Inverse scattering transform for the integrable nonlocal nonlinear

Schrodinger equation

Mark J. Ablowitz1 and Ziad H. Musslimani2

1Department of Applied Mathematics, University of ColoradoCampus Box 526, Boulder, Colorado 80309-0526

2Department of Mathematics, Florida State University, Tallahassee, FL 32306-4510

June 17, 2015

Abstract

A nonlocal nonlinear Schrodinger (NLS) equation was recently introduced and shown to be an integrableinfinite dimensional Hamiltonian evolution equation. In this paper a detailed study of the inverse scatteringtransform of this nonlocal NLS equation is carried out. The direct and inverse scattering problems are analyzed.Key symmetries of the eigenfunctions and scattering data and conserved quantities are obtained. The inversescattering theory is developed by using a novel left-right Riemann-Hilbert problem. The Cauchy problem forthe nonlocal NLS equation is formulated and methods to find pure soliton solutions are presented; this leads toexplicit time-periodic one and two soliton solutions. A detailed comparison with the classical NLS equation isgiven.

1 Introduction

Exactly solvable models and integrable evolution equations are ubiquitous in nonlinear science and play an essentialrole in many branches of physics. Generally speaking, many of these systems can be derived from basic principlesand arise as universal models in diverse physical phenomena. For example, the Korteweg-deVries (KdV) and modi-fied Korteweg-deVries (mKdV) equations describe the evolution of weakly dispersive and small amplitude waves inquadratic and cubic nonlinear media respectively. Physically speaking, the KdV equation is well known in terms ofits application to shallow water waves [19, 5].

The integrable cubic nonlinear Schrodinger (NLS) equation [28] is also a universal model. It describes the evo-lution of weakly nonlinear and quasi-monochromatic wave trains in media with cubic nonlinearities. In the contextof nonlinear optics, the NLS equation is a key model describing optical wave propagation in Kerr media [9, 18].Moreover, in the limit of deep water, the NLS equation can be derived from the classical irrotational and inviscidwater waves equations [5]. The KdV, mKdV and NLS equations share the mathematical property that all are inte-grable and exactly solvable evolution equations.

There are also numerous continuous and discrete exactly integrable evolution equations that are physically rele-vant and apply to diverse problems in fluid mechanics, electromagnetics, gravitational waves, elasticity, fundamentalphysics and lattice dynamics, to name but a few [3, 5, 6]. Recently integrable continuous and discrete nonlocalnonlinear Schrodinger equations describing wave propagation in nonlinear PT symmetric media were also found[1, 2].

Interestingly there are multi-dimensional extensions of many of these equations. The best known integrable multi-dimsional equation is the Kadomtsev-Petviashvili (KP) equation [17]. It is too a universal model describing theevolution of weakly dispersive and small amplitude waves with additional weak transverse variation. It arises in

1

water waves, plasma physics and internal waves [3, 5]. Remarkably, multi-line soliton solutions of the KP equationare observed virtually daily on shallow flat beaches [8].

Generally speaking integrability is established once an infinite number of constants of motion or an infinite numberof conservation laws are obtained. However considerably more information about the solution can be obtained ifthe inverse scattering transform (IST) can be carried out [4]. Corresponding to rapidly decaying initial data, ISTprovides a linearization and a class of explicit solutions–e.g. solitons.

The method associates a compatible pair of linear equations (i.e. a Lax pair) with the integrable nonlinear equa-tion. One of the equations, termed the scattering problem, is used to determine suitably analytic eigenfunctionsand provides a mechanism to transform the initial data to appropriate scattering data. The other linear equationserves to determine the evolution of the scattering data. Using the analytic behavior of the eigenfunctions an in-verse scattering problem is developed. It is convenient to transform the inverse scattering system to a generalizedlinear Riemann-Hilbert (RH) problem to solve for the underlying meromorphic functions. With the time depen-dence of the scattering data one can find the solution of the nonlinear evolution equation from the inverse or RHproblem. The IST method is viewed as an extension of Fourier transforms to a class of nonlinear systems [3, 5, 24, 12].

Here we investigate in detail the following nonlocal nonlinear Schrodinger (NLS) equation

iqt(x, t) = qxx(x, t)± 2q2(x, t)q∗(−x, t) , (1.1)

where ∗ denotes complex conjugate. This equation was first introduced in [1] but due to size limitations many detailshad to be omitted. In the above equation we see that the nonlocality occurs in a remarkably simple way; namelyone of the nonlinear terms has the dependent variable evaluated at −x, i.e. note the term q∗(−x, t). Another wayto write the above equation is

iqt(x, t) = qxx(x, t) + V (x, t)q , (1.2)

where V (x, t; q) = ±2q(x, t)q∗(−x, t). In this latter form the equation is viewed as a Schrodinger equation witha nonlinear “PT” symmetric potential [10]: V (x, t) = V ∗(−x, t). It is remarkable that such a simple nonlocalnonlinear Schrodinger equation turns out to be integrable. We also note that this equation is Galilean Invariant.PT -symmetric equations have been extensively studied in recent years [15, 22, 23, 27, 16, 25, 26, 20, 21]. This isdiscussed further in Section 2.

There are other nonlocal integrable systems which have been analyzed; two that are well-known are the Benjamin-Ono and Intermediate Long Wave equations [3, 5]. These equations were found in the study of long internal waveswith a deep bottom layer. The IST for these systems has been known since the 1980s.

In this paper we develop the IST associated with nonlocal nonlinear Schrodinger (NLS) equation (1.1). It turns outthat the general method of AKNS [5, 6] can be applied but with a new symmetry relationship imposed. This cru-cial symmetry relation has major consequences; it leads to new symmetries amongst the eigenfunctions, scatteringdata and a novel inverse problem which can be related to a Riemann-Hilbert problem formulated via eigenfunctionsdefined at both plus and minus infinity; here we refer to the method as a left/right Riemann-Hilbert approach. Toour knowledge this is the first time that such left-right Riemann-Hilbert problems have been employed in the ISTassociated with nonlinear wave problems.

The paper is organized as follows. In Sec. 2 we employ the AKNS procedure to find a coupled NLS type sys-tem in terms of two potentials: q(x, t) and r(x, t). With the symmetry

r(x, t) = σq∗(−x, t), σ = ∓1 (1.3)

the nonlocal nonlinear Schrodinger (NLS) equation (1.1) results. In Sec. 3 we provide a method to derive an infinitenumber of local and global conservation laws associated with the nonlocal nonlinear Schrodinger equation (1.1).This establishes its integrability as an infinite dimensional Hamiltonian dynamical system. In Sec. 4 certain keyasymptotic properties of the eigenfunctions and scattering data are discussed; this is followed by Sec. 5 where thesymmetries of the eigenfunctions as well as of the scattering data are established. The basic inverse scatteringproblem is developed in Sec. 6 along with a detailed analysis of the left-right Riemann-Hilbert problem. Thereconstruction formula of the potentials is presented in Sec. 7. The time periodic one and two soliton solutionsand some of their properties are given in Sec. 8. Comparisons with the classical NLS equation is in Sec. 9; thecalculation for the scattering data associated with Eq. (1.1) for special box initial condition is given in Sec. 10. Weconclude in Sec. 11.

2

2 The nonlocal nonlinear Schrodinger equation: Linear pair and com-patibility condition

We begin our discussion by considering the AKNS scattering problem [5, 6]

vx = Xv , (2.4)

where v = v(x, t) is a two-component vector, i.e., v(x, t) = (v1(x, t), v2(x, t))T and q(x, t), r(x, t) are (in general)complex valued functions that vanish rapidly as |x| → ∞ and k is a complex spectral parameter. The matrix Xdepends on the functions q and r as well as on the spectral parameter k

X =

(−ik q(x, t)r(x, t) ik

). (2.5)

The time evolution of the eigenfunctions vj , j = 1, 2 is given by

vt = Tv , (2.6)

where

T =

(A BC −A

), (2.7)

and A,B and C are scalar functions of q(x, t), r(x, t) given by

A = 2ik2 + iq(x, t)r(x, t) , (2.8)

B = −2kq(x, t)− iqx(x, t) , (2.9)

C = −2kr(x, t) + irx(x, t) . (2.10)

The compatibility condition of system (2.4) and (2.6), i.e., vjxt = vjtx, j = 1, 2 yields

iqt(x, t) = qxx(x, t)− 2r(x, t)q2(x, t) , (2.11)

−irt(x, t) = rxx(x, t)− 2q(x, t)r2(x, t) . (2.12)

Under the symmetry reductionr(x, t) = σq∗(−x, t) , σ = ∓1 , (2.13)

system (2.11) and (2.12) are compatible and leads to the nonlocal nonlinear Schrodinger equation first introducedin [1] and mentioned above in the introduction. We write the resulting equation (1.1) again for the convenience ofthe reader:

iqt(x, t) = qxx(x, t)± 2q2(x, t)q∗(−x, t) ,

where again ∗ denotes complex conjugation and q(x, t) is a complex valued function of the real variables x and twith corresponding Lax pairs given by

X =

(−ik q(x, t)σq∗(−x, t) ik

), T =

(2ik2 + iσq(x, t)q∗(−x, t) −2kq(x, t)− iqx(x, t)−2σkq∗(−x, t)− iσq∗x(−x, t) −2ik2 − iσq(x, t)q∗(−x, t)

). (2.14)

The crucial symmetry reduction (2.13) was first noted in [1] and leads to a novel class of nonlocal integrable nonlinearevolution equations including a nonlocal NLS hierarchy. This is a special and remarkably simple reduction of themore general AKNS system [4] which has not been previously found. A list of few important properties of Eq. (1.1)are shown below:

• Time-reversal symmetry: If q(x, t) is a solution so is q∗(x,−t).

• Invariance under the transformation x→ −x : If q(x, t) is a solution so is q(−x, t).

• Gauge invariance: If q(x, t) is a solution so is eiθ0q(x, t) with real and constant θ0.

• Complex translation invariance: If q(x, t) is a solution so is q(x+ ix0, t) for any constant and real x0.

3

• Equation (1.1) is a Hamiltonian dynamical system and is obtained using the variational formulation

iqt(x, t) =δH

δq∗(−x, t), (2.15)

where δHδq∗(−x,t) is the variational derivative of the Hamiltonian with respect to q∗(−x, t) and is given by

H =

∫ +∞

−∞

[−qx(x, t)q∗x(−x, t)− σq2(x, t)q∗2(−x, t)

]dx , σ = ∓1 . (2.16)

• PT symmetry: If q(x, t) is a solution so is q∗(−x,−t). As mentioned in the Introduction, calling the quantity

V (x, t) = ±2q(x, t)q∗(−x, t) , (2.17)

which, in classical optics is referred to as a self-induced potential, implies that V ∗(−x, t) = V (x, t).; and withthis at hand, Eq. (1.1) takes the form

iqt(x, t) = qxx(x, t) + V (x, t)q(x, t) . (2.18)

This equivalent formulation allows one to connect Eq. (1.1) with PT symmetric optics for which V (x, t)represents a “waveguide” and the resulting equation remains invariant under the joint transformation of x→−x, t→ −t and a complex conjugate. Thus, the nonlocal equation (1.1) is PT symmetric [10]. We remark thatwave propagation in PT symmetric coupled waveguides or photonic lattices has been observed in experimentsin classical optics [15, 22, 23, 27, 16, 25, 26, 20, 21].

• Galilean invariance: If q(x, t) solves Eq. (1.1) with initial condition q(x, 0) then

q(x, t) = q(x+ 2iξt, t)e−ξxe−iξ2t , (2.19)

also solves to Eq. (1.1) with corresponding initial condition q(x, 0) = q(x, 0)e−ξx for any real constant ξ. If onerestricts the class of solutions to Eq. (1.1) to rapidly decaying solutions at |x| → ∞ then this would put somerestriction on the parameter ξ to guarantee that q(x, t) itself remains within that class. We point out that thegeneral q, r system (2.11) and (2.12) is also Galilean invariant: If q(x, t), r(x, t) solve system (2.11) and (2.12)so does

q(x, t) = q(x− iV t, t)e−K1xe−iω1t , (2.20)

r(x, t) = r(x− iV t, t)e−K2xe−iω2t , (2.21)

with K1 = −K2 and ω2 = −ω1 all being real parameters satisfying the relations V = −2K1 and ω1 = K21 .

An infinite number of local and global conservation laws associated with Eq. (1.1) can derived hence it is an in-tegrable evolution equation. Using a left-right Riemann-Hilbert formulation, the inverse scattering transform iscarried out and general solution to Eq. (1.1) corresponding to rapidly decaying initial data, is obtained includingpure one and two solitons solutions. Key important properties of Eq. (1.1) are also contrasted with the classicalNLS equation where the nonlocal nonlinear term q∗(−x, t) is replaced by q∗(x, t). In particular, the symmetries ofthe eigenfunctions and scattering data associated with the classical NLS equations with even initial data are shownto coincide with symmetries of (2.4) under the symmetry reduction (2.13).

3 Infinite number of conserved quantities and conservation laws

3.1 Global Conservation Laws

The infinite number of conserved quantities of (1.1) is derived as follows. We assume that q(x, t) decays rapidlyat infinity. As a result, solutions of the scattering problem (2.4) can be defined subject to the following boundaryconditions

limx→−∞

φ(x, k) =

(10

)e−ikx , lim

x→−∞φ(x, k) =

(01

)eikx , (3.22)

4

limx→+∞

ψ(x, k) =

(01

)eikx , lim

x→+∞ψ(x, k) =

(10

)e−ikx . (3.23)

Throughout the paper, φ is not the complex conjugate of φ. We shall instead use φ∗ to denote the complex conjugateof φ. If φ(x, t) = (φ1(x, t), φ2(x, t))T is the solution to (2.4) that satisfies the boundary conditions (3.22) then, forIm k ≥ 0, the scattering data a(k) ≡ φ1(x, t)eikx is analytic in the upper half complex k plane and approaches 1as x → ±∞. Substituting φ1(x, t) = exp [−ikx+ ϕ(x, t)] into (2.4) we find (after eliminating φ2) that the functionµ(x, t) ≡ ϕx(x, t) satisfies the Riccati equation

q∂

∂x

(µ

q

)+ µ2 − qr − 2ikµ = 0 . (3.24)

Since for Imk > 0 , lim|k|→∞ ϕ(x, k) = 0 we substitute the asymptotic expansion

µ(x, k) =

∞∑n=0

µn(x, t)

(2ik)n+1, (3.25)

in Eq. (3.24) and equate powers of k to find

µ0(x, t) = −q(x, t)r(x, t) , (3.26)

µ1(x, t) = −q(x, t)rx(x, t) , (3.27)

and for any integer n ≥ 1

µn+1 = q∂

∂x

(µnq

)+

n−1∑m=0

µmµn−m−1 . (3.28)

From the boundary conditions (3.22) it follows limx→−∞ φ1(x, k)eikx = 1 and limx→−∞ ϕ(x, k) = 0. Combiningthis result together with the definition of µ we have

ln a(k) =

∞∑n=0

Cn(2ik)n+1

, (3.29)

where we have defined

Cn =

∫ +∞

−∞µn(x, t)dx . (3.30)

Since for all k with Im k > 0, a(k) is time independent then it follows that Cn is also time independent. Below is alist of the first few conserved quantities under the symmetry reduction r(x, t) = σq∗(−x, t) , σ = ∓1 :

C0 = −σ∫ +∞

−∞q(x, t)q∗(−x, t)dx , (3.31)

C1 =σ

2

∫ +∞

−∞[qx(x, t)q∗(−x, t) + q(x, t)q∗x(−x, t)] dx , (3.32)

C2 = −σ∫ +∞

−∞

[qx(x, t)q∗x(−x, t)− σq2(x, t)q∗2(−x, t)

]dx , (3.33)

C3 =σ

2

∫ +∞

−∞{qxxx(x, t)q∗(−x, t) + q(x, t)q∗xxx(−x, t)

−2σq(x, t)q∗(−x, t) [4q(x, t)q∗x(−x, t)− qx(x, t)q∗(−x, t)]}dx , (3.34)

C4 = −σ∫ +∞

−∞{qxx(x, t)q∗xx(−x, t)− 6σq(x, t)q∗(−x, t)qx(x, t)q∗x(−x, t)

+σ [q∗(−x, t)qx(x, t)− q(x, t)q∗x(−x, t)]2 + 2q3(x, t)q∗3(−x, t)}dx , (3.35)

5

3.2 Local Conservation Laws

In this section we derive all local conservation laws. We start with the time-dependent problem (2.6)

φ1t = Aφ1 + Bφ2 . (3.36)

Substituting the expression for µ and ϕ into (3.36) and taking the x derivative of the resulting equations we find

∂tµ(x, t) = ∂x

(Anonloc +

µ(x, t)Bnonloc

q(x, t)

). (3.37)

Recall that the nonlocal nonlinear Schrodinger equation (1.1) is obtained when

Anonloc = 2ik2 + iσq(x, t)q∗(−x, t) , (3.38)

Bnonloc = −2kq(x, t)− iqx(x, t) , (3.39)

Cnonloc = −2kσq∗(−x, t)− iσq∗x(−x, t) . (3.40)

Substituting the power series expansion for µ from (3.25) and (3.38)-(3.39) into (3.37) we obtain

∂t

( ∞∑n=0

µn(x, t)

(2ik)n+1

)= i∂x

[σq(x, t)q∗(−x, t) +

(2ik − qx(x, t)

q(x, t)

) ∞∑n=0

µn(x, t)

(2ik)n+1

]. (3.41)

The coefficients of (2ik)−n are trivial for n ≤ −1. For n ≥ 0 we find

∂tµn(x, t) + i∂x

[qx(x, t)

q(x, t)µn(x, t)− µn+1(x, t)

]= 0 , n = 0, 1, 2, 3, · · · . (3.42)

We can write the conservation laws in the form

∂T∂t

= −i∂X∂x

, (3.43)

where T and X are the so-called densities and fluxes respectively. The first few conservation laws are given by

1. T = q(x, t)q∗(−x, t) , X = q(x, t)q∗x(−x, t) + q∗(−x, t)qx(x, t) .

2. T = q(x, t)q∗x(−x, t) , X = q∗x(−x, t)qx(x, t) + q(x, t)q∗xx(−x, t)− σq2(x, t)q∗2(−x, t) ,

3. T = q(x, t)q∗xx(−x, t)−σq2(x, t)q∗2(−x, t) , X = qx(x, t)q∗xx(−x, t)+q(x, t)q∗xxx(−x, t)−4σq2(x, t)q∗(−x, t)q∗x(−x, t) .

4 Direct scattering problem

In this section we study the scattering problem (2.4) subject to the boundary conditions (3.22). It is expedient toreformulate the scattering problem in terms of eigenfunctions having constant boundary conditions (the so-called“Jost functions”) defined by (note: hereafter, for simplicity of notation, we often suppress the time dependence)

M(x, k) = eikxφ(x, k) , M(x, k) = e−ikxφ(x, k) , (4.44)

N(x, k) = e−ikxψ(x, k) , N(x, k) = eikxψ(x, k) . (4.45)

With this in mind, one can show that the Jost functions satisfy a linear implicit integral equation that in turn isused to establish the following important results: M(x, k), N(x, k) are analytic functions in the upper half complexk plane whereas M(x, k), N(x, k) are analytic functions in the lower half complex k plane [6]. From the integralrepresentation one can also derive the large k asymptotics of the Jost functions used in the inverse problem, (c.f.[6])

M(x, k) =

1− 12ik

∫ x−∞ r(z)q(z)dz

− r(x)2ik

+O(k−2) , (4.46)

6

N(x, k) =

q(x)2ik

1− 12ik

∫ +∞x

r(z)q(z)dz

+O(k−2) , (4.47)

M(x, k) =

q(x)2ik

1 + 12ik

∫ x−∞ r(z)q(z)dz

+O(k−2) , (4.48)

N(x, k) =

1 + 12ik

∫ +∞x

r(z)q(z)dz

− r(x)2ik

+O(k−2) . (4.49)

The solutions φ(x, k) and φ(x, k) of the scattering problem (2.4) with the boundary conditions (3.22) are linearlyindependent for all k satisfying a(k) 6= 0 – see below. This follows from the fact that the Wronskian W (u, v) is givenby

W (u, v) ≡ u1v2 − u2v1 , (4.50)

of any two solutions u and v to (2.4) is independent of x. Similar arguments hold for ψ(x, k) and ψ(x, k). Thereforebecause the scattering problem (2.4) is a second order linear ODE, the pairs {φ, φ} and {ψ,ψ} are linearly dependentand one can express one basis set in terms of the other:

φ(x, k) = a(k)ψ(x, k) + b(k)ψ(x, k) , (4.51)

φ(x, k) = a(k)ψ(x, k) + b(k)ψ(x, k) . (4.52)

With this result, the scattering coefficients are thus given by

a(k) = W (φ, ψ) , (4.53)

a(k) = W (ψ, φ) . (4.54)

b(k) = W (ψ, φ) , (4.55)

b(k) = W (φ, ψ) . (4.56)

Note that the scattering data satisfy the relation

a(k)a(k)− b(k)b(k) = 1 . (4.57)

If the potentials q, r → 0 as |x| → ∞ then from the analyticity properties of the Jost functions, it can be shown thata(k) is analytic in the upper half complex k plane whereas a(k) is analytic in the lower half complex k plane [6]. Ingeneral, b(k) and b(k) cannot be extended off the real k axis. Therefore, Eq.(4.57) is defined only for Imk = 0. Thelarge k asymptotics of the scattering data a(k) and a(k) is given by [6]

a(k) ∼ 1− 1

2ik

∫ +∞

−∞r(z)q(z)dz +O(k−2) , (4.58)

a(k) ∼ 1 +1

2ik

∫ +∞

−∞r(z)q(z)dz +O(k−2) . (4.59)

7

5 Symmetry reduction r(x, t) = σq∗(−x, t), σ = ∓1: eigenfunctions andscattering data

5.1 Symmetry of the eigenfunctions

In this section we establish important symmetry properties of the eigenfunctions of the eigenvalue problem (2.4)valid under the symmetry reduction r(x, t) = σq∗(−x, t) where σ = ∓1. To do so, let v(x, k) ≡ (v1(x, k), v2(x, k))T

be a solution to system (2.4) with r(x, t) = σq∗(−x, t). If we take the complex conjugation of (2.4) and letx→ −x, k∗ → −k∗ one reaches the following conclusion:

If

(v1(x, k)v2(x, k)

)solves (2.4) then

(v∗2(−x,−k∗)−σv∗1(−x,−k∗)

)solves (2.4) . (5.60)

Therefore, because the solutions of the scattering problem (2.4) are uniquely determined by their respective boundaryconditions (3.22) we obtain the important symmetry relations

ψ(x, k) =

(0 −σ1 0

)φ∗(−x,−k∗) , (5.61)

ψ(x, k) =

(0 1−σ 0

)φ∗(−x,−k∗) . (5.62)

With the definitions (4.44) and (4.45) one can readily write down the corresponding symmetry conditions of theJost functions:

N(x, k) =

(0 −σ1 0

)M∗(−x,−k∗) , (5.63)

N(x, k) =

(0 1−σ 0

)M∗(−x,−k∗) . (5.64)

Thus we see that the symmetry r(x, t) = σq∗(−x, t) leads to symmetry between eigenfunctions defined at ±∞.

5.2 Symmetry of the scattering data

The symmetry in the eigenfunctions in turn imposes a very important symmetry in the scattering data a(k), a(k) andb(k), b(k). From the Wronskian representations for the scattering data and the above symmetry relations togetherwith the fact that the Wronskian does not depend on x it follows for both signs σ = ∓1

a(k) = a∗(−k∗) , (5.65)

a(k) = a∗(−k∗) , (5.66)

b(k) = σb∗(−k∗) . (5.67)

It therefore follows from Eq. (5.65) that if kj = ξj + iηj is a zero (eigenvalue) of a(k) in the upper half k plane then−k∗j = −ξj + iηj is a zero of a(k) in the upper half k plane. Similarly, if kj is a zero of a(k) in the lower half k plane

so is −k∗j in the upper half k plane.

6 Inverse scattering: A left-right Riemann - Hilbert approach

The inverse problem consists of constructing the potential functions r(x, t) and q(x, t) from the scattering data, i.e.,reflection coefficients ρ(k, t) and ρ(k, t) defined on Imk = 0 as well as the eigenvalues kj , kj and norming constants

8

(in x) Cj(t), Cj(t). Our approach is based on solving two Riemann-Hilbert problems associated to what we refer asleft and right scattering problems and use the symmetry conditions established in Section 5 to relate between thetwo pieces. In doing so we will make frequent use of the so-called projection operators defined as follows. For anyintegrable function f(k) , k ∈ C that rapidly decays to zero as |k| → ∞ we define the projection operators P± as

P±f = limε→0

1

2πi

∫ ∞−∞

f(ξ)

ξ − (k ± iε). (6.68)

One of the most important properties of the projection operators are

P±f± = ±f± , P±f∓ = 0 , (6.69)

where f+(z) and f−(z) are analytic functions in the upper and lower complex half plane respectively satisfyingf±(z)→ 0 as |z| → ∞.

6.1 Left scattering problem

Our starting point is the “left” scattering problem (4.51) and (4.52). Divide Eq. (4.51) by a(k); (4.52) by a(k) anduse the definition of the Jost functions given in (4.44) and (4.45) gives the equivalent formulation

M(x, k)

a(k)= N(x, k) + ρ(k)e2ikxN(x, k) , (6.70)

M(x, k)

a(k)= N(x, k) + ρ(k)e−2ikxN(x, k) , (6.71)

where we have defined the left reflection coefficients

ρ(k) ≡ b(k)

a(k), ρ(k) ≡ b(k)

a(k). (6.72)

Taking into account the corresponding boundary conditions as well as the asymptotic behavior of a(k) at large k wefind

lim|k|→∞

[M(x, k)

a(k)−(

10

)]=

(00

). (6.73)

The functions M(x, k) and a(k) are both analytic functions in the upper half complex k plane. Since a(k) has simple

zeros at k = k` then M(x,k)a(k) is not analytic and has simple poles at k = k` (note that M(x, k) has no zeros in the

upper half plane). Let k = k0 be a simple zero of a(k). Then around k0 the Laurent series expansion for the functionM(x,k)a(k) is given by

M(x, k)

a(k)=

M(x, k0)

a′(k0)(k − k0)+ analytic function . (6.74)

Let k` be an eigenvalue of a(k) in the upper half complex k plane, i.e., a(k`) = 0. Then Eq. (4.51) gives φ(x, k`) =b(k`)ψ(x, k`) which, in terms of the Jost functions, reads

M(x, k`) = b(k`)N(x, k`)e2ik`x . (6.75)

We subtract from both sides of Eq. (6.70) the contributions from all poles and use (6.75) to find

M(x, k)

a(k)−(

10

)−

J∑`=1

C`N`(x)e2ik`x

k − k`= N(x, k)−

(10

)−

J∑`=1

C`N`(x)e2ik`x

k − k`+ ρ(k)e2ikxN(x, k) , (6.76)

where we have defined the left norming constant C` as

C` ≡b(k`)

a′(k`). (6.77)

9

The left hand side of Eq. (6.76) is an analytic function in the upper half plane and goes to zero as |k| → ∞ hence

it forms a “+” function. Also, the function N(x, k)−(

10

)is an analytic function in the lower half plane and goes

to zero as |k| → ∞ therefore, it forms an “-” function. Apply P− on (6.76) to find

N(x, k) =

(10

)+

J∑`=1

C`N(x, k`)e2ik`x

k − k`+

1

2πi

∫ ∞−∞

ρ(ξ)e2iξxN(x, ξ)

ξ − (k − i0)dξ . (6.78)

Similarly, the functions M(x, k) and a(k) are both analytic functions in k in the lower half plane. Since a(k) has

simple zeros at k = k` then M(x,k)a(k) is not analytic and has simple poles at k = k` (the function M(x, k) has no zeros

in the lower half plane). Let k = k0 be a simple zero for a(k). Then around k0 the Laurent series expansion for the

function M(x,k)a(k) is given by

M(x, k)

a(k)=

M(x, k0)

a′(k0)(k − k0)+ analytic function . (6.79)

At an eigenvalue k` we have a(k`) = 0. Then Eq. (4.52) gives φ(x, k`) = b(k`)ψ(x, k`) which imply

M(x, k`) = b(k`)N(x, k`)e−2ik`x . (6.80)

Subtract all the poles from (6.71) and use the relation (6.80) to find

M(x, k)

a(k)−(

01

)−

J∑`=1

C`N `(x)e−2ik`x

k − k`= N(x, k)−

(01

)−

J∑`=1

C`N `(x)e−2ik`x

k − k`+ ρ(k)e−2ikxN(x, k) , (6.81)

where as before the left norming constant is given by

C` ≡b(k`)

a′(k`). (6.82)

Note that the left hand side of Eq. (6.81) is an analytic function in the lower half plane and goes to zero as |k| → ∞

hence it forms a “-” function. Also, the function N(x, k)−(

01

)is an analytic function in the upper half plane and

goes to zero as |k| → ∞ therefore, it forms an “+” function. Apply P+ on (6.81) to find

N(x, k) =

(01

)+

J∑`=1

C`N(x, k`)e−2ik`x

k − k`− 1

2πi

∫ ∞−∞

ρ(ξ)e−2iξxN(x, ξ)

ξ − (k + i0)dξ . (6.83)

6.2 Time evolution of the scattering data: left scattering problem

The time evolution of the scattering data is derived from the evolution equation (2.6) and is given by (see ABLOWITZ)

a(k, t) = a(k, 0) , a(k, t) = a(k, 0) (6.84)

b(k, t) = e−4ik2tb(k, 0) , b(k, t) = e4ik

2tb(k, 0) . (6.85)

Equation (6.84) implies that the zeros of the scattering data kj and kj (the soliton eigenvalues) are time independent.The time-evolution of the reflection coefficients and norming constants follows from (6.84) and (6.85) and arerespectively given by

ρ(k, t) = e−4ik2tb(k, 0)/a(k, 0) , ρ(k, t) = e4ik

2tb(k, 0)/a(k, 0) , (6.86)

C` = C`(0)e−4ik2` t , C` = C`(0)e4ik

2` t , (6.87)

10

6.3 Right Scattering Problem

To account for the symmetry conditions (5.64) and (5.63), we view the system (4.51) and (4.52) as a left scatteringproblem and supplement it with the right scattering problem

ψ(x, k) = α(k)φ(x, k) + β(k)φ(x, k) , (6.88)

ψ(x, k) = α(k)φ(x, k) + β(k)φ(x, k) . (6.89)

where α(k), α(k), β(k) and β(k) are the right scattering data. We can also write system (6.88) and (6.89) in thematrix form

Ψ(x, k) = SR(k)Φ(x, k) , (6.90)

where the right scattering matrix is

SR(k) =

(α(k) β(k)β(k) α(k)

), (6.91)

and Φ(x, k) ≡ (φ(x, k), φ(x, k))T , Ψ(x, k) ≡ (ψ(x, k), ψ(x, k))T where superscript T denotes matrix transpose.Following the same steps as for the left RH above, we can formulate the corresponding RH problem on the rightand find the following linear integral equations which govern the functions M(x, k),M(x, k):

M(x, k) =

(01

)+

J∑`=1

B` M(x, k`)e−2ik`x

k − k`+

1

2πi

∫ ∞−∞

R(ξ)e−2iξxM(x, ξ)

ξ − (k − i0)dξ , (6.92)

M(x, k) =

(10

)+

J∑`=1

B` M(x, k`)e2ik`x

k − k`− 1

2πi

∫ ∞−∞

R(ξ)e2iξxM(x, ξ)

ξ − (k + i0)dξ , (6.93)

where R(k) and R(k) are the right reflection coefficients defined by

R(k) =β(k)

α(k), R(k) =

β(k)

α(k), (6.94)

and B`, B` are the right norming constants defined by

B` ≡β(k`)

α′(k`), B` ≡

β(k`)

α′(k`). (6.95)

6.4 Time evolution of the scattering data: right scattering problem

The derivation of the time evolution of the right scattering data follows closely that of the left case and are given by

α(k, t) = α(k, 0) , α(k, t) = α(k, 0) (6.96)

β(k, t) = e+4ik2tβ(k, 0) , β(k, t) = e−4ik2tβ(k, 0) . (6.97)

The time evolution of the norming constants and reflection coefficients follows from (6.96) and (6.97)

B` = B`(0)e+4ik2` t , B` = B`(0)e−4ik2` t . (6.98)

6.5 Relation between the reflection coefficients

The left scattering problem (4.51) and (4.52) can be rewritten in the matrix form

Φ(x, k) = SL(k)Ψ(x, k) , (6.99)

11

where Φ(x, k) ≡ (φ(x, k), φ(x, k))T , Ψ(x, k) ≡ (ψ(x, k), ψ(x, k))T (here superscript T stands for matrix transpose)and SL(k) is the so-called left scattering matrix

SL(k) =

(a(k) b(k)

b(k) a(k)

). (6.100)

With this notation at hand, the scattering matrix (6.100) is related to the right scattering matrix (6.91) throughoutthe relation SR(k) = S−1L (k) which explicitly gives

α(k) = a(k) , α(k) = a(k) ,

β(k) = −b(k) , β(k) = −b(k) . (6.101)

From the definition of the right reflection coefficient R given in (6.94) we have

R(k) =β(k)

α(k)= − b(k)

a(k)= −σ b

∗(−k∗)a∗(−k∗)

= −σρ∗(−k∗) . (6.102)

In obtaining the result (6.102) we used the definition of the left reflection coefficient ρ given in (6.72) as well asthe symmetry relation between the scattering data a, a and b given in (5.65), (5.66) and (5.67) respectively. Similarsymmetry result hold between R and ρ, i.e.,

R(k) = −σρ∗(−k∗) . (6.103)

With this at hand we have reduced the number of scattering data (reflection coefficients) in half. Next we aim atachieving the same goal for the norming constants.

6.6 Closing the System

To close the system we substitute k = kj in Eqns. (6.78), (6.92) and k = kj in Eqns. (6.83), (6.93) then impose thesymmetry condition r(x) = σq∗(−x) where σ = ∓1 throughout the symmetry relation between the Jost functions(5.63) and (5.64) as well as the reflection coefficients to find(

M∗2(−x,−k∗j )

−σM∗1(−x,−k∗j )

)=

(10

)+

J∑`=1

C`e2ik`x

kj − k`

(N1(x, k`)N2(x, k`)

)+

1

2πi

∫ ∞−∞

ρ(ξ)e2iξx

ξ − (kj − i0)

(N1(x, ξ)N2(x, ξ)

)dξ , (6.104)

(M1(x, kj)

M2(x, kj)

)=

(01

)+

J∑`=1

B`e−2ik`x

kj − k`

(N∗2 (−x,−k∗` )−σN∗1 (−x,−k∗` )

)+

1

2πi

∫ ∞−∞

ρ∗(−ξ)e−2iξx

ξ − (kj − i0)

(−σN∗2 (−x,−ξ)N∗1 (−x,−ξ)

)dξ ,(6.105)

(N1(x, kj)N2(x, kj)

)=

(01

)+

J∑`=1

C`e−2ik`x

kj − k`

(M∗2(−x,−k∗` )

−σM∗1(−x,−k∗` )

)− 1

2πi

∫ ∞−∞

ρ(ξ)e−2iξx

ξ − (kj + i0)

(M∗2(−x,−ξ)

−σM∗1(−x,−ξ)

)dξ ,(6.106)

(N∗2 (−x,−k∗j )−σN∗1 (−x,−k∗j )

)=

(10

)+

J∑`=1

B`e2ik`x

kj − k`

(M1(x, k`)

M2(x, k`)

)+

1

2πi

∫ ∞−∞

σρ∗(−ξ)e2iξx

ξ − (kj + i0)

(M1(x, ξ)M2(x, ξ)

)dξ , (6.107)

Equations (6.104) – (6.107) are integro-algebraic equations that (in principle) solve the inverse problem. As weshall see later, the above system will be used to construct breathing one soliton solution. To reduce the numberof norming constants we first establish a symmetry relation between the reflection coefficients, thus reducing themfrom four to two then use the symmetry relation between the eigenfunctions to establish the desired conditions onthe left and right norming constants.

12

6.7 Eigenvalues on the imaginary axis

When formulating the left and right RH problem for the Jost eigenfunctions we have subtracted only the contributionfrom all poles the are located at kj , kj the zeros of the scattering data a(k) and a(k) respectively. However it followsfrom the symmetry of the scattering data (5.65) and (5.66) that there are two more contribution to the pole arising

from −k∗j , k∗j . Thus, in order for the above analysis to be consistent we restrict ourselves to the case for which all

eigenvalues are located on the imaginary axis, i.e.,

kj = −k∗j , kj = −k∗j . (6.108)

To obtain the symmetry condition that relates the left norming constants Cj and Cj to the right norming constantsBj and Bj we restrict the equations (6.104), (6.105), (6.106) and (6.107) to eigenvalues satisfying (6.108) and obtainafter some algebra the following set of equations(

M1(x, kj)

M2(x, kj)

)=

(01

)+

J∑`=1

C∗` e−2ik`x

−kj + k`

(−σN∗2 (−x, k`)N∗1 (−x, k`)

)+

1

2πi

∫ ∞−∞

ρ∗(−ξ)e−2iξx

ξ − (kj − i0)

(−σN∗2 (−x,−ξ)N∗1 (−x,−ξ)

)dξ,(6.109)

(M1(x, kj)

M2(x, kj)

)=

(01

)+

J∑`=1

B`e−2ik`x

kj − k`

(N∗2 (−x, k`)−σN∗1 (−x, k`)

)+

1

2πi

∫ ∞−∞

ρ∗(−ξ)e−2iξx

ξ − (kj − i0)

(−σN∗2 (−x,−ξ)N∗1 (−x,−ξ)

)dξ ,(6.110)

(N1(x, kj)N2(x, kj)

)=

(01

)+

J∑`=1

C`e−2ik`x

kj − k`

(M∗2(−x, k`)

−σM∗1(−x, k`)

)− 1

2πi

∫ ∞−∞

ρ(ξ)e−2iξx

ξ − (kj + i0)

(M∗2(−x, ξ)

−σM∗1(−x, ξ)

)dξ ,(6.111)

(N1(x, kj)N2(x, kj)

)=

(01

)+

J∑`=1

B∗`e−2ik`x

−kj + k`

(−σM∗2(−x, k`)M∗1(−x, k`)

)− 1

2πi

∫ ∞−∞

ρ(ξ)e−2iξx

ξ − (kj + i0)

(M∗2(−x, ξ)

−σM∗1(−x, ξ)

)dξ , (6.112)

Comparing Eq. (6.109) with (6.110) one finds

B` = σC∗` , (6.113)

Similarly, one obtains from (6.111) and (6.112) the result

B` = σC∗` . (6.114)

It should be noted that both relations are valid under the symmetry reduction r(x) = σq∗(−x).

6.8 Additional symmetry between the eigenfunctions

At an eigenvalue k = k`, ` = 1, 2, · · · , N (a zero of a(k) in the upper half complex k plane), the eigen functionsφ(x, k`) and ψ(x, k`) are linearly dependent, i.e., φ(x, k`) = b(k`)ψ(x, k`) which, in terms of the Jost functions, itreads (see Eq. (6.75))

M1(x, k`) = b`e2ik`xN1(x, k`) , (6.115)

M2(x, k`) = b`e2ik`xN2(x, k`) . (6.116)

Equations (6.115) and (6.116) yields the product relation

M1(x, k`)N2(x, k`) = M2(x, k`)N1(x, k`) , (6.117)

between the Jost functions Nj and Mj , j = 1, 2 valid at an eigenvalue k`, ` = 1, 2 · · ·N. Now use the symmetrycondition (5.63) and (5.64) between the Jost functions in (6.117) to find

N2(x, k`)N∗2 (−x, k`) = N1(x, k`)N

∗1 (−x, k`) , ` = 1, 2, · · · , N . (6.118)

13

Similar relation can be derived for the other set of the Jost functions N j and M j , j = 1, 2 valid at an eigenvaluek`, ` = 1, 2 · · ·N. Indeed, starting from the right scattering problem (6.90) one finds that at an eigenvalue k` theJost functions N j and M j , j = 1, 2 are linearly dependent and satisfy

N1(x, k`) = β` e2ik`x M1(x, k`) , (6.119)

N2(x, k`) = β` e2ik`x M2(x, k`) . (6.120)

Eliminating β` e2ik`x from Eqns. (6.119), (6.120) and make use of the symmetry condition (5.63), (5.64) between

the Jost functions gives

M2(x, k`)M∗2(−x, k`) = M1(x, k`)M

∗1(−x, k`) , ` = 1, 2, · · · , N . (6.121)

The above relations (6.118) and (6.121) are useful in finding explicit soliton solutions.

The above relations (6.118) and (6.121) are useful in order to find the norming constants Cj and Cj and theirdependence on the soliton eigenvalues kj and kj .

7 Recovery of the Potentials r(x) and q(x)

Recall from Eq. (6.78 ) that

N(x, k) =

(10

)+

J∑`=1

C`N`(x)e2ik`x

k − k`+

1

2πi

∫ ∞−∞

ρ(ξ)e2iξxN(x, ξ)

ξ − (k − i0)dξ . (7.122)

The large k behavior of N2(x, k) is thus given by

N2(x, k) ∼|k|→∞

1

k

J∑`=1

C`N2(x, k`)e2ik`x − 1

k

1

2πi

∫ ∞−∞

ρ(ξ)e2iξxN2(x, ξ)dξ . (7.123)

From (4.49) we have

N2(x, k) ∼ −r(x)

2ik, (7.124)

therefore, we have the result:

r(x) = −2i

(J∑`=1

C`N2(x, k`)e2ik`x − 1

2πi

∫ ∞−∞

ρ(ξ)e2iξxN2(x, ξ)dξ

). (7.125)

We use the symmetries from (5.64) to find for r(x) = ∓q∗(−x)

M1(x, k) = ±N∗2(−x,−k∗) (7.126)

together with the asymptotic relation (4.48)

M1(x, k) ∼ q(x)

2ik, (7.127)

to find thatq(x) ∼ ±2ikN

∗2(−x,−k∗) . (7.128)

From (7.123) we have

N∗2(x, k) ∼

|k|→∞

1

k∗

J∑`=1

C∗`N∗2 (x, k`)e

−2ik∗` x +1

k∗1

2πi

∫ ∞−∞

ρ∗(ξ)e−2iξxN∗2 (x, ξ)dξ . (7.129)

14

N∗2(−x,−k∗) ∼

|k|→∞− 1

k

J∑`=1

C∗`N∗2 (−x, k`)e2ik

∗` x − 1

k

1

2πi

∫ ∞−∞

ρ∗(ξ)e2iξxN∗2 (−x, ξ)dξ . (7.130)

Therefore, comparing (7.130) with (7.128) we find (σ = ∓1)

q(x) = 2iσ

J∑`=1

C∗`N∗2 (−x, k`)e2ik

∗` x +

σ

π

∫ ∞−∞

ρ∗(ξ)e2iξxN∗2 (−x, ξ)dξ . (7.131)

8 Soliton Solutions

8.1 Norming constants

Pure soliton solutions correspond to zero reflection coefficients, i.e., ρ(ξ) = 0 and ρ(ξ) = 0 for all real ξ. In thiscase the system (6.104), (6.105), (6.106) and (6.107) reduces to an algebraic equations that would determine thefunctional form of the soliton(s). Unlike the case of the classical NLS equation for which the two norming constantsCj and Cj are related through a symmetry condition (similarly the soliton eigenvalues), here, as we shall see, thenorming constants are determined either using the symmetry condition (6.118) and (6.121) which is tedious to applyor via a trace formula [6, 11] that leads to a simple formula relating the norming constants to the soliton eigenvalues.We start the derivation from Eq. (5.63) whose first component satisfy (at an eigenvalue kj)

N1(x, kj) = −σM∗2 (−x,−k∗j ) . (8.132)

At an eigenvalue k = kj , the eigenfunctions M(x, kj and N(x, kj) are related via Eq. (6.75) which in our case give

M2(x, kj) = bje2ikjxN2(x, kj) , (8.133)

where bj ≡ b(kj). Equation (8.133) combined with (8.132) gives

N1(x, kj) = −σb∗je−2ikjxN∗2 (−x,−k∗j ) . (8.134)

On the other hand, the eigenfunctions N2(x, kj) and M1(x, kj) are related thru the symmetry condition (5.63), i.e.,

N2(x, kj) = M∗1 (−x,−k∗j ) , (8.135)

which together with (see Eq. (6.75))

M1(x, kj) = bje2ikjxN1(x, kj , (8.136)

one finds

N2(x, kj) = b∗je−2ikjxN∗1 (−x,−k∗j ) . (8.137)

By complex conjugating Eq. (8.137); making the transformation x → −x, kj → −k∗j and substituting the resultingexpression into (8.134) we find the important relation

−σ|bj |2 = 1 . (8.138)

This result imply that the phase of bj is arbitrary and bj = eiθj . Moreover we see that these types of soliton solutionscan only be obtained when σ = −1. Similar derivation holds for the functions M(x, kj) and N(x, kj). Indeed startingfrom the symmetry (5.64) and (6.80) one arrives after some calculations to a similar result as the one shown above

−σ|bj |2 = 1 , (8.139)

where bj ≡ b(kj). The phase of the scattering data bj is also free and we thus write bj = eiθj . To find the normingconstants

Cj =bja′j

, Cj =bja′j

, (8.140)

15

we need to compute a′j and a′j . This is accomplished with the help of the trace formula [6, 11] which, for a general

N eigenvalues kj , kj , j = 1, 2, · · ·N takes the form

a(k) =

N∏j=1

k − kjk − kj

. (8.141)

By taking the derivative of a(k) with respect to k we find

a′(k) =

N∏j=1

k − kjk − kj

[N∑`=1

(1

k − k`− 1

k − k`

)]. (8.142)

Evaluating Eq. (8.142) at the poles k = kn we find

a′(kn) = limk→kn

∏Nj=1(k − kj)∏Nj=1(k − kj)

N∑`=1

(k` − k`)(k − k`)(k − k`)

. (8.143)

For the special case for which all the eigenvalues reside on the imaginary axis we let kj = iηj , kj = −iηj forj = 1, 2, · · ·N with all ηj , ηj being real positive numbers. In this case, Eq. (8.143) simplifies to

a′(kn) = limη→ηn

∏Nj=1(η − ηj)∏Nj=1(η + ηj)

N∑`=1

(η` + η`)

i(η − η`)(η + η`), (8.144)

where we defined k ≡ iη with η > 0. Similar results can be obtained for the scattering data a(k). Indeed, startingfrom

a(k) =

N∏j=1

k − kjk − kj

, (8.145)

one can derive the relation

a′(k) =

N∏j=1

k − kjk − kj

[N∑`=1

(1

k − k`− 1

k − k`

)]. (8.146)

As before, evaluating Eq. (8.146) at the poles k = kn we find

a′(kn) = limk→kn

∏Nj=1(k − kj)∏Nj=1(k − kj)

N∑`=1

k` − k`(k − k`)(k − k`)

. (8.147)

Again, when all the eigenvalues reside on the imaginary axis, Eq. (8.147) simplifies to

a′(kn) = limη→ηn

∏Nj=1(η − ηj)∏Nj=1(η + ηj)

N∑`=1

i(η` + η`)

(η − η`)(η + η`). (8.148)

where now k ≡ −iη with η > 0 and kj = iηj , kj = −iηj for j = 1, 2, · · ·N with all ηj , ηj being positive real numbers.Below we list the norming constants associated with the one and two soliton solutions:

• One soliton solution. Here we take N = 1 in Eqns. (8.144), (8.148) and find

a′(k1) =1

i(η1 + η1), a′(k1) =

i

η1 + η1. (8.149)

The norming constants are readily obtained from (8.140) and are given by

|C1| = η1 + η1 , |C1| = η1 + η1 . (8.150)

• Two soliton solution. Substituting N = 2 in Eqns. (8.144), (8.148) we find

|C1| =(η1 + η1)(η1 + η2)

|η2 − η1|, |C2| =

(η2 + η2)(η2 + η1)

|η2 − η1|, (8.151)

|C1| =(η1 + η1)(η2 + η1)

|η2 − η1|, |C2| =

(η2 + η2)(η1 + η2)

|η2 − η1|. (8.152)

16

8.2 1-Soliton Solution

In this section we give an explicit form for the one soliton solution by setting J = J = 1. In this case, system (6.105)and (6.107) give (σ = −1)

N∗2 (−x, k1) = 1− C∗1e

2ik1x

k1 − k1M1(x, k1) , (8.153)

N∗1 (−x, k1) = −C∗1e

2ik1x

k1 − k1M2(x, k1) , (8.154)

M∗2(−x, k1) = 1 +

C1e2ik1x

k1 − k1N1(x, k1) , (8.155)

M∗1(−x, k1) =

C1e2ik1x

k1 − k1N2(x, k1) . (8.156)

Now let k1 = iη1 and k1 = −iη1 with η1, η1 both being real and positive. Substituting these quantities into system(8.153), (8.154), (8.155), and (8.156) gives after lengthy algebra

N1(x, iη1) =C1e

−2η1x

i(η1 + η1)

/(1− C1C1e

−2(η1+η1)x

(η1 + η1)2

), (8.157)

N2(x, iη1) =1

1− C1C1e−2(η1+η1)x

(η1+η1)2

, (8.158)

M2(x,−iη1) = 1 +C∗1e

2η1x

i(η1 + η1)N∗1 (−x, iη1) , (8.159)

M1(x,−iη1) =C∗1e

2η1x

i(η1 + η1)N∗2 (−x, iη1) . (8.160)

The explicit form of the 1-soliton solution follows from (7.131) by setting the reflection coefficient ρ to zero,

q(x) = −2iC∗1N∗2 (−x, k1)e2ik

∗1x . (8.161)

The norming constants C1 and C1 are given in Eq. (8.150), i.e., |C1| = (η1 + η1) and |C1| = (η1 + η1) with arbitraryphases. Note that we would have arrived at the same results had we used the symmetry conditions (6.118) and(6.121) between the Jost functions. After some algebra we obtain

q(x) = − 2iC∗1 (η1 + η1)2e2η1x

(η1 + η1)2 − C∗1C∗1e2(η1+η1)x. (8.162)

The time evolution of the norming constants C1 and C1 is found from Eq. (6.87) and is given by

C1 = C1(0)e+4iη21t = |c|ei(ϕ+π/2)e+4iη21t , |c| = η1 + η1 , (8.163)

C1 = C1(0)e−4iη21t = |c|ei(ϕ+π/2)e−4iη

21t , |c| = η1 + η1 . (8.164)

Substituting (8.163) and (8.164) back in (8.162) we find after some algebra the most general two parameter familyof a breathing one soliton solution

q(x, t) = − 2(η1 + η1)eiϕe−4iη21te−2η1x

1 + ei(ϕ+ϕ)e4i(η21−η21)te−2(η1+η1)x

. (8.165)

17

To obtain the one soliton solution for the corresponding local NLS equation we let η1 = η1 and C1 = −C∗1 . Thisimplies ϕ+ϕ = 0. Next we show that the one soliton solution (8.165) is indeed Galilean invariant. To establish thisresult, we use the transformation (2.19) and obtain

q(x, t) = −2(η1 + η1)eiϕe−4iη21te−2η1(x+2iξt)e−ξxe−iξ

2t

1 + ei(ϕ+ϕ)e4i(η21−η21)te−2(η1+η1)(x+2iξt)

. (8.166)

By defining the new soliton eigenvalues η1 and η1 via the transformation

η1 = η1 + ξ/2 , (8.167)

η1 = η1 − ξ/2 , (8.168)

the one soliton solution then reads

q(x, t) = − 2(η1 + η1)eiϕe−4iη21te−2η1x

1 + ei(ϕ+ϕ)e4i(η21−η

21)te−2(η1+η1)x

. (8.169)

In order for the soliton q(x, t) the remain within the class of rapidly decaying functions one requires

ξ > −2η1 , (8.170)

where −2η1 is exactly the asymptotic decay rate of the soliton as x → +∞. One interesting feature of the onesoliton solution (8.165) is the formation of singularity at a finite time. Indeed, at the origin (x = 0) the solution(8.165) becomes singular when

tn =(2n+ 1)π − (ϕ+ ϕ)

4(η21 − η21), n ∈ Z . (8.171)

8.3 2-Soliton Solution

In this section we construct a 2-soliton solution to the PT invariant nonlinear Schrodinger equation (1.1) withσ = −1. Such solution corresponds to soliton eigenvalues

k1 = iη1 , k2 = iη2 , ηj > 0 j = 1, 2 ,

k1 = −iη1 , k2 = −iη2 , ηj > 0; j = 1, 2 .(8.172)

Substituting J = 2 in Eq. (7.131) and set the reflection coefficient ρ to zero we find

q(x) = 2iσC∗1N∗2 (−x, k1)e2η1x + C∗2N

∗2 (−x, k2)e2η2x , (8.173)

where Cj , Cj , j = 1, 2 are the norming constants (in x) whose time evolution is determined by

C1(t) = C1(0)e4iη21t , C2(t) = C2(0)e4iη

22t , (8.174)

C1(t) = C1(0)e−4iη21t , C2(t) = C2(0)e−4iη

22t , (8.175)

and |Cj |, |Cj | are given in (8.151). To determine the functional form of the eigenfunctions N∗2 (−x, k1) and N∗2 (−x, k2)we solve the system of equations given in (6.104) and (6.107) with J = J = 2, and `, j = 1, 2. After some lengthycalculations we arrive at the following set of algebraic equations satisfied by the soliton eigenfunctions

N∗2 (−x, k1) = 1− γ1e2η1x M1(x, k1)− γ2e2η2x M1(x, k2) , (8.176)

N∗2 (−x, k2) = 1− δ1e2η1x M1(x, k1)− δ2e2η2x M1(x, k2) , (8.177)

N∗1 (−x, k1) = −γ1e2η1x M2(x, k1)− γ2e2η2x M2(x, k2) , (8.178)

N∗1 (−x, k2) = −δ1e2η1x M2(x, k1)− δ2e2η2x M2(x, k2) , (8.179)

18

M∗2(−x, k1) = 1 + α1e

−2η1x N1(x, k1) + α2e−2η2x N1(x, k2) , (8.180)

M∗2(−x, k2) = 1 + β1e

−2η1x N1(x, k1) + β2e−2η2x N1(x, k2) , (8.181)

M∗1(−x, k1) = α1e

−2η1x N2(x, k1) + α2e−2η2x N2(x, k2) , (8.182)

M∗1(−x, k2) = β1e

−2η1x N2(x, k1) + β2e−2η2x N2(x, k2) , (8.183)

where the quantities αj , βj , γj and δj , j = 1, 2 are time dependent and defined as

α1 =iC1(0)e4iη

21t

η1 + η1, α2 =

iC2(0)e4iη22t

η1 + η2(8.184)

β1 =iC1(0)e4iη

21t

η2 + η1, β2 =

iC2(0)e4iη22t

η2 + η2(8.185)

γ1 =C∗1(0)e4iη

21t

i(η1 + η1), γ2 =

C∗2(0)e4iη

22t

i(η1 + η2)(8.186)

δ1 =C∗1(0)e4iη

21t

i(η2 + η1), δ2 =

C∗2(0)e4iη

22t

i(η2 + η2)(8.187)

Solving system (8.176) - (8.183) we get

N∗2 (−x, k1) =α∗2 (δ1 − γ1) e2η1x + β∗2 (δ2 − γ2) e2η2x + e−2η2x

G1(x)− G2(x), (8.188)

where we defined the functions:

G1(x) =[γ1α

∗1e

2(η1+η1)x + γ2β∗1e

2(η1+η2)x + 1] [δ1α∗2e

2η1x + δ2β∗2e

2η2x + e−2η2x], (8.189)

and

G2(x) =[δ1α∗1e

2(η1+η1)x + δ2β∗1e

2(η1+η2)x] [γ1α

∗2e

2η1x + γ2β∗2e

2η2x]. (8.190)

The solution for N∗2 (−x, k2) is given by

N∗2 (−x, k2) =α∗1 (γ1 − δ1) e2η1x + β∗1 (γ2 − δ2) e2η2x + e−2η1x

H1(x)− H2(x), (8.191)

where we defined the functions

H1(x) =[γ1α

∗1e

2η1x + γ2β∗1e

2η2x + e−2η1x] [δ1α∗2e

2(η2+η1)x + δ2β∗2e

2(η2+η2)x + 1], (8.192)

and

H2(x) =[δ1α∗1e

2η1x + δ2β∗1e

2η2x] [γ1α

∗2e

2(η2+η1)x + γ2β∗2e

2(η2+η2)x]. (8.193)

Substituting eq.(8.188-8.191) into eq.(8.173) yields the 2-soliton solution.

19

9 Comparison with the classical NLS equation

In this part we briefly contrast the properties of the nonlocal NLS (1.1) with that of the classical (local) NLSequation:

iqt(x, t) = qxx(x, t)± 2|q(x, t)|2q(x, t) , (9.194)

obtained from Eq. (1.1) by replacing the term q∗(−x, t) with q∗(x, t). Three different scenarios will be addressed allof which concerning Eq. (9.194):

1. general initial conditions posed on the whole real line,

2. even initial conditions posed on the whole real line and,

3. general initial conditions on the semi-infinite interval (x ≥ 0.)

In [28], it was shown that (9.194) is integrable on the whole real line. Furthermore, it was found that the symmetriesof the eigenfunctions of the associated Zakharov-Shabat scattering problem are such that the eigenfunctions in theupper half complex plane are related to those in the lower half plane. This is in sharp contrast to the nonlocal casewhere the eigenfunctions at the upper/lower half plane are not related. On the other hand, if one restricts the class ofinitial conditions to be even (in x) then one obtains extra symmetry conditions on the scattering data that resemblesthe one we find. This leads us to the important conclusion that soliton solutions to (1.1) will have a classical NLSlimit so long (9.194) admits an even soliton solution. Finally, we point out that similar symmetry results wereobtained in [7] for the classical NLS (9.194) on the semi-infinite interval. Finally, in Table 1 we summarize andhighlight the key differences between the classical and nonlocal NLS equations.

10 Some special potentials: Box initial conditions

In the previous sections we obtained pure soliton solutions for the nonlocal NLS equation corresponding to reflec-tionless potentials. However, more general choices of initial data q(x, 0) give rise to non zero reflection coefficientsin which case it is not (generally speaking) solvable by inverse scattering transform. As an example, we consider inthis section few special cases of a box initial conditions and compute the eigenvalues and conserved quantities.

10.1 Single box function

The first example we consider the box initial condition

q(x, 0) =

0 for −∞ < x < 0h for 0 < x < L0 for x > 0

(10.195)

where h, L are both real and positive. The initial condition for r(x, 0) is obtained from (10.195) by imposing thesymmetry condition r(x, 0) = −q∗(−x, 0),i.e.,

r(x, 0) =

0 for −∞ < x < −L−h for −L < x < 0

0 for x > 0(10.196)

We use the scattering problem

v1x = −ikv1 + q(x, t)v2 , (10.197)

v2x = ikv2 + r(x, t)v1 , (10.198)

20

Symmetry property Classical NLS equation Nonlocal NLS equationiqt(x, t) = qxx(x, t)± 2|q(x, t)|2q(x, t) iqt(x, t) = qxx(x, t)± 2q2(x, t)q∗(−x, t)

Scattering data:a(k) = a∗(k∗)

b(k) = σb∗(k∗)

a(k) = a∗(−k∗)a(k) = a∗(−k∗)b(k) = σb∗(k∗)

a and a are not related

Eigenvalues: kj = k∗j

If kj , kj are eigenvalues then

−k∗j ,−k∗j are also eigenvalues.

no relations between kj and kj .

Eigenfunctions

N(x, k) =

(N (2)(x, k∗)σN (1)(x, k∗)

)∗

M(x, k) =

(σM (2)(x, k∗)M (1)(x, k∗)

)∗N(x, k) = ΛM∗(−x,−k∗)

N(x, k) = Λ−1M∗(−x,−k∗)

Λ =

(0 −σ1 0

)

Norming constantsCj = σC∗j free parameters;

independent of kj and kj

Cj , Cj depend on kj and kjvia Eqns. (8.140), (8.144), (8.148)

Galilean invariance q(x, t) = q(x− 2ζt, t)e−iζxeiζ2t q(x, t) = q(x+ 2iξt, t)e−ξxe−iξ

2t

Table 1: Symmetry relations between various quantities related to the classical and nonlocal integrable NLS equa-tions. Here σ = ∓1.

with the scattering functions satisfying the boundary conditions (3.22) and (3.23). Since this is a linear, constantcoefficient, second order differential equation, solution to system (10.197) and (10.198) is thus given by(

v1(x, k)

v2(x, k)

)=

(h2ik c1e

ikx + c2e−ikx

c1eikx

)for 0 < x < L . (10.199)

(v1(x, k)

v2(x, k)

)=

(c1e−ikx

c2eikx + h

2ik c1e−ikx

)for − L < x < 0 . (10.200)

Matching the eigenfunctions at x = 0 and x = −L using the boundary conditions (3.22) and (3.23) gives

c1 = 1 , c2 = − h

2ike2ikL , (10.201)

c1 =h

2ik

(1− e2ikL

), c2 = 1 +

(h

2ik

)2 (e2ikL − 1

). (10.202)

On the other hand for x > L we have from (3.23) and (4.51)

φ(x, t) =

(a(k)e−ikx

b(k)eikx

)for x > L . (10.203)

21

Matching the eigenfunctions at x = L gives the formula for the scattering data

a(k) = 1 +

(h

2ik

)2 (e2ikL − 1

)−(h

2ik

)2 (e4ikL − e2ikL

)(10.204)

b(k) = −heikL sin(kL)

k. (10.205)

The asymptotic behavior of the scattering coefficient a(k) for large and small k are readily obtained from (10.204)as

a(k) ∼ 1− h2

(2ik)2as k →∞ (10.206)

a(k) ∼ 1− h2L2 +O(k) as k → 0 . (10.207)

The zeros of the scattering data a(k) are implicitly given by solutions to

eiz = 1± iz

hL, (10.208)

where z = 2kL. Note that k = 0 is an improper eigenvalue for ant h, L. If Imz > 0 then we choose the negativesign whereas positive sign for Imz > 0. The conserved quantities can be derived from (3.29). Using the large kasypmtotic of the scattering data a(k) leads to C2n = 0

C1 = −h2 , C3 = −h4/2 (10.209)

10.2 Separated boxes

As a second example we consider two separated boxes initial condition

q(x, 0) =

0 for −∞ < x < L1

H for L1 < x < L2

0 for x > L2

(10.210)

where H,Lj , j = 1, 2 are real and positive. The initial condition for r(x, 0) is obtained from (10.195) by imposingthe symmetry condition r(x, 0) = −q∗(−x, 0),i.e.,

r(x, 0) =

0 for −∞ < x < −L2

−H for −L2 < x < −L1

0 for x > −L1

(10.211)

Solution to the scattering problem (10.197) and (10.198) satisfying the boundary conditions (3.22) and (3.23) is thusgiven by (

v1(x, k)

v2(x, k)

)=

(c1e−ikx

c2eikx + h

2ik c1e−ikx

)for − L2 < x < −L1 . (10.212)

(v1(x, k)

v2(x, k)

)=

(˜c1e−ikx

˜c2eikx

)for − L1 < x < L1 . (10.213)

(v1(x, k)

v2(x, k)

)=

(h2ik c1e

ikx + c2e−ikx

c1eikx

)for L1 < x < L2 . (10.214)

Matching the eigenfunctions at x = −L2,−L1 and x = L1 using the boundary conditions (3.22) and (3.23) gives

c1 = 1 , c2 = − h

2ike2ikL2 , (10.215)

22

c1 =h

2ik

(1− e2ikL2

), c2 = 1 +

(h

2ik

)2 (e2ikL2 − 1

)e2ikL1 . (10.216)

˜c1 = 1 , ˜c2 =h

2ik

(1− e2ikL2

). (10.217)

Now for x > L2 the eigenfunctions are given by (10.203) Matching the eigenfunctions at x = L gives the formula forthe scattering data

a(k) = 1 +

(h

2ik

)2 (e2ikL − 1

)−(h

2ik

)2 (e4ikL − e2ikL

)(10.218)

b(k) = −heikL sin(kL)

k. (10.219)

The asymptotic behavior of the scattering coefficient a(k) for large and small k are readily obtained from (10.204)as

a(k) ∼ 1− h2

(2ik)2as k →∞ (10.220)

a(k) ∼ 1− h2L2 +O(k) as k → 0 . (10.221)

The zeros of the scattering data a(k) are implicitly given by solutions to

eiz = 1± iz

hL, (10.222)

where z = 2kL. Note that k = 0 is an improper eigenvalue for ant h, L. If Imz > 0 then we choose the negativesign whereas positive sign for Imz > 0. The conserved quantities can be derived from (3.29). Using the large kasypmtotic of the scattering data a(k) leads to C2n = 0

C1 = −h2 , C3 = −h4/2 (10.223)

11 Conclusion

A detailed study of the inverse scattering transform for the recently introduced integrable nonlocal nonlinearSchrodinger (NLS) equation is carried out. The direct and inverse scattering problems are analyzed and key symme-tries of the eigenfunctions and scattering data and conserved quantities obtained. The inverse scattering theory isformulated using a new left-right Riemann-Hilbert problem and the Cauchy problem for the nonlocal NLS equationis formulated and methods to find pure soliton solutions presented. An explicit time-periodic one and two solitonsolutions are given. A detailed comparison with the classical nonlinear Schrodinger (NLS) equation is presented asis scattering data for special box type initial conditions.

12 Acknowledgements

The authors appreciate very useful comments of G. Biondini. The research of M.J.A. was partially supported byNSF 253 under Grant No. DMS-1310200 and the U.S. Air Force Office 254 of Scientific Research, under Grant No.FA9550-12-1-0207. The research of Z.H.M. was partially supported by NSF Grant No. DMS-0908599.

References

[1] Ablowitz M J and Musslimani Z H, Phys. Rev. Lett. 110 064105 (2013)

[2] Ablowitz M J and Musslimani Z H, Phys. Rev. E 90 032912 (2014)

23

[3] Ablowitz M J and Clarkson P A, Solitons, Nonlinear Evolution Equations and Inverse Scattering, CambridgeUniversity Press, Cambridge, (1991).

[4] Ablowitz M J, Kaup D J, Newell A C, and Segur H, Stud. Appl. Math. 53, 249 (1974).

[5] Ablowitz M J and Segur H, Solitons and Inverse Scattering Transform, SIAM Studies in Applied MathematicsVol. 4, SIAM, Philadelphia, PA, (1981)

[6] Ablowitz M J, Prinari B and Trubatch A D, Discrete and Continuous Nonlinear Schrodinger Systems, Cam-bridge University Press, Cambridge, (2004).

[7] Ablowitz M J and Segur H, J. Math. Phys. 16, 1054 (1975).

[8] Ablowitz M J and Baldwin D E, Phys. Rev. E, . 86 036305 (2012).

[9] Agrawal G P, Nonlinear Fiber Optics, Academic Press (1995).

[10] Bender C M and Boettcher S, Phys. Rev. Lett. 80 5243 (1998).

[11] Biondini G and Wang H, ”Solitons, BVPs and a nonlinear method of images”, J. Phys. A 42, 205207: 1-18(2009).

[12] Calogero F and Degasperis A, Spectral Transform and Solitons I, North Holand (1982).

[13] Clarkson P A J. of Comp. and Applied Math. 153,127, (2003).

[14] Clarkson P A and Mansfield E L, Nonlinearity 16, R1-R26 (2003).

[15] El-Ganainy E, Makris K G, Christodoulides D N, and Musslimani Z H, Opt. Lett., bf 32, 2632 (2007).

[16] Guo A et. al., Phys. Rev. Lett. 103, 093902 (2009).

[17] Kadomtsev B B and Petviashvili V I, Sov. Phys. Dokl. 15, 539 (1970).

[18] Kivshar Y and Agrawal G P, Optical Solitons: From Fibers to Photonic Crystals, Academic Press (2003).

[19] Korteweg D J and de Vries G, Philos. Mag. Ser. 5, 39, 422, (1895).

[20] Lazarides N and Tsironis G P, Phys. Rev. Lett. 110, 053901 (2013).

[21] Lumer Y, Plotnik Y, Rechtsman M C, and Segev M Phys. Rev. Lett. 111, 263901 (2013).

[22] Makris K G, El Ganainy R, Christodoulides D N, and Musslimani Z H, Phys. Rev. Lett. 100, 103904 (2008).

[23] Musslimani Z H, Makris K G, El Ganainy R, and Christodoulides D N, Phys. Rev. Lett. 100, 030402 (2008).

[24] Novikov S P, Manakov S V, Pitaevskii L P and Zakharov V E, Theory of Solitons. The inverse ScatteringMethod, Plenum (1984).

[25] Regensburger A et. al. Nature, 488 167 (2012).

[26] Regensburger A, Miri M A, Bersch C, Nager J, Onishchukov G, Christodoulides D N, and Peschel U Phys. Rev.Lett. 110, 223902 (2013).

[27] Ruter C E et. al., Nature Phys. 6, 192 (2010).

[28] Zakharov V E and Shabat A B, Sov. Phys. JETP 34, 62 (1972).

24