anjan biswas- adiabatic dynamics of non-kerr law vector solitons

8/3/2019 Anjan Biswas- Adiabatic Dynamics of Non-Kerr Law Vector Solitons

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Dynamics of Continuous, Discrete and Impulsive SystemsSeries A: Mathematical Analysis 12 (2005) 363-373Copyright c2005 Watam Press

ADIABATIC DYNAMICS OF NON-KERR LAWVECTOR SOLITONS

Anjan Biswas

Department of Mathematics & Center of Excellence in ISEMTennessee State University

Nashville, TN 37203-3401, USA

Abstract. The soliton perturbation theory is used to obtain the adiabatic parameter

dynamics of the perturbed vector solitons that are governed by the generalized nonlinear

Schrodingers equation. The perturbation terms considered in this paper are of dispersive,

dissipative as well of nonlocal type.

AMS (MOS) Subject classification: 35Q51, 35Q55, 78A60

1 Introduction

The nonlinear Schrodingers equation (NLSE) plays a vital role in various ar-eas physical, biological and engineering sciences. It appears in many appliedareas like Fluid Dynamics, Nonlinear Optics, Plasma Physics and ProteinChemistry just to name a few. In this paper we are going to study animportant generalization of the NLSE known as the generalized nonlinearSchrodingers equation (GNLSE) that is given by:

iqt +1

2qxx + F|q|

2

q = 0 (1)where, F is a real valued algebraic function. This is a nonlinear parabolicequation that is not integrable, in general. The nonintegrability is not nec-essarily related to the nonlinear term in (1). Higher order dispersion orbirefringence, for example, can also make the system nonintegrable while itstill remains Hamiltonian. The special case, F(s) = s, also known as theKerr law of nonlinearity, is integrable by the method of Inverse ScatteringTransform (IST) first discovered by Zakharov and Shabat[1]. The IST is thenonlinear analog of Fourier Transform that is used for solving linear partialdifferential equations. Schematically, the technique of IST and the Fouriertransform are similar [1]. This special case falls in the category of S-integrableequations [12]. In this case (1) is known as the cubic NLSE. The solutionsare known as solitons. It arises in Fluid Dynamics, Nonlinear Optics and-helix proteins in Protein Chemistry and many other areas.

The general case where F(s) = s takes us away from the IST picture asit is not of Painleve type [1]. In a rigorous sense, the pulses of the noninte-grable systems are not solitons. However, the term solitons has been used


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364 A. Biswas

broadly for the solution of the nonintegrable system as well. Unlike the cu-bic NLSE which has an infinite number of conserved quantities, the GNLSEgiven by (1) has only a few. Although stationary pulses exist, and some so-

lutions can be written in the analytic form, their behaviour is different fromthat of the solutions of the cubic NLSE.

As we have mentioned that (1), unlike the Kerr law case, does not haveinfinitely many conserved quantities. In fact, it has as few as three. Theyare

E =

|q|2 dx (2)

M =i

2

(qqx q

qx) dx (3)

H =

1

2

|qx|2 f|q|2 (4)

where

f(s) =

s0

F(s) ds

The first conserved quantity (E), given by (2), has various definitions de-pending on the context in which equation (1) arises. It is commonly knownas the wave energy, but it is also known as the mass, wave action or plasmonnumber; in optics, however, it is called the wave power while mathematicallyspeaking it is known as the L2 norm. The second (M) and the third (H)integrals of motion given by (3) and (4), are respectively known as the mo-mentum and the Hamiltonian.

One can see that (1) can be written in the cannonical form as:

iqt =H

q(5)

iqt = H

q(6)

Now, (5) and (6) define a Hamiltonian dynamical system on an infinite di-mensional phase space of two complex functions which decrease to zero atinfinity. It can be analyzed using the theory of Hamiltonian systems. Thismeans that a behaviour of the solution is defined, to a large extent, by thesingular points of the system, namely the stationary solutions of (1) anddepends on the nature of these points as determined by the stability of its

stationary solutions.

There are various types of non-Kerr law nonlinearity that are studied. Someof those common forms are as follows:


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Adiabatic Dynamics of Non-Kerr Law Vector Solitons 365

1. Power law: F(s) = sp

In this case we need to have 0 < p < 2 to avoid wave collapse. In

fact, we need to have p = 2 to avoid the self-focussing singularity [1]. Adetailed analysis of this law is studied [6]. This law of nonlinearity arisesin nonlinear plasmas that solves the problem of small K-condensationin weak turbulence theory. it also arises in the context of nonlinearoptics. Physically, various materials including semiconductors, exhibitpower law nonlinearities [23, 27]. This case of nonlinearity has beenstudied including the perturbation term by multiple-scale analysis [6].

2. Square-Root Law: F(s) =

s

This is a special case of the power law nonlinearity with p = 12

. Re-cently there has been a growing interest in this special case. This law iscurrently under study and is under continous investigation. It is used

to study soliton turbulence. Besides the analytical approach, this lawis also being investigated numerically.

3. Parabolic Law: F(s) = s + s2

This law for constant is commonly known as the cubic-quintic non-linearity. The second term is large for the case of p-toluene sulfonatecrystals. This law arises in the nonlinear interaction between Langmuirwaves and electrons. It describes the nonlinear interaction between thehigh frequency Langmuir waves and the ion-acoustic waves by ponder-motive forces. This case of cubic-quintic nonlinearity was also studiedby the multiple-scale analysis [29].

4. Saturating law: F(s) = s1+s

Here, the case with > 0 acurately describes the variation of dielec-tric constant of gas vapours through which a laser beam propagates[30]. It also arises in the case of fiber optics in the case of soliton prop-agation. In semiconductor doped fibers the soliton propagation havebeen modeled using the saturable nonlinearity rather than the usualKerr law nonlinearity. The main motivation behind such attemptsis the observation of such nonlinearity at not too high intensities insemiconductor-doped glass and other composite materials. This casewas studied numerically [25].

5. Exponential Law: F(s) = 12

1 e2s

This case of exponential nonlinearity where is a constant serves as anuseful model in homogenous unmagnetized plasmas and laser producedplasmas. When the phase velocity of the slow plasma oscillations muchsmaller than the ion thermal velocity, one can obtain the adiabatic


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366 A. Biswas

or quasistatic electron density under the quasineutral approximation.Now, combining the coupling equation that exhibits the slowly varyingcomplex amplitude interacting with the low frequency plasma motion,

one obtains (1) with exponential nonlinearity [30].

6. Log law: F(s) = a ln(b2s)

This law arises in various fields of contemporary physics for constants aand b. It allows closed form exact expressions for stationary Gaussianbeams (Gaussons) as well as for periodic and quasiperiodic regimesof the beam evolution. The advantage of this model is that the radia-tion from the periodic soliton is absent as the linearized problem has adiscrete spectrum only [19, 20].

7. Dual-Power law: F(s) = sp + s2p

This model is used to describe the saturation of the nonlinear refractiveindex and its exact soliton solutions are known []. Here is a constant.The effective GNLSE with its dual power law nonlinearity serves as abasic model to describe spatial solitons in photovoltaic-photorefractivematerials such as LiNbO3. The solitons of this model become unstableand decaying in the unstable region 1 p < 2 while for p 2 thesolitons collapse in a finite time [4].

8. Higher order polynomial law: F(s) = s + s2 + s3.

This is an extension of the parabolic law that is given in item number3. Here alos and are constants. The Hamiltonian-Energy diagramsfor this law are already studied [4]. This law is also observed in variousphysical systems [5].

9. Triple-Power law: F(s) = sp + s2p + s3p

For constants and , this is also an extension of the dual powerlaw and is a generalization of the higher order polynomial law. In thiscase also the Hamiltonian-Energy diagrams are studied [5].

10. Threshold law:

F(s) =

n1 : s < I0n2 : s I0

A smooth transition of this kind can be modeled as

F(s) = As1 + tanh s2 I20where for s I0, F(s) n1s, where n1 = A

1 tanh2 I20,

and for s I0, F(s) n2s, where n2 = 1 + . Although examplesof nonlinear optical materials with such law are not yet known, the


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bistable solitons have interesting properties that is useful for the futureapplications in all-optical logic and switching devices [19, 20].

2 Vector Solitons

The vector solitons that we are going to study in this paper are given by thecoupled generalized nonlinear Schrodingers equation of the following form

iut +1

2uxx +

F|u|2

+ F

|v|2

u = 0 (7)

ivt +1

2vxx +

F|v|2

+ F

|u|2

v = 0 (8)

Here, is a constant. Equations (7) and (8) are known as the vector general-ized nonlinear Schrodingers equation. These vector solitons arise in variousareas of Physics and Mathematical Physics. One of the most common areaswhere we see these coupled nonlinear Schrodingers equation is in nonlinearoptics in the context of propagation of solitons through a birefringent opticalfiber. In this case, the constant is known as the cross-phase modulation(XPM) coefficient [8]. The case of Kerr law solitons in birefringent opticalfibers has been extensively studied [8]. Under special circumstances, equa-tions (7) and (8) are integrable for the special Kerr law case and they areknown as the Manakov Equations [8].

Equations (7) and (8) as it appears are not integrable. It does not haveinfinitely many conserved quantities either. They have at least two inte-grals of motion that are the energy (E) and the momentum (M) that arerespectively given by

E =

|u|

2

+ |v|

2dx (9)

M =i

2

{(uux u

ux) + (vv

x v

vx)} dx (10)

As we have mentioned that equations (7) and (8) are not integrable we shall takea look at them from a different perspective. We assume that the solitons arerespectively given by:

u(x, t) = A1(t)g [B1(t) {x x1(t)}]exp[i1(t) {x x1(t)} + i1(t)] (11)

v(x, t) = A2(t)g [B2(t) {x x2(t)}]exp[i2(t) {x x2(t)} + i2(t)] (12)

where g represents the shape of the solitons described by the vector GNLSE andit depends on the type of nonlinearity in (7) and (8). Also here, the parametersAj(t), Bj(t), j(t), xj(t) and j(t) for j = 1, 2 respectively represent the solitonamplitude, the width of the soliton, frequency, the center of the soliton and the


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368 A. Biswas

phase of the soliton respectively. For convenience, we now define the followingintegrals.

I(k)l,m,n =

lkg

m

(k) dg

dkn

dk (13)

for non-negative integers l, m, and n while k = 1, 2, where k = Bk(t)(x xk(t)).For such general forms of the solitons given by (11) and (12) , the integrals ofmotion given by (9) and (10) respectively reduce to:

E =

|u|2 + |v|2

dx =

A21B1

I(1)0,2,0 +

A22B2

I(2)0,2,0 (14)

M =i

2

{(uux u

ux) + (vv

x v

vx)} dx

= 1 A21

B1

I(1)0,2,0 + 2

A22

B2

I(2)0,2,0 (15)

The soliton parameters for (11) are defined as follows:

A1(t) =

I(1)0,2,0

|u|4 dx

I(1)0,4,0

|u|2 dx

12

(16)

B1(t) =

I(1)2,2,0

|u|2 dx

I(1)0,2,0

x2 |u|2 dx

12

(17)

1(t) =

i

2 (uux uux) dx

|u|2 dx (18)

x1(t) =

x |u|2 dx

|u|2 dx(19)

We shall now use these definitions of the soliton parameters given by (16)through (19) to obtain the parameter dynamics of the soliton given by (11).Now, differentiating these parameters in (16) through (19) with respect to tand using (7) and (8), while treating the XPM terms as perturbation terms,we arrive at the following evolution equations for the soliton parameters [9]:

dE

dt = 0 (20)

dA1

dt= 0 (21)


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dB1

dt= 0 (22)

d1

dt = 0 (23)

dx1

dt= 1 (24)

d1

dt=

212

B212

I(1)0,0,2

I(1)0,2,0

F

A22g

2(2)

+1

I(1)0,2,0

g2(1)F

A21g

2(1)

d1

(25)

Here (25) is obtained by differentiating (11) with respect to t and subtracting fromits conjugate while using (7). Similarly, the soliton parameters for (12) are definedas

A2(t) =

I(2)0,2,0

|v|4 dx

I(2)0,4,0

|v|2 dx

12

(26)

B2(t) =

I(2)2,2,0

|v|2 dx

I(2)0,2,0

x2 |v|2 dx

12

(27)

2(t) =i

2

(vvx vvx) dx

|v|2 dx(28)

x2(t) = x |v|

2

dx

|v|2 dx(29)

Again, using these definitions for the parameters of the soliton of (12) wesimilarly arrive at the following parameter dynamics for the soliton given by(12).

dA2

dt= 0 (30)

dB2

dt= 0 (31)

d2

dt = 0 (32)

dx2

dt= 2 (33)


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370 A. Biswas

d2

dt=

222

B222

I(2)0,0,2

I(2)0,2,0

F

A21g

2(1)

+1

I(2)0,2,0

g2(2)F

A22g

2(2)

d2

(34)

Thus, from (20) through (23) and (30) through (32), we notice that theenergy, amplitude, width and the frequency of the solitons remains constant.However, the center of mass and the phase of the soliton undergo a changeas governed by the pairs (24), (25) and (33), (34) respectively. These are theparameter dynamics for the vector solitons due to non-Kerr law nonlinearity.In particular, the case of Kerr-law nonlinearity where F(s) = s, we have thesech profile of the soliton that has already been studied elsewhere using thevariational principle [7, 8].

3 Perturbation Terms

We shall now consider the GNLSE along with its perturbation terms that isgiven by

iut +1

2uxx +

F|u|2

+ F

|v|2

u = iR1 [u, u

; v, v] (35)

ivt +1

2vxx +

F|v|2

+ F

|u|2

v = iR2 [v, v

; u, u] (36)

Here is a perturbation parameter with 0 < 1 while R1 and R2 are spatio-differential operators. The perturbation parameter depends on the type of nonlin-earity. For example, in the context of fiber optics, where F(s) = sp, in general, thisperturbation parameter is called the relative width of the spectrum that arises dueto quasi-monochromaticity [7, 8]. In presence of the perturbation terms we havethe adiabatic dynamics of the soliton parameters as

dE

dt= 2

A1

B1

g(1)

R1ei1

d1 +

A2

B2

g(2)

R2ei2

d2

(37)

dA1

dt=

2

I(1)0,4,0

g3(1)

R1e

i1

d1 (38)

dB1

dt=

2

I(1)0,4,0

B1

A1

2

g3(1)

R1e

i1

d1

g(1)

R1ei1

d1

(39)

d1

dt = 2

I(1)0,2,0

1

A1

g(1)

R1ei1

d1

2

I(1)0,2,0

1

A1

B1

dg

d1

R1ei1

+ 1g(1)

R1ei1

d1 (40)


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dx1

dt= 1 +

2

I(1)0,2,0

1

A1

xg(1)

R1ei1

d1 (41)

d1

dt=

212

B212

I(1)0,0,2

I(1)0,2,0

F

A22g

2(2)

+1

I(1)0,2,0

g2(1)F

A21g

2(1)

d1

I(1)0,2,0

1

A1

g(1)

R1ei1

d1

(42)

Similarly, for the perturbed equation given by (36) the adiabatic parameter dy-namics is given by

dA2

dt=

2

I(2)0,4,0

g3(2)

R2e

i2

d2 (43)

dB2

dt= 2

I(2)0,4,0

B2

A2

2

g3(2)

R2e

i2

d2

g(2)

R2ei2

d2

(44)

d2

dt=

2

I(2)0,2,0

2

A2

g(2)

R2ei2

d2

2

I(2)0,2,0

1

A2

B2

dg

d2

R2ei2

+ 2g(2)

R2ei2

d2 (45)

dx2

dt= 2 +

2

I(2)0,2,0

1

A2

xg(2)

R2ei2

d2 (46)

d2

dt=

222

B222

I(2)0,0,2

I(2)0,2,0

F

A21g

2(1)

+1

I(2)0,2,0

g2(2)F

A22g

2(2)

d2

I(2)0,2,0

1

A2

g(2)

R2ei2

d2

(47)

Here, equations (37) through (41) are obtained by differentiating (14) and(16) through (19) with respect to t and using (35). Similarly, one obtains(43) through (46).

4 Conclusions

This paper studies the adiabatic evolution of vector soliton parameters inpresence of the perturbation terms. The Dynamical System that was ob-tained in (38) through (47) is very useful in various areas of applied nonlinear


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372 A. Biswas

science like Fluid Dynamics, Nonlinear Optics, Protein Chemistry. For ex-ample, in the case of Nonlinear Optics, one can use these parameter dynamicsto study the coherent energy coupling, the frequency and timing jitter due

to collision of solitons in a wavelength-division-multiplexed system and manymore.

For an arbitrary initial input one gets solitons along with the small am-plitude dispersive waves that is commonly known as the radiation. We havenot included those results in this paper although such studies are under way.Moreover, besides the deterministic perturbation terms that are consideredhere, one encounters, in reality, the stochastic type perturbation. The adia-batic soliton parameter dynamics due to such type of perturbations will bereported in a future publication.

5 Acknowledgement

This research was partially supported by NSF Grant No: HRD-970668 andthe support is very much appreciated with thanks.

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Received March 2003; revised March 2004.

anjan biswas- adiabatic dynamics of non-kerr law vector solitons

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