inverse scattering transform for vector defocusing ... · straightforward way to derive explicit...

Inverse Scattering Transform for vectordefocusing Nonlinear Schrödinger equation

with nonvanishing boundary conditionsBarbara Prinari, Mark J. Ablowitz, Gino Biondini

Nonlinear Physics: Theory and Experiment. IV

June 22- July 1, 2006

B.P, M.J. Ablowitz, G. Biondini,J. Math. Phys. 47 (2006)

Inverse Scattering Transform for vector defocusing Nonlinear Schrodinger equationwith nonvanishing boundary conditions – p. 1/29

Overview

We construct the IST for the vector defocusing nonlinear Schrödinger (VNLS)equation with non-vanishing boundaries.The direct scattering problem is first formulated on a two-sheeted covering of thecomplex plane. Two out of the six Jost eigenfunctions do not admit an analyticextension on either sheet of the Riemann surface⇒ a suitable modification of boththe direct and the inverse problem formulations is necessary.

X On the direct side, we constructtwo additional analytic eigenfunctionsusing theadjoint problem. Discrete spectrum and bound states are discussed.

X The inverse scattering is formulated in terms of ageneralized Riemann-Hilbert(RH) problem in the upper/lower half planes of a suitableuniformizationvariable.

X Special soliton solutions are constructed from the poles inthe RH problem(dark-darkanddark-brightsolitons)

X Soliton interactions are discussed.

X The linear limit is obtained from the RH problem and is shown to correspond tothe Fourier transform solution obtained from the linearized equation.


Scalar Nonlinear Schrödinger (NLS) equation

The IST for the scalar defocusing NLS equation

iqt = qxx −2|q|2q

with nonvanishing boundary conditions was first studied byZakharov and Shabatin1973.Subsequently generalized by:Kulish et al[1976],Manakov and Faddeev[1976],Gerdjikov and Kulish[1978],Leon[1980],Boiti and Pempinelli[1982],Asano andKato [1984] etc. An extensive study:Faddeev and Takhtajan[1987].

In this dispersion regime, NLS admits soliton solutions on abackground

q(x, t) = q0e2iq20t [cosα+ isinα tanh(q0sinα(x−2q0cosαt − x0))]

withq(x, t) → q±(t) = q0e2iq2

0te±iα as x →±∞

A greysoliton appears as a localized intensity dip of amplitudeq0|cosα| on thebackground fieldq0.


Vector Nonlinear Schrödinger equation

While the scalar problem was solved many years ago, IST had not been completelydeveloped for the vector nonlinear Schrödinger (VNLS) equation

iqt = qxx −2‖q‖2 q

where nowq(x, t) is a vector and‖.‖ the Euclidean norm.

The IST with zero boundary conditions for the two component (focusing) equationwas solved byManakov[1973]

Partial results for the case of non vanishing boundaries in:Gerdjikov and Kulish[1985].

More recently, some direct methods have been applied to the VNLS as a morestraightforward way to derive explicit bright and dark solitons:Kivshar and Turitsyn[1993],Radhakrishnan and Lakshmanan[1995],Sheppard andKivshar[1997],Kivshar and Luther Davies [1998], Nakkeeran[2001] etc.


Lax pairThe 2-component VNLS equation

iqt = qxx −2‖q‖2 q (1)

q = (q(1)(x, t),q(2)(x, t))T is associated to the Lax pair:

∂xv = (ikJ+Q)v (2a)

∂tv =

(

2ik2 + iqT r −2kqT − iqTx

−2kr+ irx −2ik2I− irqT

)

v (2b)

v is a 3-component vector

k is the so-calledscattering (or spectral) parameter

J = diag(−1,1,1), Q =

(

0 qT

r 0

)

≡

0 q(1)(x) q(2)(x)

r(1)(x) 0 0

r(2)(x) 0 0


The compatibility of the Lax pair, i.e. the equality of the mixed derivatives of the3-component vectorv with respect tox andt, is equivalent to the statement thatqsatisfies the VNLS equation withr = q∗.

We consider potentials withboundary valuesasx →±∞ with the samet-independentamplitudes at both space infinities, i.e.

q( j)(x, t) ∼ q( j)± (t) = q( j)

0 eiθ±j (t)

r( j)(x, t) ∼ r( j)± (t) = q( j)

0 e−iθ±j (t)


Direct Problem: eigenfunctionsEigenfunctions for the scattering problem are introduced by fixing the largeasymptotics asx →±∞

φ1 (x,k) ∼ w−1 e−iλx, φ2 (x,k) ∼ w−

2 eikx, φ3 (x,k) ∼ w−3 eiλx

ψ1 (x,k) ∼ w+1 e−iλx, ψ2 (x,k) ∼ w+

2 eikx, ψ3 (x,k) ∼ w+3 eiλx

where

λ =√

k2−q20, q2

0 ≡ ‖q0‖2 = |q(1)

± |2 + |q(2)± |2

w±1 =

λ+ k

ir(1)±

ir(2)±

, w±

2 =

0

−iq(2)±

iq(1)±

, w±

3 =

λ− k

−ir(1)±

−ir(2)±


Direct problem: EigenfunctionsThe solutions with fixed (with respect tox) boundaries

M1(x,k) = eiλxφ1(x,k), M2(x,k) = e−ikxφ2(x,k), M3(x,k) = e−iλxφ3(x,k)

N1(x,k) = eiλxψ1(x,k), N2(x,k) = e−ikxψ2(x,k), N3(x,k) = e−iλxψ3(x,k)

can be represented in terms ofintegral equations

M j(x,k) = w−j +

+∞R

−∞G−

j (x− x′,k)(Q(x′)−Q−)M j(x′,k)dx′

N j(x,k) = w+j +

+∞R−∞

G+j (x− x′,k)(Q(x′)−Q+)N j(x

′,k)dx′

with appropriate Green’s functionsG±j andQ± being the asymptotic values of the

potential matrix:

Q± =

0 q(1)± q(2)

±

r(1)± 0 0

r(2)± 0 0


To extend the eigenfunctions to complex values ofk: define the Riemann surface ofλ2 = k2−q2

0 by gluing together two copies of the extended complex plane,cut alongthe semilines(−∞,−q0) and(q0,∞) :

+8+8

8--

8--

0 I-- q

0q

0

aI

cI d

I

bI

Sheet I: Im λ>0

Im k>0

Im k<0

+8+8

8--

8-- 0 II-- q

0q

0

aII

cII d

II

bII

Sheet II: Im λ<0

Im k>0

Im k<0


Investigating the properties of the Green’s functions one can show that theeigenfunctions

φ1 and ψ3 are analytic on the upper sheet

φ3 and ψ1 are analytic on the lower sheet

φ2 and ψ2 neither

Also, from the asymptotic behavior of the solutions

W (φ1,φ2,φ3) = W (ψ1,ψ2,ψ3) = −2λq20eikx

i.e. [φ1,φ2,φ3] and[ψ1,ψ2,ψ3] are each a set of linearly independent solutions of thethird order scattering problem.


Direct Problem: Scattering Data

Introduce the scattering coefficientsai j expressing one set of eigenfunctions as alinear combination of the other one

φ j(x,k) =3∑

i=1a ji(k)ψi(x,k), j = 1,2,3 (3a)

or, equivalently,

ψ j(x,k) =3∑

i=1b ji(k)φi(x,k), j = 1,2,3 (3b)

where(bi j) is the inverse matrix of(ai j) (both have determinant equal to 1.)ai j,bi j are in general defined only on the banks of the cuts of the Riemann surface.However,

a11(k) and b33(k) are analytic on the upper sheet

a33(k) and b11(k) are analytic on the lower sheet


Problem: how to substitute the non-analytic eigenfunction s?

Generalize the approach introduced by Kaup [1976] for the three wave interaction toobtain a representation of the nonanalytic eigenfunctionsφ2 andψ2 in terms ofanalytic eigenfunctions and scattering data.Key idea:consider, together with the scattering problem

∂xv = (ikJ+Q)v (4a)

theadjoint eigenvalue problem

∂xvad =(

−ikJ+QT)

vad (4b)

and use that ifuad andwad are two arbitrary solutions of the adjoint problem (4b), then

v = −J(

uad∧wad)

eikx (5)

is a solution of the original scattering problem (4a).Note∧ denotes standard exterior product (recall the eigenfunctions are 3-componentvectors.)


Adjoint eigenfunctions

Then one can introduce two sets of solutions of the adjoint problem by theirasymptotic conditions asx →±∞, i.e.

[

φad1 ,φad

2 ,φad3

]

and[

ψad1 ,ψad

2 ,ψad3

]

.By the same techniques as above, one shows

φad3 (x,k) and ψad

1 (x,k) are analytic in the upper sheet

φad1 (x,k) and ψad

3 (x,k) are analytic on the lower sheet

From the adjoint states, we can now constructtwo new solutionsχ andχ

χ = −eikxJ(φad1 ∧ψad

3 ) (6a)

χ = −eikxJ(φad3 ∧ψad

1 ) (6b)

which by construction areanalyticin the lower and upper sheet respectively.

[recallJ = diag(−1,1,1)]


Moreover, one can show that

χ(x,k) = 2λ [b33(k)ψ2(x,k)−b23(k)ψ3(x,k)] (7a)

χ(x,k) = 2λ [b21(k)ψ1(x,k)−b11(k)ψ2(x,k)] (7b)

and thus obtains decompositions of the nonanalytic eigenfunctionψ2

ψ2(x,k) =b21(k)b11(k)

ψ1(x,k)−12λ

χ(x,k)b11(k)

(8a)

ψ2(x,k) =b23(k)b33(k)

ψ3(x,k)+12λ

χ(x,k)b33(k)

(8b)

Similar relations hold for the eigenfunctionφ2.


Uniformization

Following an idea proposed by Faddeev and Takhtajan [1987] for the scalar NLS, weintroduce auniformization variablez defined by the conformal mapping

z = z(k) = k +λ(k) (9a)

The inverse mapping is given by

k = k(z) =12

(

z+q2

0

z

)

(9b)

λ(k) = z− k =12

(

z−q2

0

z

)

, λ(k)− k = −q2

0

z


i The mapping takes the cuts on the Riemann surface onto the real z axis

ii The sheetsC1/C2 are mapped onto the upper/lower half planes ofz

iii k = ∞ on either sheet is mapped ontoz = ∞ or z = 0 depending on the sign ofkIm

iv (−q0,q0) on each sheet mapped into the upper/lower semicircles of radiusq0

Im z

Re z

Sheet II: Im λ<0

Sheet I: Im λ>0

-- q 0

8--

Im k>0

Im k>0

q0

Im k<0

+8

Im k<0

dI

aIIc

IId

IIb

II

aI

bI

cI

0I

0II


Direct problem: Symmetries

When the potentials are such thatr∗(x) = q(x), the scattering coefficients satisfycertain symmetry relations.

For the scalar NLS:symmetries+ self-adjointness+ unitarity of the scattering matrix

confine the zerosof a11(z) etc on the circle|z| = q0

For the vector NLS this is no longer generally valid.

In fact, we exclude zeros on the realz-axis, and since

b33(zn) = 0 ⇔ a33(z∗n) = 0 ⇔ a11(q

20/z∗n) = 0 ⇔ b11(q

20/zn) = 0

it follows:zeroson the circleof radiusq0 come intopairs{ζn,ζ∗n}

zerosoff the circlecome intoquartets{

zn,z∗n,q20/zn,q2

0/z∗n}


Re z

Im z

zn

zn*

zn==q02/zn*

^

zn**= q02/zn

^

ζn

ζn*

-q0 0 q

0

Location of the discrete eigenvalues in the complexz-plane


Discrete eigenvalues

Wr(φ1(x,z),χ(x,z),ψ3(x,z)) = −4q20λ2(z)a11(z)b33(z)e

ik(z)x

Wr(ψ1(x,z), χ(x,z),φ3(x,z)) = −4q20λ2(z)a33(z)b11(z)e

ik(z)x

henceφ1,χ,ψ3 linearly dependent at the zeros ofa11(z) andb33(z)

ψ1, χ,φ3 linearly dependent at the zeros ofa33(z) andb11(z)

If b33 has a zero atz = ζn on the circle of radiusq0, sinceq20/ζ∗n ≡ ζn, then thena11(z)

vanish at the same point and the Wronskian has a double zero.

If b33 vanishes at a pointzn off the circle then such zeros appear in quartets and theWronskian will have a simple zero both atz = zn and atz = zn ≡ q2

0/z∗n in the UHP.

Similarly for the lower half plane.


Inverse ProblemIn order to formulate the inverse problem in terms ofanalytic eigenfunctions, one canuse, for instance, the expression ofψ2 in terms ofψ1 andχ

ψ2(x,z) =b21(z)b11(z)

ψ1(x,z)−12λ

χ(x,z)b11(z)

into the equations defining the scattering coefficients

φ3(x,z) = a31(z)ψ1(x,z)+a32(z)ψ2(x,z)+a33(z)ψ3(x,z)

and getφ3(x,z)a33(z)

= ψ3(x,z)−

[

b31(z)b11(z)

ψ1(x,z)+χ(x,z)

2λ(z)b11(z)a32(z)a33(z)

]

and similar relations for the eigenfunctionsφ3 andχ.

One obtains a system which can be considered as a Riemann-Hilbert problem withpoles on the realz-axis. The aim is to convert the RH problem stated above into alinear system of algebraic-integral equations.


ψ3(x,z)e−iλ(z)x = −

q20/z

ir+

−N1∑

n=1C(1)

nψ1(x,ζ∗n)e−iλ(ζ∗n)x

z−ζ∗n−

N2∑

n=1C(2)

nχ(x,z∗n)e

−iλ(z∗n)x

z− z∗n

−1

2πi

∞R

−∞

dζζ− (z+ i0)

[

ρ1 (ζ)ψ1(x,ζ)−ρ2(q20/ζ)

χ(x,ζ)

2λ(ζ)b11(ζ)

]

e−iλ(ζ)x

ψ1(x,z)eiλ(z)x =

z

ir+

+N1∑

n=1

zC(1)n

z−ζnψ3(x,ζn)e

iλ(ζn)x +N2∑

n=1

zC(2)n

z− znχ(x,z∗n)e

iλ(zn)x

+z

2πi

∞R

−∞

dζζ− (z− i0)

[

−ρ1(q20/ζ)ψ3(x,ζ)+ρ2(ζ)

χ(x,q20/ζ)

2λ(q20/ζ)b11(q2

0/ζ)

]

eiλ(ζ)x

ζ

χ(x,z)e−ik(z)x

2λ(z)b11(z)=

0

iq⊥+

+N1∑

n=1

χ(x,ζ∗n)e−ik(ζn)x

ζ∗nb′11(ζ∗n)z

(z−ζn)(z−ζ∗n)

−N2∑

n=1b(2)

nψ1(x, z∗n)e

−ik(zn)x

z∗nb′11(z∗n)

z(z− zn)(z− z∗n)

−1

2πi

∞R−∞

dζζ− (z− i0)

[

ρ2(ζ)ψ1(x,ζ)+ ρ2(q20/ζ)ψ3(x,ζ)

]

e−ik(ζ)x


We obtained a formal solution for the inverse problem as a linear system ofthreealgebraic-integral equationsin thethree unknownsψ1, ψ3, χ/(2λb11), with reflectioncoefficients

ρ1(z) =b31(z)b33(z)

ρ2(z) =a12(z)a11(z)

and theC j’s, C j ’s etc are the norming constants, corresponding to a set ofN1 pairs ofzeros

{

ζ j,ζ∗j}

on the circle or radiusq0 andN2 quartets of zeros{

z j,z∗j ,q20/z j,q2

0/z∗j}

off the circle.

If C( j)n = C( j)

n = 0, the above system is a linear system ofintegralequations.

In the so-calledreflectionlesscase, i.e. when the coefficientsρ j, ρ j are identicallyzero on the real axis, it becomes a linear algebraic system.

In general, the system is consistently closed by evaluatingthe equations at the properpoints of the discrete spectrum.

The associated linear problem allows then to determine the time evolution ofeigenfunctions and scattering data.


Explicit solutions: dark-dark solitonOne can explicitly solve the system in the reflectionless case with one (or more)discrete eigenvalues. In the case ofone eigenvalueζ1 on the circle of radiusq0 onegets thedark-dark soliton

q(x, t) = q+(0)e2iq20t

[

1+(e2iα −1)e2q0 sinα(x−2q0 cosα t−x0)

1+ e2q0sinα(x−2q0 cosα t−x0)

]

ζ1 = k1 + iν1 = q0e−iα, e2ν1x0 = γ(0) ≡ |b(1)1 (0)|

whereb(1)1 plays the role of the norming constant associated to the eigenvalueζ1.


Explicit solutions: dark-bright solitonOne can solve the equations of the inverse problem with one quartet of eigenvaluesoff the circleof radiusq0 (with principal eigenvaluez1 = k1 + iν1). If, for simplicity,

one assumesq(1)+ = 0 andq(2)

+ 6= 0, the solution takes the form

q(1)(x, t) = −ν1 sinα√

q20−|z1|2 sech[ν1(x−2k1t)+ x0]e−ik1x+i[2q2

0+(k21−ν2

1)]t−iϕ1

q(2)(x, t) = q0{cosα+ isinα tanh[ν1(x−2k1t)+ x0]}e2iq20t−iϕ2

wherek1 = |z1|cosα , ν1 = −|z1|sinα

e2x0 =q2

0

4ν21

(q20−|z1|

2)|C(2)1 (0)|2, ϕ1 = argC(2)

1 (0)+θ(2)+ (0), ϕ2 = −θ(2)

+ (0)

C(2)1 is the norming constant associated to the eigenvaluez1. Note that the condition

k21 +ν2

1 ≡ |z1|2 < q2

0 is necessary and sufficient to ensure the regularity of the solutionat all times.


0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

Amplitude of the “bright” component as a function ofν1, for severalvalues of the soliton velocitiesk2

1 = .7, .5, .2, .1, .001 bottom to top.


-10 -5 5 10x

0.2

0.4

0.6

0.8

1

P

-6 -4 -2 2 4 6x

0.2

0.4

0.6

0.8

1

P

-6 -4 -2 2 4 6x

0.2

0.4

0.6

0.8

1

P

Amplitudes of the “dark” and “bright” components as a function ofξ = ν1(x−2k1t), for several values of the soliton parametersk1,ν1.


Soliton interactions

When two dark-dark or dark-bright solitons interact, one expects no change except adisplacement of the center of each soliton.Assume the first soliton travels faster (k1 > k2), so that it crosses soliton 2 from left toright as time increases. Long-time asymptotic analysis then gives for the positionshift of the first soliton (both components)

∆x1 =1ν1

|z1− z2|2|q0− z1z2|

|z∗1− z2|2|q0− z∗1z2|

and an overall phase shift which is given byz∗2/z2 for the dark component and

z1

|z1|

z∗2z2

(z1− z2)2

|z1− z2|4|z∗1− z2|

4

(z∗1− z2)2

|q20− z∗1z2|

q20− z∗1z2

q20− z1z2

|q20− z1z2|

.

Note that, unlike the “bright” Manakov solitons, there is nopolarization shift, i.e. theamount of energy in each component (dark-dark or dark-bright) remains unchanged.


Explicit solutions: small amplitude limitOne can solve the inverse problem in thesmall amplitude limit, i.e., withρ1,ρ2 · · · ≪ 1 and in absence of solitons.One can substitute forψ1,ψ3, χ their leading order expansions, and neglecting, at firstorder, those terms which are “quadratic” in the small quantities, one gets a solution

(

q(1)(x, t)q(2)(x, t)

)

=

(

q(1)+ (t)

q(2)+ (t)

)

[

1+1

2πi

∞R−∞

dζζ

ρ∗1(ζ, t)e2iλ(ζ)x

]

−

(

r(2)+ (t)

−r(1)+ (t)

)

12πi

∞R−∞

dζζ

ρ∗2(q

20/ζ, t)e−i(k(ζ)−λ(ζ))x

which is found consistent with the linear limit of the vectorequation.


ConclusionsAn IST for VNLS systems with non-vanishing boundary conditions as|x| → ∞ hasbeen presented.

Suitable formulations (wrt to scalar case) both of the direct and of the inverseproblems have been obtained.

On the direct side, this has been achieved via theadjoint scattering problem, whichprovides two additional analytic solutionsof the scattering problem.

The inverse problem has formulated as a Riemann-Hilbert (RH) problem for theanalytic eigenfunctions on the complex plane of theglobal uniformizing parameterz.

Assuming the discrete (proper) eigenvalues of the scattering problem are suitablyconfined in thez plane, the RH problem has been transformed into a closed linearsystem of algebraic-integral equations.

Explicit solutions(dark-dark, dark-bright solitons, small amplitude limit solutions)have beenobtained.


inverse scattering transform for vector defocusing ... · straightforward way to derive explicit...

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