �(t)V � (at)2

2+ U(a) + �(t)

2

�2a; (19)

for the uctuating nonlinear term (13).

8

4 Action-angle variables for the Kepler problem

In order to continue the analysis of the random Kepler problem, we must�rst transform the unperturbed Kepler problem to action-angle variables. Wesummarize here the relevant facts that we need, with details given in the textsby Landau and Lifschitz[21] or Goldstein[22].

The minimum of the potential U(a) occurs at ac = 1 and is equal to U0 =�2=�2. The frequency !0 of small oscillations about the minimum is given by!0 = 2=�. This is the frequency of the width oscillations of an unperturbedsoliton as it propagates down a homogeneous �ber.

For large oscillations the qualitative behavior of the orbits of the Kepler prob-lem is determined by the energy

E0 = 2a20�

20 +

2

�2a20� 4�2a0

;

where �0 is the initial chirp and a0 is the initial width. When E0 < 0 (i.e�2a20�

20 <

2a0� 1

a20, corresponding to a suÆciently weak initial chirp) the or-

bits are closed, corresponding to oscillatory motion. In the the case E0 > 0,corresponding to suÆciently large initial chirp, the orbits are unbounded, andthe asymptotic motion is qualitatively like the motion of a free particle. Inthis regime the soliton does not persist, but instead spreads out and is lost(a ! 1). The interesting questions in soliton propagation arise when theunpertubed motion is oscillatory, and so we consider the regime E0 < 0.

We now make a change to action-angle variables. For the Kepler problem,where the phase space is two dimensional, there is one action variable J andthe conjugate angle variable �. The action variable is given by

J =1

2�

Iatda:

This integral is easily evaluated by using the energy to express the momentumat in terms of the position. The resulting integral is elementary, and gives

J =2p2

�2p�E �

2

�: (20)

Solving for the total energy E in terms of the action J gives the unperturbedHamiltonian

H0 = E = � 8�2(�J + 2)2

; (21)

9

which is a function of J only. The change of variables to action-angle variablesis canonical, so that Hamilton's equations retain their form

dJ

dt=�@H0(J)

@�= 0;

d�

dt=@H0(J)

@J= !(J);

where

!(J) =16

�(�J + 2)3: (22)

The following implicit representation for the orbits is useful in the pertur-bation calculations. The position a and the time t can both be expressedparametrically in terms of a variable � (called the Kepler parameter) by

a = b(1� e0 cos �);� = !(J)t = � � e0 sin �;

(23)

where the eccentricity e0 and the semi-major axis b are given in terms of theconstant E as follows

e0=(1� �2 j E j2

)1=2 =

1� 4

(�J + 2)2

!1=2(24)

b =2

�2 j E j =(�J + 2)2

4

This parametric representation determines a implicitly as a function of t. Notethat as � varies over (0; 2�), � also varies over (0; 2�). Figure (2) illustratesthe geometric meaning of the Kepler parameter. The solid curve representsthe Kepler ellipse while the dashed curve represents the circumscribing circle.The Kepler parameter, interpreted geometrically, is the angle of a point onthe circumscribing circle, as measured from the center of the circle, to a pointon the circle with the same x coordinate as the given point on the ellipse.

From equation (23) it follows that the width of the soliton oscillates between aminimum value of amin = b(1� e0) and a maximum value of amax = b(1+ e0).The value of the action J varies from 0, for oscillations near the bottom ofpotential well, to 1 for oscillations near the separatrix E = 0. The frequency!(J) varies from 2=� for oscillations near the bottom of the potential well to0 for oscillations near the separatrix.

10

Semimajor Focus

a(t)

Kepler Orbit

KeplerAngle

Semiminor Axis

Axis

ξ

Fig. 2. The geometric interpretation of the Kepler parameter.

It is also useful to relate the initial action J0 to the initial chirp �0 and initialwidth a0:

�J0 + 2 =2q

4a0� 2

a20� 2�2�20a20

: (25)

5 The random Kepler problem - white noise case.

In the previous section we considered the oscillations in width of an unper-turbed soliton, where the coeÆcients of the NLS are constant. In this sectionwe address the case where the coeÆcients of the NLS vary randomly. In thiscase the soliton parameters evolve according to a set of random ODE's, andwe can derive a Fokker-Planck equation for the evolution of the probabilitydistribution for the soliton parameters.

The uctuating quadratic potential perturbation (12) and the uctuating non-linear term perturbation (13) can be analyzed with white noise uctuationsdirectly, using the Stratonovich interpretation. In the case of uctuating groupvelocity dispersion (14) the treatment is slightly more subtle, and is presentedin a later section.

In action-angle variables, the perturbed Hamiltonian has the form

H = H0(J) + �(t)V (J;�); (26)

where V (J;�) is obtained from (18) and (19). In general V , while simple in theoriginal variables, is quite complicated in the action-angle variables. However

11

for weak disorder we do not need the explicit form of V , but only averages over�. These averages will be simple to calculate using the implicit representationgiven in (23).

In the presence of weak disorder the action-angle variables for the unperturbedKepler problem are a convenient framework for the analysis of the perturbedproblem because the change in the action is proportional to the small param-eter �. Since the change to action-angle variables is canonical the Hamiltonianstructure is preserved, and the perturbed equations become

dJ

dt=��(t)@V

@�; (27)

d�

dt=!(J) + �(t)

@V

@J:

Here V (�; J) is the perturbation Hamiltonian de�ned in equations (18,19).We assume that (t) is Gaussian white noise, so the Stratonovich form ofthe Fokker-Planck equation for the evolution of the probability distribution ofP (J;�; t), is given by

@P

@t=!(J)

@P

@�+ �2

@

@J(V�

@

@J(V�P ))� �2 @

@�(VJ

@

@J(V�P )) (28)

� �2 @@J

(V�@

@�(VJP )) + �

2 @

@�(VJ

@

@�(VJP ))

with initial condition P (0; J;�) = Æ(J � J0)Æ(�� �0).

In the absence of randomness, when � = 0, this is a �rst order PDE and canbe solved simply by the method of characteristics. The characteristics are, ofcourse, exactly the orbits of the Kepler problem, and the distribution is carriedalong these classical orbits with no spreading. When � is small, so that thedi�usion is weak, the Fokker-Planck equation can be simpli�ed by averagingover the periodic orbits of the Kepler problem. Since the advective term !@�acts on an O(1) time scale, while the di�usion acts on the slower O(�2) timescale we introduce a relative angle variable �0 de�ned by

�0=�� !(J)t (29)� = �2t:

This transformation removes the advective term, giving a di�usion equationwith rapidly varying coeÆcients. After averaging over the rapidly varying co-eÆcients we are left with the simpli�ed Fokker-Planck equation

12

@P

@�=

@

@J(A(J)

@P

@J)� @

@JB(J)

@P

@�0� (30)

@

@�0B(J)

@P

@J+

@

@�0C(J)

@P

@�0:

The coeÆcients A(J); B(J); C(J) are given by averages of products of deriva-tives of V over the angular variable:

A(J)=1

2�

2�Z0

(V�0(J;�0)2d�0;

B(J)=1

2�

2�Z0

(V�0(J;�0)VJ(J;�

0))d�0; (31)

C(J)=1

2�

2�Z0

(VJ(J;�0))2d�0:

These di�usion coeÆcients depend only on the action J , since the angulardependence has been averaged out.

The �rst of these terms is the most interesting, because if we assume that theinitial probability distribution is independent of �0 then the radial part of thedistribution evolves according to

@P

@�=

@

@J(A(J)

@P

@J) (32)

with P (0; J) = Æ(J � J0).

We note that this averaging calculation breaks down very near the separatrix,since !(J) vanishes there, and the perturbations can no longer be consideredsmall.

5.1 Exit Times

Since we are interested in calculating the distribution of exit times we willbriey review the relevant theory of exit times for a Fokker-Planck equation.For details we refer to the books of Feller[23] or Gardner[24].

Statistics of exit time can be understood easily in terms of the moment gen-

13

erating function (or Laplace transform) P̂ (J; s), which is de�ned as

P̂ (J; s) =

1Z0

e�s�P (J; �)d�

and which satis�es the transformed Fokker-Planck equation

sP̂ (J; s) + Æ(J � J0) = @@J

A(J)@

@JP̂ (J; s): (33)

The moment generating function provides a direct way to derive the di�erentialequations satis�ed by the moments of the exit time. The mean exit time isgiven by

�(1)(J) =

1Z0

�P (J; �)d� = �@P̂@s

; (34)

with the last expression evaluated at s = 0. It is easy to show by di�erentiating(33) that �

(1)(�;�)(J), the mean time to exit the interval (�; �) starting from the

point J , satis�es the ordinary di�erential equation

@

@JA(J)

@

@J�(1)(a;b)(J) = �1;

subject to vanishing boundary conditions �(1)(�;�)(�) = �

(1)(�;�)(�) = 0: Similarly

the nth moment satis�es

@

@JA(J)

@

@J�(n)(J) = �n�(n�1):

From these relations it is straightforward to calculate moments of the exittime. The solution for the equation for the mean time to exit the interval(�; �) is given by

�(1)(�;�)(J) =

R J�

d�A(�)

R ��

�d�A(�)

� R J� �d�A(�) R �� d�A(�)R ��

d�A(�)

; (35)

and more generally the nth moment of the exit time from (�; �) is given by

�(n)(�;�)(J) = n

R J�

d�A(�)

R ��

R�(n�1)(�)d�

A(�)� R J�

R�(n�1)(�)d�

A(�)

R ��

d�A(�)R �

�d�A(�)

: (36)

14

The averaged di�usivity A(J) vanishes at J = 0, so the integrals divergethere. Upon taking the limits � ! 0 and � ! 1 we �nd that the followingexpressions for the mean time to reach in�nity and the nth moment of thetime to reach in�nity starting at J = J0:

�(1)(J0)=

1ZJ0

JdJ

A(J); (37)

�(n)(J0)=n

1ZJ0

R�(n�1)(J)dJ

A(J)(38)

This is the exit time as measured in the rescaled variable � . In the originalvariable, t, the expression is

�(1)(J0) =1

�2

1ZJ0

dJJ

A(J); (39)

which is still dimensionless. In the next section we apply this formula to thespeci�c perturbations which we have considered.

5.2 Fluctuating quadratic potential

For the case of a uctuating quadratic potential the Hamiltonian is given by

H =a2t2+ U(a) +

�(t)

2a2;

so that the perturbation Hamiltonian V is given by

V (a) =a2

2:

Here � = �1 as de�ned by (3) in terms of the physical parameters. The quantitywe are interested in is the di�usion coeÆcient A(J) de�ned by

A(J) =1

2�

2�Z0

(V�)2d�:

Rather than make the relatively complicated transformation to action-anglevariable to evaluate this integral we make use of the following procedure. We

15

have an implicit representation of solutions to the Kepler problem given inequation (21):

a= b(1� e0 cos �);�=!(J)t = � � e0 sin �:

As the angle variable � varies over (0; 2�) the Kepler anomaly � also variesover (0; 2�), and thus we can make a change of variable in the integral from� to � by making the replacements

V (a)=a2

2=b2(1� e0 cos(�))2

2;

d

d�=(

d�

d�)�1

d

d�;

d�=d�

d�d�:

After this change of variables the expression for A(J) becomes

A(J)=b4e202��2

Zsin2(�)(1� e0 cos �)d� (40)

=b4e202�2

=�J

29�2(�J + 4)(�J + 2)6:

Note the behavior near the origin is given by A0(0) = �29, so the Fokker-Planck

equation is well-behaved, and that for large J the dispersion grows like J8, sothat the expected time to reach in�nity is �nite. Substituting into (37) whenJ0 = 0, we �nd that the mean exit time for a soliton with no initial chirp isgiven

�(1)(0)=29

�8�2

1Z0

dJ

(J + �4)(J + 2

�)6

(41)

= 247� 60 ln(2)

15�2�2� :073

�2:

In an optical setting the mean exit time corresponds to the mean distanceuntil the soliton breaks up. Recall that here � = �1, given by (3).

More generally we can consider the expected time to soliton breakup in thecase where the initial soliton has a non-zero chirp �0. In this case the initialaction J0 is not equal to zero. Using equation (37) we can express the initial

16

action in terms of the initial chirp. In this case the expected time to solitondissolution is given by

�(1)(J0) =29

�8�2

1ZJ0

dJ

(J + �4)(J + 2

�)6: (42)

This integral is elementary, since it is the integral of a rational function, andcan be done relatively easily using the identity

1

(1 + x)n(2 + x)=

n�1Xk=0

(�1)k(1 + x)n�k

+(�1)nx + 2

:

giving the general result

�(1)(J0) =8

�2�2

5X

k=1

(�1)k+1k(�

2J + 1)k

� ln (�2J + 2)

(�2J + 1)

!!

In the case where the initial chirp J0 is large, so that the soliton begins nearthe separatrix, the mean exit time is approximately given by

�(1)(J0) � 2563�8�2J60

J0 >> 1:

It is also interesting to use the explicit formulas for the second moment tocalculate the standard deviation of the escape time. If we substitute the aboveexpressions for �(1)(J) and A(J) into Equation (37) we �nd that the integralwhich gives the second moment is no longer elementary and involves the poly-logarithm function. However the integral can easily be evaluated numerically.For the unchirped case, J0 = 0, we �nd that

�(2)(0) � :0073�4

:

The standard deviation is given by

q�(2) � (�(1))2 � 0:044

�2

so the standard deviation is about 60% of the mean. For the rest of the caseswe consider we do not calculate the variances/standard deviations, thoughof course they could be computed in exactly the same manner. It would bevery interesting to calculate the asymptotic behavior of the moments, sinceintuition suggests that quantities such as exit times should have very large

uctuations.

17

5.3 Fluctuating nonlinearity.

For the case of a uctuating nonlinear term the perturbation Hamiltonian isgiven by

V =4

�2a; (43)

and � = �2 with �2 given by (3) in terms of the physical parameters. We proceedin the same way as in the previous section, and make a change of variablesfrom integration over the angle to integration over the Kepler parameter. Theexpression for A(J) is

A(J)=8e20�5b2

2�Z0

sin2 (�)

(1� e0 cos �)5d� (44)

=8e20�4b2

4 + e208(1� e20)7=2

(45)

=1

16�4((�J + 2)2 � 4)(5(�J + 2)2 � 4)(�J + 2): (46)

For the mean exit time �(1) for a particle starting from J = 0 we obtain

�(1)(0)=1

�2

1Z0

JdJ

A(J)(47)

=�3

4�2

5 ln(5)�

p5 ln(

p5 + 1p5� 1)� ln(256)

!:

In this example, as in the previous example, the di�usion A(J) is a rationalfunction of the action variable, so that the mean escape time can be computedfor any initial chirp. Again the general result is somewhat lengthy, so we donot give it here. For large values of the initial chirp, however, the mean exittime is given by

�(1)(J0) =1

�2

1ZJ0

JdJ

A(J)(48)

� 8�3

�2

1ZJ0

JdJ

5(�J)5

� 815�2�2

J�30

18

6 Other problems

The techniques we have used in the previous section are quite general, andare applicable to a wide variety of di�erent perturbations. In particular theassumption that the perturbations be Hamiltonian and white noise can berelaxed. In this section we consider a number of other perturbations which canbe be handled with techniques similar to the ones presented in the previoussection.

6.1 Fluctuating dispersion.

The case of uctuating dispersion is somewhat more subtle, and is treated inthis section. For the �rst part of this calculation we do not assume that iswhite noise, but is instead some process with �nite time correlations. Afterapplying the averaged Lagrangian approach we �nd the following equations ofmotion for the soliton parameters:

at� 2(1 + �(t))a� = 0; (49)a�t� 2(1 + �(t))

�2a3+ 2(1 + �(t))a�2 +

2

�2a2= 0: (50)

Here � = �2 given by (3). Upon eliminating the variable � we are led to asecond order equation for a, the soliton width:

d

dt

at2(1 + �(t))

� 2(1 + �(t))�2a3

+2

�2a2= 0:

This equation comes from a Hamiltonian of the form

H =a2t

2(1 + �(t))+

2(1 + �(t))

�2a2� 2�2a

:

We can write this as

H =a2t2(1� �(t) + �

2

2+O(�2)) +

2(1 + �(t))

�2a2� 2�2a

;

where the O(�2) terms have mean zero and thus do not contribute to the orderwe consider. We can eliminate the the �2=2 term by making the rescaling oftime � = (1 + �2=2)1=2t = �t. We get

19

H =a2�2(1� �((��1�) +O(�2)) + 2(1 + �((�

�1�))

�2a2� 2�2a

(51)

=a2�2(1� �(�)) + 2(1 + �(�))

�2a2� 2�2a

+O(�2);

where, again, the O(�2) terms have mean zero and thus do not contributeto leading order. At his point we can pass to the white noise limit and thearguments of the previous section apply. In particular the perturbation partof Hamiltonian in the random Kepler problem has the form

V =�(ax)2

2+

2

�2a2(52)

=V1 + V2:

The di�usion coeÆcient A(J) is given by

A(J) =1

2�

Z(V�)

2d� (53)

=1

2�

Z(V1�)

2 + 2V1�V2� + (V2�)2d�:

When we make the change of variable to the Kepler parameter � we obtain

V1 = � !2b2e20 sin �

2

(1� e0 cos �)2 ;

V2 =2

�2b2(1� e0 cos �)2 : (54)

The integral de�ning A(J) is, as in the previous cases, elementary. We donot write the full result here, but we do note that for large J the dominantcontribution to the integral comes from V2. If we ignore the contributions ofV1 and the cross terms to the di�usion coeÆcient A(J) we obtain the result

A(J)=e20(8 + 12e

20 + e

40)

�4b4(1� e20)11=2(55)

=J(�J + 4)(128 + 448�J + 448�2J2 + 168�3J3 + 21�4J4)

8�3(�J + 2)3(56)

It is routine, though tedious, to calculate the contributions of V1 and the crossterms, though we do not do this here. However we note that in the case wherethe soliton begins near the separatrix, so that J0 is large we �nd that

A(J) � 218�J3: (57)

20

Integrating from J0 in (37), we obtain

�(1) � 8�21�2J0

: (58)

6.2 Exit times for the damped Kepler problem

The presence of damping terms in a stochastic di�erential equation can dra-matically change the behavior of the solutions. The presence of damping candramatically increase exit times, or even lead to in�nite exit times, if thedamping creates a steady-state.

It might be thought that radiative e�ects lead to damping in the Kepler prob-lem, however this is not the case. The main e�ect of radiation is an adiabaticchange in the L2 norm, the mass of the soliton. However there do exist physi-cal perturbations which can produce an e�ective damping term in the Keplerproblem. One such perturbation arises in the work of Malomed [25]. Malomedconsiders the e�ects of weak ampli�cation and �ltering with the situationwhere the ampli�cation is a constant plus a periodic array of delta functionampli�ers. In the special case where the strength of the array of delta functionampli�ers is zero, corresponding to constant ampli�cation, the unperturbedmodel equation reduces to

iUt +1

2Uxx + jU j2U = i�2�0

�1

3U + Uxx

�:

Here we are working in dimensionless variables scaled on the natural scalesof the unperturbed problem, so that U has unit L2 norm. The factor of 1=3in front of the ampli�cation guarantees that a soliton of unit L2 norm willpropagate without change. For our analysis it is important that the dampingscales like �2, while the random uctuations scale like �; so that the modelproblem becomes

iUt +1

2Uxx + jU j2U = i�2�0

�1

3U + Uxx

�+ �f(U; x; t):

The standard variational calculation, as presented by Malomed[25], gives thefollowing ODE for the soliton width

att = a�3 � a�2 � �2�at + �(t)f(a);

which corresponds to the random Kepler problem with a linear damping term.

Here � is related to �0 by � =8(6+�2)3�2

�0 � 4:3�0. In the action-angle formula-tion of the problem this damping term leads to the presence of an extra term

21

in the evolution equation for the action J , which again takes the form of lineardamping.

�t=!(J)� �(t)@V@J

;

Jt= �(t)@V

@�� �2�J:

Note that these equations are no longer Hamiltonian, as the damping termbreaks the Hamiltonian structure. The presence of this term leads to the pres-ence of an additional term in the Fokker-Planck equation:

@P

@�=

@

@JA(J)

@P

@J+ �

@

@JJP:

The di�erential equation governing the mean exit time for this Fokker-Planckequation is easily seen to be

�2 @

@J�(1) + �

@

@JJ�(1)

!= �1: (59)

Integrating this equation once leads to

�2 A(J)

@

@J�(1) + �J�

!= �J + C: (60)

This equation has an integrating factor exp(�R J

A(J))=A(J). We thus �nd the

following expression for �(1)(J0), the mean exit time to reach in�nity startingat the point J0:

�(1)(J0) =exp[�

R1J0

JdJA(J)

]� 1��2

:

Several commments are in order. First we note that since the escape time inthe problem without ampli�cation and �ltering is given by

RJdJ=A(J), there

exists a surprisingly simple functional relationship between the escape time inthe damped/driven case and the escape time in the undamped case. Since theexponential is convex it is easy to see that the escape time in the presence ofweak ampli�cation and �ltering is strictly longer than the escape time in theabsence of these terms. When J0 is large or the damping coeÆcient � is smallthis expression is approximately the same as the one derived in the previoussection. Thus when the soliton is near the separatrix the e�ect of damping isrelatively unimportant. This is to be expected on physical grounds - for a low,

22

wide soliton the e�ect of �ltering is expected to be small. However when J0is not large and � is not small the expected escape time is exponentially long.Physically it is clear why this should be the case - the additional term in theFokker-Planck equation corresponds to a mean drift towards the origin andaway from the separatrix. It is well-known that the expected time to randomwalk a given distance against a mean ow is exponentially long. In an opticalcontext this is a mathematical justi�cation of the stabilizing e�ect of �lters.

6.3 Colored Noise Perturbations

It is also possible to treat perturbatively the case where the randomness hasa small but �nite correlation time by expanding in a series in powers of thecorrelation time. We briey sketch the derivation. Using standard argumentsthe e�ective di�usion coeÆcient A(J) is given by

A(J) =1

�c

1Z0

dsB(s

�c) ; (61)

where ��1c B(s�c) =<(s)(0)> is the correlation function of the colored noise

process, and the brackets denote averaging over the angular variable. In thewhite-noise limit, when the correlation time �c is short the correlation functionB( s

�c

)

�creduces to a delta function, and the above integral reduces to the previous

expression. If we make the change of variable � = s�c

in the above expressionfor the di�usion coeÆcient A(J) we get

A(J) =ZB(�) :

Expanding V�(� + �c!(J)�) in a Taylor series and integrating we get

A(J)� �m2�2c !(J)

2

2 + : : :

+(�1)n �2nc !(J)

2nm2n2n!

+ : : : ;

Here mn denotes the nth moment of the correlation function mn =

RsnB(s)ds.

Note that the odd terms in the perturbation series vanish, since they areproportional to angular averages of terms which are exact derivatives in the� variable.

Note also that the �rst order correction to the e�ective di�usivity is negative,leading to increased exit times. If this perturbation problem is to be regular we

23

require that the �rst order correction to the white-noise result grow no fasterthan the leading order term for large J , otherwise the e�ective di�usivitywould be negative for large J . Physically there is good reason to expect thisto be true. When the action J is large the period of the orbit is large, and thereare many correlation times in one period. Thus we expect that the e�ects ofthe �nite time correlations should become less important near the separatrix.Of course this should still be checked for each example.

We calculate the form of the correction for the simplest case is the quadraticpotential, which has a perturbation Hamiltonian V (a) = a2=2. The calculationis essentially the same as the one for the leading order term. The secondderivative V��, when expressed in terms of the Kepler parameter, is given by

V�� =b2e0cos(�)

1� e0 cos(�)

and thus the angular average of V 2�� is given by

=b4e202�

Z cos2(�)1� e0 cos(�)d�:

After doing the � integration we derive the following expression for the di�u-sion coeÆcient:

A(J)�A0(J)� � 2cA1(J)A0(J)=

�J

29�2(�J + 4)(�J + 2)6 ; A1(J) =

m2(�J)(�J + 2)2

4�2�2:

Note that the correction, which grows as J3, does so much more slowly thanthe leading order term, which grows as J8. Thus the mean exit time for thenear-white-noise problem is given by

�1(J0)�1Z

J0

JdJ

A0(J)� � 2cA1(J)

�1Z

J0

JdJ

A0(J)+ � 2c

1ZJ0

JA1(J)dJ

A20(J):

As in previous examples these integrals involve rational functions, and can bedone explicitly.

24

6.4 The external white noise.

It is interesting to consider the Kepler problem with a white noise driver. Thisis the equivalent of the standard random walk for a particle which moves ina Kepler potential rather than moving as a free particle. This particular casedoes not arise in a natural way from any soliton problem, but since the randomKepler problem is interesting in and of itself we include it.

The perturbation Hamiltonian takes the form

V = (t)a; (62)

where (t) is a white noise. The calculation gives the following expression forA(J)

A(J) =�J(�J + 2)3

16: (63)

For the mean exit time we �nd

�(1) =8

�7�2(J0 +

2

�)�2: (64)

We note that for the white noise driver the mean exit time is proportional tothe square of the initial power.

7 Conclusions

We have investigated the propagation of chirped solitons in Kerr media withrandomly varying parameters. We have used a variational approach to reducethe analysis to that of a randomly perturbed Kepler problem. The fact thatthe underlying ODE is the Kepler problem means that the transformation toaction-angle variables is known implicitly, and this transformation can be used,in the weak noise limit, to reduce the problem to a one-dimensional di�usion.For this one dimensional di�usion we can compute explicitly quantities suchas the mean exit time, which corresponds physically to the expected time untilthe soliton disintegrates. This analysis is very general, and can be applied tomany di�erent types of perturbations, since the underlying ODE, the Keplerproblem, is the same in all cases.

This same general theory can be applied to compute higher order moments ofthe escape time. One interesting computation, which we have not done, would

25

be to calculate the behavior of the escape time for large moment number,a kind of large deviations theory. It would also be interesting to test thepredictions of this theory with some numerical experiments.

Finally the most signi�cant drawback of this approach is that it completelyneglects radiation e�ects, which are expected to be important in many cases. Itwould be extremely interesting to do a similar exit time calculation includingradiation e�ects, perhaps using an IST based perturbation theory, or the av-eraged Lagrangian approach ro radiation developed by Kath and Smythe[20].However such a calculation seems extremely diÆcult.

8 Acknowledgments

We acknowledge partial support by the U.S. Civilian Research & DevelopmentFoundation (Award ZM1-342), by the NSF Postdostoral Fellowship Programgrant DMS 94-07473, by AFOSR grant F49620-98-1-0211 and by NSF grantDMS-9622854.

26

References

[1] F.Kh. Abdullaev, S.A. Darmanyan and P.K. Khabibullaev, Optical Solitons,(Springer-Verlag, Heidelberg, 1993).

[2] J.C. Bronski, J. Nonlinear Sci., 8, 161 (1998).

[3] J.C. Bronski, J. Stat. Phys., 92, 5/6 (1998).

[4] J. Garnier, SIAM J. Appl. Math, In Press.

[5] J. Garnier, Ph.D. Thesis, Ecole Polytechnique.

[6] R. Knapp, G. Papanicolaou, and B. White, In Nonlinearity and DisorderA.R.Bishop, D.Campbell, St.Pnevmatikos, (Springer-Verlag, Heidelberg, 1989).

[7] F.Kh. Abdullaev, In \Fluctuations:Nonlinearity and Disorder", A.R.Bishop,L.Vasquez (Eds.) p. (Singapore, WS, 1995);F.Kh.Abdullaev,A.A.Abdumalikovand B.B.Baizakov, Opt.Commun. 138,49 (1997).

[8] G. Fibich and G. Papanicolaou, SIAM, 1998, In Press.

[9] P.K. Wai, C.R Menyuk and H.H Chen, Opt. Lett. 16, 1231 (1991).

[10] J.N. Elgin, Opt.Lett. 18, 10 (1993).

[11] Yu. Kodama and A. Hasegawa, Solitons in Optical Communications,(Clarendon Press, Oxford, 1995).

[12] W. Kath and T. Ueda, Physica D,55, 166 (1992).

[13] W. Kath and T. Ueda, JOSA B,11, 818 (1994).

[14] S.A. Gredeskul and Yu.S. Kivshar, Phys.Rep., 216, 1 (1992).

[15] J.N. Kutz, P.K. Wai, Opt. Lett. 23, 13 (1998).

[16] G. Whitham, Linear and Nonlinear Waves, (Wiley-Interscience).

[17] D. Anderson, M. Lisak, and T. Reichel, JOSA B5, 207 (1988).

[18] V.I. Karpman, Physica Scripta, 20, 462 (1978).

[19] J.P. Keener and D.W. McLaughlin, 16, 777 (1977).

[20] W.L. Kath and N.F. Smyth, Phys. Rev. E 51, 1484 (1995).

[21] L.D.Landau and E.M.Lifshitz, Mechanics, (Pergamon, New York, 1975).

[22] H. Goldstein, Classical Mechanics, (Wiley, New York, 1965).

[23] W. Feller, Probability Theory and Its Applications, (Wiley, New York, 1950)

[24] C.W. Gardner,Handbook of Stochastic Methods, (Springer-Verlag, Berlin, 1983).

[25] B.A. Malomed, J. Opt. Soc. Am. B, 13, 677 (1996).

27