INVERSE SCATTERING TRANSFORM FOR THE NONLOCAL NONLINEAR SCHRÖDINGER EQUATION WITH NONZERO BOUNDARY CONDITIONS

Transcription

1 INVERSE SCATTERING TRANSFORM FOR THE NONLOCAL NONLINEAR SCHRÖDINGER EQUATION WITH NONZERO BOUNDARY CONDITIONS MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI Abstract. In 213 a new nonlocal symmetry reduction of the well-known AKNS scattering problem was found. It was shown to give rise to a new nonlocal P T symmetric and integrable Hamiltonian nonlinear Schrödinger NLS equation. Subsequently, the inverse scattering transform was constructed for the case of rapidly decaying initial data and a family of spatially localied, time periodic one soliton solutions was found. In this paper, the inverse scattering transform for the nonlocal NLS equation with nonero boundary conditions at infinity is presented in four different cases when the data at infinity have constant amplitudes. The direct and inverse scattering problems are analyed. Specifically, the direct problem is formulated, the analytic properties of the eigenfunctions and scattering data and their symmetries are obtained. The inverse scattering problem, which arises from a novel nonlocal system, is developed via a left-right Riemann-Hilbert problem in terms of a suitable uniformiation variable and the time dependence of the scattering data is obtained. This leads to a method to linearie/solve the Cauchy problem. Pure soliton solutions are discussed and explicit 1-soliton solution and two 2-soliton solutions are provided for three of the four different cases corresponding to two different signs of nonlinearity and two different values of the phase difference between plus and minus infinity. In the one other case there are no solitons. 1. Introduction Solitons are unique type of nonlinear wave that arise as a solution to integrable infinite dimensional Hamiltonian dynamical systems. They were first discovered by Zabusky and Kruskal while conducting numerical experiments on the Korteweg-deVries KdV equation. To their surprise, such solitons revealed an unusual particle-like behavior upon collisions despite the fact that they are inherently nonlinear objects. Their results sparked intense research interest on two parallel fronts. One is related to the physics and applications of solitons or solitary waves while the other is focused on the mathematical structure of integrable evolution equations. From the physics point of view, solitons, or solitary waves, represent finite energy spatially localied structures that generally form as a balance between dispersion and nonlinearity. They have been theoretically predicted and observed in laboratory experiments in various settings in the physical and optical sciences see [3], [31], [42] for extensive reviews. 21 Mathematics Subject Classification. Primary: 37K15; 35Q51; 35Q15. Key words and phrases. Inverse scattering transform, the nonlocal NLS equation, PT symmetry. 1

2 2 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI In fluid mechanics, they have been shown to appear as isolated humps in shallow water whereas in nonlinear optics they occur as a diffraction-free self guided nonlinear modes of a self-induced optical potential. Both disciplines provide exceptional situations where mathematical analysis, numerical simulations, mathematical modeling and laboratory experiments go hand in hand. Mathematically speaking, exactly solvable models play an essential role in the study of nonlinear wave propagation. There are many integrable equations that arise as important models in diverse physical phenomena. For example, the KortewegdeVries KdV and the Kadomtsev-Petviashvili KP equations describe weakly nonlinear shallow water waves [32, 7] propagating in one and two dimensions respectively. The cubic nonlinear Schrödinger NLS equation is also a physically important integrable model [43]. It describes the evolution of weakly nonlinear and quasi-monochromatic wave trains in media with cubic nonlinearities [8]. Solitons appear as a special class of solutions to these models which are integrable in the sense that they admit an infinite number of conserved quantities. The KdV, KP and NLS equations share the mathematical property that they all are exactly solvable evolution equations with many explicit solutions and lineariations known. There are many other continuous and also discrete integrable evolution equations that are physically relevant. Applications are diverse and include problems in fluid mechanics, electromagnetics, gravitational waves, elasticity, fundamental physics and lattice dynamics, to name but a few [5, 7, 9]. Recently, continuous and discrete integrable nonlinear nonlocal Schrödinger equations describing wave propagation in certain nonlinear P T symmetric media were also found; remarkably, they have a very simple structure [1, 2]. Furthermore, reverse space-time and reverse time only nonlocal NLS equations have also been reported [4]. Generally speaking integrability is established once an infinite number of constants of motion or an infinite number of conservation laws are obtained. However considerably more information about the solution can be obtained if the inverse scattering transform IST can be carried out [6]. The inverse scattering transform IST to solve the initial-value problem with rapidly decaying data for the nonlocal NLS equation 1.1 iq t x, t = q xx x, t 2σq 2 x, tq x, t, where q x, t denotes complex conjugate of qx, t, x R, t and σ = 1, has been developed in [1, 3]. The direct and inverse scattering problems arise from a novel and interesting nonlocal second order system of differential equations; see system 2.1 below. Associated with this system are symmetry relations which relate analytic eigenfunctions as x to those as x. In turn it is useful to employ Riemann-Hilbert RH problems from both the left and right in order to effectively develop the inverse scattering for both sets of eigenfunctions: i.e. those defined as x ±. We refer to this as left-right Riemann-Hilbert problems. This is different from the classical NLS equation where the inverse problem is carried out using a RH problem using corresponding symmetries at either infinity [9]. It is important to carry out the inverse scattering analysis not only to be able to solve the nonlinear equation but also because inverse scattering is important in its own right. The equation 1.1 was derived based upon on physical intuition. Recently this equation was derived in the physical context of magnetics [26].

3 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS 3 It is well-known that the IST procedure for rapidly decaying potentials must be substantially modified when one is interested in potentials that do not decay as x. This class of potentials is also relevant for the nonlocal NLS equation, since it admits soliton solutions with nonero boundary conditions NZBCs. For the classical NLS equation the first studies of NZBCs were done in [44]. The method to carry out the inverse problem employed two Riemann surfaces associated with square root branch points in the eigenfunctions/scattering data. An improvement was made with the introduction of a uniformiation variable [24]. This transforms the inverse problem to the more standard inverse problem in the upper lower/half planes in the new variable. Subsequently a number of researchers have also studied NLS problems in this manner cf. [38, 21, 22, 15]. In this paper, we consider Eq. 1.1 with the following nonero boundary conditions 1.2 qx, t q ± t = q e iθ±t, as x ±, where q > is a constant, θ ± < 2π and θ := θ + θ is either or π. If θ := θ + θ, π then the amplitude q ± t is exponentially growing/decaying at one or the other infinity. We do not consider this situation. We consider four different cases: two different signs of σ = 1 and two different values of θ =, π. First, we consider the case when σ = 1, θ = π. We find the the following 1-soliton stationary in space, oscillating in time solution 1.3 qx, t = q e i2q2 t+θ+ π tanh [q x iθ ] where θ = 1 2 θ + + θ 1 + π, θ 1 is a real constant related to the scattering data. This solution can be singular in the nongeneric case when x = and θ + +θ 1 = 2nπ, n Z. Apart from a complex phase shift, the above solution is similar to the well known black soliton solution in the standard integrable NLS equation. Second, we consider the case when σ = 1, θ =. In this case a single eigenvalue is found to be in the continuous spectrum; there is no proper exponentially decaying pure one soliton solution. The simplest decaying pure reflectionless potential generates a 2-soliton standing wave solution. This is a new property of the nonlocal NLS equation 1.1 which does not appear in the classical NLS equation. Third, we consider the case when σ = 1, θ = π. Here all solitons must arise from an even number of eigenvalues: 2N. The simplest situation occurs when N = 1 which leads to a 2- soliton traveling wave solution. Lastly, the case when σ = 1, θ = is considered; in this case we show that there are no eigenvalues/solitons. We use the uniformiation methodology mentioned earlier for the nonlocal NLS problem with the above NZBCs. We first introduce a two sheeted Riemann surface and then introduce a suitable uniformiation variable. There are a number of new features regarding the nonlocal NLS equation such as the introduction of important new symmetries which, when combined with a left-right RH problem allows us to construct the inverse scattering. There are situations when only an even number of solitons eigenvalues can be obtained and others in which there are no eigenvalues/solitons at all. In certain non-generic situations the solitons can be singular. We also study box like potentials and show that the eigenvalue spectrum is consistent with these results. The outline of this paper is as follows. In section 2 some preliminaries are developed. The equation and compatible linear pair are given and the different nonero boundary values at infinity that we will consider in this paper are presented.

4 4 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI It is also shown that the only cases in which the amplitude at infinity are not exponentially growing/decaying are when σ = 1, θ =, π. In section 3 the direct scattering theory is analyed and the analytic structure of the eigenfunctions and associated scattering data are found. From the symmetry of the potentials the corresponding symmetry of the eigenfunctions and scattering data are deduced. A suitable uniformiation variable is introduced and the inverse scattering from both the right and left is developed and pure reflectionless potentials and trace formulae are obtained. From the time dependence of the scattering data the IST is constructed; pure soliton solutions are discussed and an explicit one soliton solution is given. The methodology in the other three cases follows along similar lines. However the details, due to the underlying multivalued/branching structure of the scattering data, are quite different in each case; the analysis is developed in subsequent sections. 2. Preliminaries The nonlocal nonlinear Schrödinger NLS equation 1.1 is associated with the following 2 2 compatible system [1]: ik qx, t 2.1 v x = Xv = σq v, x, t ik 2.2 v t = T v = 2ik 2 + iσqx, tq x, t 2kqx, t iq x x, t 2kσq x, t iσqx x, t 2ik 2 iσqx, tq x, t v, where qx, t is a complex-valued function of the real variables x and t. Alternatively,the space part of the compatible system may be written in the form 2.3 v x = ikj + Qv, x R, where 2.4 J = 1 qx, t, Q = 1 σq x, t Here, qx, t is called the potential and k is a complex spectral parameter. general as x ±, q q ± t. Then equation 1.1 simplifies to 2.5 iq +,t = 2σq 2 +q as x +, 2.6 iq,t = 2σq 2 q + as x. From the above equations we find the conserved quantity 2.7 q + q = C, C is a constant. The solutions to equations are then given by. In 2.8 q + t = q,+ e iθ+ e 2iσCt as x +, 2.9 q t = q, e iθ e 2iσC t as x, where q,± > are constant. Furthermore, since 2.1 C = q,+ q, e i θ, where θ = θ + θ = const

5 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS 5 then if θ = or θ = π, C is real. Otherwise it is complex and the background either grows or decays exponentially as t. In this paper we shall only consider the cases θ = or π. For convenience we will take q,± = q. There is no material difference in the analysis if we take q,+ q,. We also note that as x ±, the eigenfunctions of the scattering problem asymptotically satisfy 2.11 v1 v 2 i.e., x = ik σq e ±2σq2 t sin θ e i2σq2 t cos θ+θ ik 2.12 v x = ikj + Q ± tv, 2.13 Q ± t = where q e 2σq2 t sin θ e i2σq2 t cos θ+θ± q e 2σq2 t sin θ e i2σq2 t cos θ+θ± σq e ±2σq2 t sin θ e i2σq2 t cos θ+θ 2.14 qx, t q ± t = q e 2σq2 t sin θ e i2σq2 t cos θ+θ±, as x ±. Here, q >, θ ± < 2π. We see that with the condition θ = or θ = π the exponentially growing terms disappear. The exponential growth is one of the additional novel aspects exhibited by the nonlocal NLS eq The case of σ = 1 with θ + θ = π 3.1. Direct Scattering. In this section we consider the nonero boundary conditions NZBCs given above in 2.14 and σ = 1, θ := θ + θ = π. With this condition, equation 2.13 conveniently reduces to 2 v j 3.1 x 2 = k2 + qe 2 i θ v j = k 2 qv 2 j, j = 1, 2. Each of the two equations has two linearly independent solutions e iλx and e iλx as x, where λ = k 2 q 2. The variable k is then thought of as belonging to a Riemann surface K consisting of two sheets C 1 and C 2 with the complex plane cut along, q ] [q, + with its edges glued in such a way that λk is continuous through the cut. We introduce the local polar coordinates 3.2 k q = r 1 e iθ1, θ 1 < 2π; k + q = r 2 e iθ2, π θ 2 < π, where r 1 = k q and r 2 = k +q. Then the function λk becomes single-valued on K, i.e., { λ 3.3 λk = 1 k = r 1 r e i θ 1 +θ 2 2, k C 1, λ 2 k = r 1 r e i θ 1 +θ 2 2, k C 2. Moreover, if k C 1, then Iλ ; and if k C 2, then Iλ. Hence, the variable λ is thought of as belonging to the complex plane consisting of the upper half plane U + :Iλ, and lower half plane U :Iλ, glued together along the real axis; the transition occurs at Iλ =. The transformation k λ maps C 1 onto U +, C 2 onto U, the cut, q ] [q, + onto the real axis, and the points ±q to ; see also the figure 1 below., v1 v 2,

6 6 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI Figure 1. a The two-sheeted Riemann surface K. b The genus surface is topologically equivalent to K Eigenfunctions. It is natural to introduce the eigenfunctions defined by the following boundary conditions 3.4 φx, k we iλx, φx, k we iλx, as x, 3.5 ψx, k ve iλx, ψx, k ve iλx, as x +. We substitute the above into 2.11, obtaining λ + k iq 3.6 w = iq+, w = λ + k, v = iq+ λ + k λ + k, v = iq which satisfy the boundary conditions, but they are not unique. In the following analysis, it is convenient to consider functions with constant boundary conditions. We define the bounded eigenfunctions as follows: 3.7 Mx, k = e iλx φx, k, Mx, k = e iλx φx, k, 3.8 Nx, k = e iλx ψx, k, Nx, k = e iλx ψx, k. The eigenfunctions can be represented by means of the following integral equations + λ + k 3.9 Mx, k = iq+ + G x x, kq Q Mx, kdx, 3.1 Mx, k = 3.11 Nx, k = iq λ + k iq+ λ + k G x x, kq Q Mx, kdx, G + x x, kq Q + Nx, kdx,

7 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS 7 λ + k 3.12 Nx, k = iq + + Using the Fourier transform method, we get G + x x, kq Q + Nx, kdx G x, k = θx 2λ [1 + e2iλx λi ie 2iλx 1ikJ + Q ], 3.14 G x, k = θx 2λ [1 + e 2iλx λi + ie 2iλx 1ikJ + Q ], 3.15 G + x, k = θ x 2λ [1 + e 2iλx λi + ie 2iλx 1ikJ + Q + ], 3.16 G + x, k = θ x 2λ [1 + e2iλx λi ie 2iλx 1ikJ + Q + ], where θx is the Heaviside function, i.e., θx = 1 if x > and θx = if x <. Definition 3.1. We say f L 1 R if + fx dx <, and f L1,2 R if + fx 1 + x 2 dx <. Then we have the following result see also [21]. Theorem 3.2. Suppose the entries of Q Q ± belong to L 1 R, then for each x R, the eigenfunctions Mx, k and Nx, k are continuous for k C 1 \ {±q } and analytic for k C 1, Mx, k and Nx, k are continuous for k C 2 \ {±q } and analytic for k C 2. In addition, if the entries of Q Q ± belong to L 1,2 R, then for each x R, the eigenfunctions Mx, k and Nx, k are continuous for k C 1 and analytic for k C 1, Mx, k and Nx, k are continuous for k C 2 and analytic for k C 2. The proof of Theorem 3.2 employs Neumann series, it is given in the Appendix. Definition 3.3. The Schwart space or space of rapidly decreasing functions on R n is the function space 3.17 SR n := {f C R n : f α,β <, α, β Z n +}, where α, β are multi-indices, C R n is the set of smooth functions from R n to C, and f α,β = sup x R n x α D β fx. In particular, x j e a x 2 SR n, where j is a multi-index and a is a positive real number. Then we have the following result. Theorem 3.4. Suppose the entries of Q Q ± do not grow faster than e ax2, where a is a positive real number, then for each x R, the eigenfunctions Mx, k, Nx, k, Mx, k and Nx, k are analytic in the Riemann surface K. The proof of Theorem 3.4 is given in the Appendix.

8 8 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI 3.3. Scattering data. The two eigenfunctions with fixed boundary conditions as x are linearly independent, as are the two eigenfunctions with fixed boundary conditions as x +. Indeed, if ux, k = u 1 x, k, u 2 x, k T and vx, k = v 1 x, k, v 2 x, k T are any two solutions of 2.3, we have d 3.18 W u, v =, dx where the Wronskian of u and v, W u, v is given by W u, v = u 1 v 2 u 2 v 1. From the asymptotics 3.4 and 3.5, it follows that 3.19 W φ, φ = lim W φx, k, φx, k = 2λλ + k, x 3.2 W ψ, ψ = lim W ψx, k, ψx, k = 2λλ + k, x + which proves that the functions φx, k and φx, k are linearly independent, as are ψ and ψ, with only the branch points ±q being excluded. Hence, we can write φx, k and φx, k as linear combinations of ψx, k and ψx, k, or vice versa. Thus, the relations 3.21 φx, k = bkψx, k + akψx, k, 3.22 φx, k = akψx, k + bkψx, k hold for any k such that all four eigenfunctions exist. Combining 3.19 and 3.2, we can deduce that the scattering data satisfy the following characteriation equation 3.23 akak bkbk = 1. The scattering data can be represented in terms of Wronskians of the eigenfunctions, i.e., W φx, k, ψx, k 3.24 ak = W ψx, k, ψx, k W φx, k, ψx, k 3.25 ak = W ψx, k, ψx, k W φx, k, ψx, k 3.26 bk = W ψx, k, ψx, k W φx, k, ψx, k 3.27 bk = W ψx, k, ψx, k W φx, k, ψx, k =, 2λλ + k φx, k, ψx, k = W, 2λλ + k φx, k, ψx, k = W, 2λλ + k W φx, k, ψx, k =. 2λλ + k Then from the analytic behavior of the eigenfunctions we have the following theorem see also [21]. Theorem 3.5. Suppose the entries of Q Q ± belong to L 1 R, then ak is continuous for k C 1 \ {±q } and analytic for k C 1, and ak is continuous for k C 2 \ {±q } and analytic for k C 2. Moreover, bk and bk are continuous in k, q q, +. In addition, if the entries of Q Q ± belong to L 1,2 R, then akλk is continuous for k C 1 and analytic for k C 1, and akλk is continuous for k C 2 and analytic for k C 2. Moreover, bkλk and bkλk are continuous for k R. If the entries of Q Q ± do not grow faster than e ax2,

9 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS 9 where a is a positive real number, then akλk, akλk, bkλk and bkλk are analytic for k K. Note that 3.21 and 3.22 can be written as 3.28 µx, k = ρke 2iλx Nx, k + Nx, k, µx, k = Nx, k + ρke 2iλx Nx, k, where µx, k = Mx, ka 1 k, µx, k = Mx, ka 1 k, ρk = bka 1 k and ρk = bka 1 k. Introducing the 2 2 matrices 3.29 m + x, k = µx, k, Nx, k, m x, k = Nx, k, µx, k, which are meromorphic in C 1 and C 2 respectively. Hence, we can write the Riemann- Hilbert problem or jump conditions in the k-plane as ρkρk ρke 2iλx 3.3 m + x, k m x, k = m x, k ρke 2iλx on the contour Σ : k, q ] [q, +. Remark 3.6. For the Riemann-Hilbert problem 3.31 F + ξ F ξ = F ξgξ on the contour Σ, where gξ is Hölder-continuously in Σ, we can consider the projection operators 3.32 P j fk = 1 λk + λξ fξ 2πi 2λξ ξ k dξ, k C j, j = 1, 2, where λk+λξ dξ 2λξ ξ k Σ is the Weierstrass kernel and denotes the integral along Σ the oriented contour in sheet I of Figure 1. One can show that [45] λk + λξ 2λξ dξ ξ k = dξ + regular terms ξ k for ξ, λξ k, λk. If f j j = 1, 2, is sectionally analytic in C j and rapidly decaying as k in the proper sheet, we have 3.33 P j f j k = 1 j 1 f j k, k C j, 3.34 P j f l k =, k C j. We obtain 3.35 F k = 1 2πi 3.36 F + k = 1 2πi Σ Σ λk + λξ 2λξ λk + λξ 2λξ F ξgξ dξ, k C 2, ξ k F ξgξ dξ, k C 1. ξ k For ξ Σ, we can take limits from proper sheet; they are connected by the analogue of Plemelj-Sokhotskii formulas 3.37 F + 1 λk + λξ ξ = lim F ξgξ dξ, k ξ Σ,k C 1 2πi 2λξ ξ k 3.38 F 1 ξ = lim k ξ Σ,k C 2 2πi Σ Σ λk + λξ 2λξ F ξgξ dξ. ξ k

10 1 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI 3.4. Symmetry reductions. The symmetry in the potential induces a symmetry between the eigenfunctions. Indeed, if vx, k = v 1 x, k, v 2 x, k T solves 2.3, then v2 x, k, v1 x, k T also solves 2.3. Moreover, if k k, according to 3.3, we have θ 1 π θ 2 and θ 2 π θ 1. Hence, λ j k = λ j k, where j = 1, 2. Taking into account boundary conditions 3.6, we can obtain 3.39 ψx, k = 1 1 φ x, k, ψx, k = 1 1 φ x, k. By 3.7 and 3.8, we can get the symmetry relations of the eigenfunctions, i.e., Nx, k = M x, k 1, Nx, k = M x, k. 1 1 From the Wronskian representations for the scattering data and the above symmetry relations, we have 3.41 a k = ak, a k = ak, b k = bk. When using a particular single sheet for the Riemann surface of the function λ 2 = k 2 q 2, the involution k, λ k, λ can only be considered across the cuts. The scattering data and eigenfunctions are defined by means of the corresponding values on the upper/lower edge of the cut; they are labeled with superscripts ± as clarified below. Explicitly, one has 3.42 a ± k = W φ± x, k, ψ ± x, k 2λ ± λ ±, k, q ] [q, +, + k 3.43 a ± k = W φ± x, k, ψ ± x, k 2λ ± λ ±, k, q ] [q, +, + k 3.44 b ± k = W φ± x, k, ψ ± x, k 2λ ± λ ±, k, q ] [q, +, + k 3.45 b ± k = W φ± x, k, ψ ± x, k 2λ ± λ ±, k, q ] [q, +. + k Using the notation λ = λ + = λ, we have the following symmetry: 3.46 φ x, k = λ + k φ ± x, k, ψ x, k = λ + k iq iq ψ ± x, k for k, q ] [q, +. Moreover, 3.47 a ± k = a k, b ± k = q2 q q + b k for k, q ] [q, +.

11 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS Uniformiation coordinates. Before discussing the properties of scattering data and solving the inverse problem, we introduce a uniformiation variable, defined by the conformal mapping: = k = k + λk, where λ = k 2 q 2 and + q2. Then λ = 1 2 q2. the inverse mapping is given by k = k = 1 2 We observe that 1 the upper sheet C 1 and lower sheet C 2 of the Riemann surface K are mapped onto the upper half plane C + and lower half plane C of the complex plane respectively; 2 the cut, q ] [q, + on the Riemann surface is mapped onto the real axis; 3 the segments [ q, q ] on C 1 and C 2 are mapped onto the upper and lower semicircles of radius q and centered at the origin of the complex plane respectively. From Theorem 3.2, we have the eigenfunctions Mx, and Nx, are analytic in the upper half plane: i.e C +, and Mx, and Nx, are analytic in the lower half plane: i.e. C. Moreover, by Theorem 3.5, we find that a is analytic in the upper half plane: C + and a is analytic in the lower half plane: C Symmetries via uniformiation coordinates. It is known that 1 when, then k, λ k, λ ; 2 when q2, then k, λ k, λ. Hence, ψx, = φ x, 1, ψx, = φ x,, φ x, q2 q 2 = φx,, iq Similarly, we can get Nx, = M x,, 1 Moreover, ψ x, q2 = iq + ψx,, I <. 1 Nx, = 1 M x, a = a, I > ; a = a, I < ; b = b, 3.52 a q 2 q 2 = a, I < ; b = q2 b. q q + We will assume that a and ā, have simple ero s in the upper/lower half planes respectively. We assume that there are no multiple ero s and no ero s on I = Asymptotic behavior of eigenfunctions and scattering data. In order to solve the inverse problem, one has to determine the asymptotic behavior of eigenfunctions and scattering data both as and as. From the integral equations in terms of Green s functions, we have

12 12 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI 3.53 Nx, iq x, Nx, qx q + iq,, 3.54 a = { 1,, 1,, a = { 1,, 1,, 3.55 lim b =, lim b 2 =. The asymptotic behavior of Mx,, Nx, and Mx, are given in the Appendix Riemann-Hilbert problem via uniformiation coordinates Left scattering problem. In order to take into account the behavior of the eigenfunctions, the jump conditions at the real axis can be written from the left end as Mx, a Mx, a Nx, Nx, = ρe i q2 x = ρe i q2 x Nx,, Nx,, so that the functions will be bounded at infinity, though having an additional pole at =. Note that Mx, /a, as a function of, is defined in the upper half plane C +, where it has by assumption simple poles j, i.e., a j =, and Mx, /a, is defined in the lower half plane C, where it has simple poles j, i.e., a j =. It follows that 3.58 Mx, j = b j e i j q2 j x Nx, j, 3.59 Mx, j = b j e i j q2 j x Nx, j. Then subtracting the values at infinity, the induced pole at the origin and the poles, assumed simple in the upper/lower half planes respectively, at a j =, j = 1, 2...J and ā j, j = 1, 2... J later we will see that J = J gives 3.6 Mx, a Nx, 1 1 = ρe i q2 x 1 iq 1 iq Nx,, J Mx, j j j a j J b j e i j q2 j x Nx, j j j a j

13 3.61 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS 13 Mx, a Nx, iq+ 1 iq+ = ρe i q2 x Nx,. We now introduce the projection operators 3.62 P ± f = 1 2πi + J Mx, j j j a j J b j e i j q2 j x Nx, j j j a j fξ ξ ± i dξ, which are well-defined for any function fξ that is integrable on the real axis. If f ± ξ is analytic in the upper/lower plane and f ± ξ is decaying at large ξ, then 3.63 P ± f ± = ±f ±, P f ± =. Applying P to 3.6 and P + to 3.61, we can obtain 3.64 Nx, = iq 3.65 Nx, = iq+ + + J J b j e i j q2 j x Nx, j j j a + + ρξ j 2πi ξξ ei ξ q2 x ξ Nx, ξdξ, b j e i j q2 j x Nx, j j j a j + 2πi Since the symmetries are between eigenfunctions defined at both ± we proceed to obtain the inverse scattering integral equations defined from the right end Right scattering problem. The right scattering problem can be written as 3.66 ψx, = αφx, + βφx,, 3.67 ψx, = αφx, + βφx,, where α, α, β and β are the right scattering data. Moreover, we can get the right scattering data and left scattering data satisfy the following relations 3.68 α = a, α = a, β = b, β = b. Thus, Nx, = amx, bmx, e i q2 x, Nx, = amx, bmx, e i q2 x. The above two equations can be written as Nx, a Nx, a Mx, Mx, = b a e i q2 x = b a ei q2 x Mx,, Mx,. ρξ ξξ e i ξ q2 x ξ Nx, ξdξ.

14 14 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI By the symmetry relations of scattering data, we have 3.73 Nx, a Mx, = ρ e i q2 x Mx,, 3.74 Nx, a Mx, = ρ e i q2 x Mx,. Using similar methods as solving left scattering problem, we obtain 3.75 Mx, = iq + J b j Mx, j e i j q2 j x j j a j + + 2πi ρ ξ ξξ e i ξ q2 x ξ Mx, ξdξ, 3.76 Mx, = iq+ + J b j Mx, j e i j q2 j x j j a j + ρ ξ 2πi ξξ ei ξ q2 x ξ Mx, ξdξ Recovery of the potentials. In order to reconstruct the potential we use asymptotics. For example from equation 3.53, we have N 1x, qx q + as. By 3.64, we can get 3.77 N 1 x, 1+ J b j e i j q2 j x j 2 N 1 x, j ρξ a j 2πi ξ 2 ei ξ q2 xn ξ 1 x, ξdξ as. Hence, 3.78 J qx = q b j e i j q2 j x 2 j a j N 1 x, j ρξ 2πi ξ 2 e i ξ q2 ξ x N 1 x, ξdξ. Note that the rapidly varying phase makes the integrals well defined at ξ =.

15 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS 15 q Closing the system. We can find J = J from a = a. To close the system, by the symmetry relations between the eigenfunctions, we have 3.79 N1 x, = N 2 x, j + J l=1 iq + J 2πi + ξ + J l=1 l=1 iq + J l=1 iq+ + J b j e i j q2 j x j j a j jb l e i l q2 x l j l l a l N 1 x, l + j + 2πi jb l e i l q2 l x j l l a l N 2 x, l + j 2πi ρξ ξξ e i ξ q2 x ξ ξb l e i l q2 x l ξ l l a l N 1 x, l + ξ + 2πi ξb l e i l q2 l x ξ l l a l + N 2 x, l + ξ + 2πi ρξ ξξ j ei ξ q2 x ξ N 1 x, ξdξ ρξ ξξ j ei ξ q2 x ξ N 2 x, ξdξ ρη ηη ξ ei η q2 x η N 1 x, ηdη ρη ηη ξ ei η q2 x η N 2 x, ηdη We note that from eq 3.78 qx is given in terms of the component N 1. We can use only the first component of eq 3.79 to find this function and hence qx. Hence to complete the inverse scattering we reduce the problem to solving an integral equation in terms of the component N 1 only Trace formula. Since a and a are analytic in the upper and lower plane respectively. As mentioned above, we assume that a has simple eros, which we call i and j, where R i = and R j, then j is also a simple ero of a by a = a. By the symmetry relation a = a, we can deduce that a has simple eros q2 j, q2 j 3.8 J 1 q2 + q2 γ = a j j j + j J 2 i=1 q2 i i, and q2 i. We define q 2 J 1 j γ = a q2 j + j + q2 j J 2 i=1 i, q2 i where 2J 1 + J 2 = J. Then γ and γ are analytic in the upper and lower plane respectively and have no eros in their respective half planes. We can get 3.81 log γ = 1 2πi + log γξ ξ dξ, 1 2πi + log γξ dξ =, I >, ξ dξ log γ = 1 + log γξ 2πi ξ dξ, 1 + 2πi log γξ dξ =, I <. ξ Adding or subtracting the above equations in each half plane respectively, yields 3.83 log γ = 1 + log γξγξ dξ, I > ; log γ = 1 + log γξγξ dξ, I <. 2πi ξ 2πi ξ

16 16 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI Hence, 3.84 J 1 log a = log j + j q2 j + q2 j J 2 i i=1 q2 i + 1 2πi + log γξγξ dξ, I >, ξ 3.85 J 1 log a = log q2 + q2 j j j + j J 2 i=1 q2 i i 1 2πi + log γξγξ dξ, I <. ξ Note that γγ = aa, from the unitarity condition aa bb = 1 and the symmetry b = b, we can obtain 3.86 J 1 log a = log j q2 j 3.87 J 1 log a = log + j + q2 j q2 + q2 j j j + j J 2 i i=1 q2 i J 2 i=1 q2 i i + 1 2πi 1 2πi + + log1 bξb ξ dξ, I >, ξ log1 bξb ξ dξ, I <. ξ Thus we can reconstruct ak, āk in terms of the eigenvalues ero s and only one function bk Discrete scattering data and their symmetries. In order to find reflectionless potentials/solitons we need to be able to calculate the relevant scattering data: b j, b j, a j, ā j, j = 1, 2,...J. The latter functions can be calculated via the trace formulae. So we concentrate on the former. Since 3.88 N 1 x, = M 2 x,, N 2 x, = M 1 x,, 3.89 M 1 x, j = b j e i j q2 j x N1 x, j, M 2 x, j = b j e i j q2 j x N2 x, j, we have 3.9 N 1 x, j = b j e i j q2 j x N 2 x, j, 3.91 N 2 x, j = b j e j q2 j x N 1 x, j. By rewriting 3.91, we obtain 3.92 N 2 x, j = b j e j q2 j x N1 x, j. Combining 3.9, we can deduce the following symmetry condition on the discrete data b j, 3.93 b j b j = 1. Similar analysis shows that b j satisfies an analogous equation b j b j = 1.

17 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS 17 Also, since a 1 as, from the trace formula on the real axis, we have the general symmetry constraint 3.94 J 1 where 2J 1 + J 2 = J. j 4 q 4 J 2 i=1 i 2 q 2 e 1 2πi log1+bξb ξ/ξdξ = Reflectioness potentials and soliton solutions. Reflectioness potentials and, soliton solutions when time dependence is added, correspond to ero reflection coefficients, i.e., ρξ = and ρξ = for all real ξ. We also note, from the symmetry relation b = b it follows that the reflection coefficients ρ = b/a, ρ = b/ā will both vanish when b = for on the real axis. By substituting = l in 3.79, the system 3.79 reduces to algebraic equation that determine the functional form of these special potentials. When time dependence is added the reflectionless potentials correspond to soliton solutions. The reduced equations take the form 3.95 N1 x, l N 2 x, l iq+ = l + J l b j e i j q2 j x l j j a j j + J l=1 iq + J l=1 The above equation is an algebraic system to solve for Nx, l, l = 1, 2...J. The potentials are reconstructed from equation 3.78 with ρξ =, ρξ = ; i.e., 3.96 qx = q J b j e i j q2 j x 2 j a j N 1 x, j. As before, since a 1 as, by the trace formula when bξ = in the real axis, we have the following symmetry constraint for the reflectionless potentials jb l e i l q2 l x j l l a l N 1 x, l jb l e i l q2 l x j l l a l N 2 x, l where 2J 1 + J 2 = J. J 1 j 4 q 4 J 2 i=1 i 2 q 2 = 1, Reflectionless potential solution: 1-eigenvalue. In this subsection, we show an explicit form for the 1 eigenvalue/1-soliton solution without time dependence, by setting J = 1. Then J 1 = and J 2 = 1. Now let 1 = iv 1, we have 1 = i q2 v 1 := iv 1, where v 1 is real and positive. Hence, we have 3.98 N 1 x, iv 1 = iq + + v1b iv1e v 1 + q2 v 1 x v 1+v 1a iv 1 1 b iv1biv1e v 1 +v 1 + q2 v1 + q2 v 1 x v 1+v 1 2 a iv 1a iv 1.

18 18 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI Therefore, 3.99 qx = q biv 1e v1 2 a iv 1 v 1+ q2 v1 x N 1 x, iv 1. By the trace formula when bξ = in the real axis, we can get 3.1 a iv 1 = 1 iv 1 + v 1, i a iv 1 =. v 1 + v 1 From 3.97, we can deduce v 1 = v 1 = q. Hence, 3.11 a iv 1 = a iq = 1 2iq, a iv 1 = a iq = i 2q. By choosing 1 = iq in 3.93, we can get biq 2 = 1. Similarly, we have b iq 2 = 1. Thus, we write biq = e iθ1 and b iq = e iθ1, where both θ 1 and θ 1 are real. Moreover, we can obtain biq = e iθ1 and b iq = e iθ1. Moreover, from 3.52, we can also get q e iθ1 = biq = b = b iq eiθ1 = iq eiθ++θ e, 2iθ+ i.e., θ 1 = θ 1 + 2θ + + 2nπ, where n Z. Thus we have the reflectionless potential corresponding to one eigenvalue: [ 3.13 qx = q e iθ eiθ1 2qx e iθ+ + e iθ1+2θ+ 2qx ]. 1 e 2iθ1+θ+ 4qx To find the corresponding soliton solution we need the time evolution of the data which we derive next Time evolution. We deduce that both at and at are time independent, and 3.14 biq, t = biq, e 2iq2 +2λkt = e iθ1 e 2iq2 t, 3.15 b iq, t = b iq, e 2iq2 +2λkt = e iθ1 e 2iq2 t = e iθ1+2θ+ e 2iq2 t. The details are shown in the Appendix. Putting all the above into the formula we had for the reflectionless potential 3.13 we obtain the following one soliton solution 3.16 qx, t = q e i2q2 t+θ+ π tanh [q x iθ ] where θ = 1 2 θ + + θ 1 + π. This is similar to the well known black soliton of the standard integrable NLS equation with only a complex phase shift difference. From this solution we see that there is a singularity only when θ + + θ 1 =, ±2π which puts a restriction on the otherwise arbitrary phases θ +, θ 1. The magnitude of the solution is stationary. In Fig. 2 below we give typical one soliton solutions. We see in Fig. 2 a the magnitude rises from a constant background, whereas in Fig. 2 b the magnitude dips from the constant background.

19 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS 19 a b Figure 2. a The amplitude of qx, t with θ + = π 4, θ 1 = and q = 2. b The amplitude of qx, t with θ + = 5π 4, θ 1 = and q = The case of σ = 1 with θ + θ = In this section we consider the nonero boundary conditions NZBCs given in 2.14 with σ = 1, θ + = θ := θ 4.1 qx, t q ± t := qt = q e 2iq2 t+iθ, as x ±, where q >, θ < 2π Direct scattering. With this condition, equation 2.13 conveniently reduces to 2 v j 4.2 x 2 = k2 + qv 2 j, j = 1, 2. Each of the two equations has two linearly independent solutions e iλx and e iλx as x, where λ = k 2 + q 2. We introduce the local polar coordinates 4.3 k iq = r 1 e iθ1, π 2 θ 1 < 3π 2 ; k + iq = r 2 e iθ2, π 2 θ 2 < 3π 2, where r 1 = k iq and r 2 = k + iq. One can write λk = r 1 r e i θ1 +θ 2 2 +imπ, and m =, 1 respectively on sheets I K 1 and II K 2. The variable k is then thought of as belonging to a Riemann surface K consisting of sheets I and II with both coinciding with the complex plane cut along Σ := [ iq, iq ] with its edges glued in such a way that λk is continuous through the cut. Along the real k axis, we have λk = ±signk k 2 + q 2, where the plus/minus signs apply, respectively, on sheet I and sheet II of the Riemann surface, and where the square root sign denotes the principal branch of the real-valued square root function. We denote C ± the open upper/lower complex half planes, and K ± the open upper/lower complex half planes cut along Σ. Then λ provides one-to-one correspondences between the following sets: 1. k K + = C + \, iq ] and λ C +, 2. k K + = R {is + : < s < q } {iq } {is + + : < s < q } and λ R, 3. k K = C \ [ iq, and λ C, 4. k K = R {is + : q < s < } { iq } {is + + : q < s < } and λ R. The two sheets of the Riemann surface K Figure 3.

20 2 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI Moreover, λ ± k will denote the boundary values taken by λk for k Σ from the right/left edge of the cut, with λ ± k = ± q 2 k 2, k = is ± +, s < q on the right/left edge of the cut. See also Fig. 3. The contours along the real axis encircling ±iq are used in a Riemann-Hilbert formulation. Figure 3. a The first sheet of the Riemann surface, showing the branch cut red, the contour green and the regions when Iλ > grey and Iλ < white, where λk = r 1 r e i θ 1 +θ 2 2. b The second sheet of the Riemann surface, showing the branch cut red, the contour green and the regions when Iλ < grey and Iλ > white, where λk = r 1 r e i θ 1 +θ 2 2. Remark 4.1. Topologically, the Riemann surface K is equivalent to a surface with genus. The eigenfunctions are defined by the following boundary conditions 4.4 φx, k we iλx, φx, k we iλx as x, 4.5 ψx, k ve iλx, ψx, k ve iλx as x +, where 4.6 w = λ + k i q i q, w = λ + k i q, v = λ + k λ + k, v = i q satisfy the boundary conditions, but they are not unique. As in Section 3, we consider functions with constant boundary conditions and define the same bounded eigenfunctions Mx, k, Nx, k, Mx, k, Nx, k except we use the boundary conditions defined above in 4.6. The bounded eigenfunctions can be represented by means of integral equations, the formulae are the same as except for the constant terms, which are defined above in 4.6.

21 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS 21 Definition 4.2. We say f L 1,1 R if + fx 1 + x dx <. Then, using similar methods as in the prior case σ = 1, θ = π we find the following result see also [22] Theorem 4.3. Suppose the entries of Q Q ± belong to L 1,1 R, then for each x R, the eigenfunctions Mx, k and Nx, k are continuous for k K + K and analytic for k K +, Mx, k and Nx, k are continuous for k K K + and analytic for k K. If we assume that the entries of Q Q ± do not grow faster than e ax2, where a is a positive real number, by similar methods as in Case 1, we have the following result. Theorem 4.4. Suppose the entries of Q Q ± do not grow faster than e ax2, where a is a positive real number, then for each x R, the eigenfunctions Mx, k, Nx, k, Mx, k and Nx, k are analytic in the Riemann surface K Scattering data. As in Section 3.3, we can define the scattering data ak, ak, bk, bk in the same way and find the same representations in terms of eigenfunctions. When k iq, iq, the scattering data and eigenfunctions are defined by means of the corresponding values on the right/left edge of the cut, are labeled with superscripts ± as clarified below. Explicitly, one has the same formulae as Then from the analytic behavior of the eigenfunctions we have the following theorem. Theorem 4.5. Suppose the entries of Q Q ± belong to L 1,1 R, then ak is continuous for k K + K \ {±iq } and analytic for k K +, and ak is continuous for k K K + \ {±iq } and analytic for k K. Moreover, bk and bk are continuous in k R iq, iq. In addition, if the entries of Q Q ± do not grow faster than e ax2, where a is a positive real number, then akλk, akλk, bkλk and bkλk are analytic for k K Symmetry reductions. The symmetry in the potential induces a symmetry between the eigenfunctions. We can obtain the same symmetry relations as 3.39 because we consider the same sign of σ in these two sections. Moreover, the symmetry between scattering data are also the same as When using a particular single sheet for the Riemann surface of the function λ 2 = k 2 +q, 2 the involution k, λ k, λ leads to relationships between eigenfunctions across the cut; namely it relates values of eigenfunctions and scattering data for the same value of k from either side of the cut. One has 4.7 φ x, k = λ± + k φ ± x, k, ψ x, k = λ± + k i q i q ψ ± x, k for k [ iq, iq ]. Similarly, 4.8 M x, k = λ± + k i q for k [ iq, iq ]. Then M ± x, k, N x, k = λ± + k i q N ± x, k 4.9 a ± k = a k, b ± k = q q b k for k [ iq, iq ].

22 22 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI 4.3. Riemann-Hilbert problem. We will develop the Riemann-Hilbert problem across K + K. 1 Riemann-Hilbert problem across the real axis see Fig. 4 a Note that 3.21 and 3.22 can be written as 4.1 µx, k = ρke 2iλx Nx, k + Nx, k, µx, k = Nx, k + ρke 2iλx Nx, k, where µx, k = Mx, ka 1 k, µx, k = Mx, ka 1 k, ρk = bka 1 k and ρk = bka 1 k. Introducing the 2 2 matrices 4.11 m + x, k = µx, k, Nx, k, m x, k = Nx, k, µx, k. Hence, we can write the jump conditions as ρkρk ρke 2iλx 4.12 m + x, k m x, k = m x, k ρke 2iλx for k R. 2 Riemann-Hilbert problem across [, iq ] see Fig. 4 b. By 3.21, 3.22, the notation λ = λ + = λ and the symmetry relations of Jost functions and scattering data, we have M + x, k a + k = b+ k a + k N + x, k e 2iλx + i q λ + k N x, k, iq λ + k M x, k a = N + x, k + q q k b k a k i q λ + k N x, k e 2iλx. Introducing the 2 2 matrices 4.15 i q m + x, k = µ + x, k, N + x, k, m x, k = λ + k N i q x, k, λ + k µ x, k, where µ + x, k = M + x, ka + k 1, µ x, k = M x, ka k 1, ρ + k = b + ka + k 1 and ρ k = b ka k 1. Then 4.16 m + x, k m x, k = m q x, k q ρ + kρ k q q ρ ke 2iλx ρ + ke 2iλx for k C +. 3 Riemann-Hilbert problem across [ iq, see Fig. 4 c. By 3.21, 3.22, the notation λ = λ + = λ and the symmetry relations of Jost functions and scattering data, we can get 4.17 λ + k i q 4.18 M x, k a x, k = q q b k a k λ + k i q N x, k e 2iλx + N + x, k, M + x, k a + k = λ + k i q N x, k + b + k a + k N + x, k e 2iλx. Introducing the 2 2 matrices 4.19 m + x, k = µ + x, k, N + λ x, k, m + k x, k = i q N x, k, λ + k µ k, i q

23 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS 23 where µ + x, k = M + x, ka + k 1, µ x, k = M x, ka k 1, ρ + k = b + ka + k 1 and ρ k = b ka k 1. Then 4.2 m + x, k m x, k = m x, k for k C. q q ρ+ kρ k q q ρ k e 2iλx ρ + ke 2iλx Figure 4. a Riemann-Hilbert problem across the real axis. b Riemann-Hilbert problem across [, iq ]. c Riemann-Hilbert problem across [ iq,. Remark 4.6. For the Riemann-Hilbert problem 4.21 F + ξ F ξ = F ξgξ, on the contour Σ := Σ 1 Σ 2 Σ 3, where Σ 1 := R, Σ 2 := [, iq ], Σ 3 := [ iq,, gξ is Hölder-continuously in Σ and gξ = g m ξ is chosen depending on which piece of the contour is considered, where m = 1, 2, 3. We can consider the projection operators 4.22 P j fk = 1 2πi where λk+λξ 2λξ Σ λk + λξ 2λξ fξ ξ k dξ, k C j, j = 1, 2, dξ ξ k is the Weierstrass kernel, Σ := Σ 1 Σ 2 + Σ 1 Σ 3 integrals along the oriented contours in Figure 3 a. One can show that [45] λk + λξ 2λξ dξ ξ k = dξ + regular terms ξ k denote the for ξ, λξ k, λk. If f j j = 1, 2, is sectionally analytic in C j and rapidly decaying as k in the proper sheet, we have 4.23 P j f j k = 1 j 1 f j k, k C j, 4.24 P j f l k =, k C j. We obtain 4.25 F k = 1 2πi Σ λk + λξ 2λξ F ξgξ dξ, k C 2, ξ k

24 24 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI 4.26 F + k = 1 λk + λξ 2πi Σ 2λξ F ξgξ dξ, k C 1. ξ k For ξ Σ, we can take limits from proper sheet and connected by the analogue of Plemelj-Sokhotskii formulas 4.27 F + 1 λk + λξ ξ = lim F ξgξ dξ, k ξ Σ,k C 1 2πi 2λξ ξ k 4.28 F 1 λk + λξ ξ = lim k ξ Σ,k C 2 2πi Σ 2λξ Σ F ξgξ dξ. ξ k In principle, the above RH problem can be analyed for the two sheeted problem. But a uniformiing coordinate makes the problem considerably more straight forward. This is discussed next Uniformiation coordinates. Before discussing the properties of scattering data and solving the inverse problem, we introduce a uniformiation variable see also [15], defined by the conformal mapping: = k = k + λk, where q2. Then λ = k 2 + q 2 and the inverse mapping is given by k = k = q2. We let C be the circle of radius q centered at the origin in λ = 1 2 plane. We observe that 1 The branch cut on either sheet is mapped onto C. In particular, ±iq = ±iq from either sheet, ± I = ±q and ± II = q. 2 K 1 is mapped onto the exterior of C, K 2 is mapped onto the interior of C. In particular, I = and II = ; the first/second quadrants of K 1 are mapped into the first/second quadrants outside C respectively; the first/second quadrants of K 2 are mapped into the second/first quadrants inside C respectively; I II = q. 2 3 The regions in k plane such that Iλ > and Iλ < are mapped onto D + = { C : 2 q 2 I > } and D = { C : 2 q 2 I < } respectively. From Theorem 4.3 and the definition of the uniformiation variable, we have that the eigenfunctions M, N are analytic for D + and the eigenfunctions M, N are analytic for in D. See Figs 5, 6. The contours green in two sheets Figure 3a, b are mapped into Figure 6 a and Figure 6 b respectively via the uniformiation coordinate Symmetries via uniformiation coordinates. It is found that 1 when, then k, λ k, λ ; 2 when q2, then k, λ k, λ. Hence, we have the same conclusions as 3.48, and 4.29 φ x, q2 q 2 = φx,, i q ψ x, q2 = i q ψx,, D.

25 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS 25 Figure 5. The complex plane, showing the branch cut red, the regions D ± where Iλ > grey and Iλ < white respectively. In particular, the first sheet is mapped onto the region outside the circle, and the second sheet is mapped onto the region inside the circle. Importantly, the eigenfunctions M, N are analytic for D + and the eigenfunctions M, N are analytic for D. Figure 6. a The contour green in the first sheet is mapped into the above curve in plane. b The contour green in the second sheet is mapped into the above curve in plane. Moreover, 4.3 a = a, D + ; a = a, D ; b = b,

26 26 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI 4.31 a q2 = a, D ; b q2 = b. q q 4.6. Asymptotic behavior of eigenfunctions and scattering data. In order to solve the inverse problem, one has to determine the asymptotic behavior of eigenfunctions and scattering data both as in K 1 and as in K 2. We have 4.32 Nx, iqx, ; Nx, i q q x q,, 4.33 a = { 1,, 1,, a = { 1,, 1,, 4.34 lim b =, lim b 2 = Riemann-Hilbert problem via uniformiation coordinates Left scattering problem. In order to take into account the behavior of the eigenfunctions, the jump conditions at Σ, where Σ :=, q q, + q, q {q e iθ, π θ 2π} clockwise,upper circle {q e iθ, π θ } anticlockwise,lower circle, can be written as 4.35 Mx, Nx, = ρe i + q2 a xnx,, Mx, Nx, = ρe i + q2 a so that the functions will be bounded at infinity, though having an additional pole at =. Note that Mx, /a, as a function of, is defined in D +, where it has simple poles j, i.e., a j =, and Mx, /a, is defined in D, where it has simple poles j, i.e., a j =. It follows that 4.36 Mx, j = b j e i j+ q2 j x Nx, j, Mx, j = b j e i j+ q2 j x Nx, j. Then subtracting the values at infinity, the induced pole at the origin and the poles, assumed simple in D + /D respectively, at a j =, j = 1, 2...J and ā j, j = 1, 2... J gives 4.37 Mx, a Nx, 1 1 = ρe i + q2 x 1 i q 1 i q Nx,, J Mx, j j j a j J b j e i j+ q2 j x Nx, j j j a j xnx,,

27 4.38 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS 27 Mx, a Nx, i q 1 i q J Mx, j j j a j J b j e i j+ q2 j x Nx, j j j a j = ρe i + q2 x Nx,. We now introduce the projection operators 4.39 P ± f = 1 fξ 2πi ξ ± i dξ, where lies in the ± regions and Σ :=, q q, + q, q {q e iθ, π θ 2π} clockwise,upper circle {q e iθ, π θ } anticlockwise,lower circle indicated by the curve in Figure 5 red with black arrows. If f ± ξ is analytic in D ± and f ± ξ is decaying at large ξ, then 4.4 P ± f ± = ±f ±, P f ± =. Applying P to 4.37 and P + to 4.38, we can obtain 4.41 Nx, = i q + J Σ b j e i j+ q2 j x Nx, j j j a + ρξ j 2πi Σ ξξ ei ξ+ q2 x ξ Nx, ξdξ, 4.42 Nx, = i q + J b j e i j+ q2 j x Nx, j j j a j 2πi Σ ρξ ξξ e i ξ+ q2 x ξ Nx, ξdξ Right scattering problem. As in Section 3.8.2, we can obtain the following Mx, and Mx, by the right scattering problem Mx, = 4.44 Mx, = i q i q + + J J b j Mx, j e i j+ q2 j x j j a j b j Mx, j e i j+ q2 j x j j a j + ρ ξ 2πi Σ ξξ e i ξ+ q2 x ξ Mx, ξdξ, ρ ξ 2πi Σ ξξ ei ξ+ q2 x ξ Mx, ξdξ Recovery of the potentials. Note that N 1 x, iqx as, and 4.45 J b j e i j+ q2 j x N 1 x, j N 1 x, i q+ j a + 1 ρξ j 2πi Σ ξ e i ξ+ q2 xn ξ 1 x, ξdξ,

28 28 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI we have 4.46 qx = q + i J b j e i j+ q2 j x N 1 x, j j a + 1 ρξ e i ξ+ q2 x ξ N 1 x, ξdξ. j 2π Σ ξ q2 = a. To 4.9. Closing the system. Similarly, we can get J = J from a close the system, by the symmetry relations between the eigenfunctions, we have 4.47 M 1 x, = M 2 x, j + J l=1 i q + J + 2πi Σ l=1 i q + J b j e i j+ q2 j x j j a j jb l e i l + q2 x l j l l a l M 1 x, l j ρ ξ 2πi Σ jb l e i l + q2 l x j l l a l M 2 x, l j 2πi ρ ξ ξξ e i ξ+ q2 x ξ ξ + J l=1 i q + J l=1 ξb l e i l + q2 x l ξ l l a l M 1 x, l ξ ρ η 2πi Σ l + q2 x l ξb l e i ξ l l a l M 2 x, l ξ 2πi Σ ξξ j ei x M 1 x, ξdξ ρ ξ ξξ j ei ξ+ q2 x ξ M 2 x, ξdξ Σ ξ+ q2 ξ ηη ξ ei x M 1 x, ηdη ρ η ηη ξ ei η+ q2 x η M 2 x, ηdη η+ q2 η We note that from eq 4.46 qx is given in only terms of the component N 1. We can use only the second component of eq 4.47 to find M 2 which via the symmetry relation 3.5 yields this function and hence qx. Hence to complete the inverse scattering we reduce the problem to solving an integral equation in terms of the component M 2 only Trace formula. In a similar manner to the prior section, we can get the Trace formula as follows J 1 log a = log j + q2 j 4.49 J 1 log a = log + j q2 j + q2 q2 j j j + j J 2 i i=1 + q2 i J 2 i=1 where R i =, R j and 2J 1 + J 2 = J. + q2 i i + 1 2πi 1 2πi Σ Σ dξ. log1 bξb ξ dξ, D +, ξ log1 bξb ξ dξ, D, ξ Reflectioness potentials and soliton solutions. Reflectioness potentials and soliton solutions correspond to ero reflection coefficients, i.e., ρξ = and ρξ = for all ξ Σ. We also note, from the symmetry relation b = b it follows that the reflection coefficients ρ = b/a, ρ = b/ā will both vanish when b = for on Σ. By substituting = l in 4.47, the system 4.47 reduces to an algebraic systems of equations that determine the functional

29 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS 29 form of these special potentials. When time dependence is added the reflectionless potentials correspond to soliton solutions. The reduced equation takes the form 4.5 M 1 x, l M 2 x, l i q = l + J l b j e i j+ q2 j x l j j a j j + J l=1 i q + J l=1 The above equations are an algebraic system to solve for either Nx, l or Mx, l, l = 1, 2...J. The potential is reconstructed from equation 4.46 with ρξ =, ρξ = ; i.e., J b j e i j+ q2 j x N 1 x, j 4.51 qx = q + i j a. j Since a 1 as, from the trace formula when bξ = in Σ, we have the following constraint for the reflectionless potentials J 1 j 4 J 2 i q 4 q 2 = 1, i=1 where 2J 1 + J 2 = J. We claim that J 2. Otherwise, if J = 1, then J 1 = and J 2 = 1, which implies that 1 2 = q 2 ; hence the eigenvalue lies on the circle, which in this case is the continuous spectrum. Such eigenvalues are not proper; they are not considered here Discrete scattering data and their symmetries. In order to find reflectionless potentials and solitons we need to be able to calculate the relevant discrete scattering data. The coefficients b j and b j, j = 1, 2,...J, can be calculated in the same way as in Section 3.12 to find 4.53 b j b j = 1, b j b j = 1, b j b j = 1, b j b j = 1. The functions a j, ā j can be calculated via the trace formulae Reflectionless potential solution: 2-eigenvalue. In this subsection, we construct an explicit 2-eigenvalue reflectionless potential by setting J = 2; this solution turns into a soliton solution when time dependence is added. The simplest reflectionless potential case obtains when J 1 = and J 2 = 2; as mentioned above when J 1 = 1, J 2 = leads to eigenvalues on the continuous spectrum. Note that 1 2 = q. 2 Now let 1 = iv 1, where v 1 > q. Then 2 = i q2 v 1, aiv 1 = a i q2 v 1 = and ai q2 v 1 = a iv 1 =. Thus, 4.54 a = iv 1 + i q2 v 1, i q2 + iv 1 v 1 we can get 4.55 a iv 1 = i v 1 + q2 v 1 2q 2 v2 1, a i q2 = v 1 a = i q 2 v 1 + iv 1 iv 1 + i q2 i q 2 v 1 + v 1 2q 2 q 2 1 v1 2 v 1,, jb l e i l + q2 l x j l l a l M 1 x, l jb l e i l + q2 l x j l l a l M 2 x, l.

30 3 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI 4.56 a iv 1 = i v 1 + q2 v 1 2v1 2 q2, a i q2 i v 1 + q2 v 1 =. v 1 2q 2 q 2 1 v1 2 Similarly, from section 4.12, we can get biv 1 = biv 1 = b i q2 v 1 = b i q2 v 1 = b iv 1 = b iv 1 = bi q2 v 1 = bi q2 v 1 = 1. So we can write biv 1 = e iϕ1 and b i q2 v 1 = e iϕ2. By the symmetry relations b = b and b q2 = q q b, we can obtain bi q2 v 1 = e i2θ+ϕ1, b iv 1 = e i2θ+ϕ2, biv 1 = e iϕ1, b i q2 v 1 = e iϕ2, b iv 1 = e i2θ+ϕ2, bi q2 v 1 = e i2θ+ϕ1. Hence, we have 4.57 M 2 x, i q2 = v 1 v 1 [2q v 1 e q 2 v +v 1 1 iq e 2v1x q 2 + v 2 1 q + v 1 e q 2 v v 1 x 1 e iθ+ϕ 2 x e iθ e iϕ1 e iϕ2 q 2 + v2 1 e ], 2v1x e 2q2 v x 1 e i2θ+ϕ1+ϕ M 2 x, iv 1 = 2q v 1 e q 2 v +v 1 1 ie v1x q 2 + v1 2 q e q 2 v x 1 e iθ+ϕ1 + v 1 e v1x x e iθ e iϕ2 e iϕ1 q 2 + v2 1 e. 2v1x e 2q2 v x 1 e i2θ+ϕ1+ϕ2 By 4.51, we have qx = q i q q 2 1 e i2θ+ϕ1 2 v v 1 x e 1 v q2 v 2 1 M 2 x, i q2 2 v 1 q2 i v 1 v 1 e i2θ+ϕ2 e q 2 v v 1 x 1 M q 2 2 x, iv 1. v 1 + v Time evolution. As in the prior section we find, 4.6 at t =, at t =, bt t = 2iq 2 2λkbt. Hence, at and at are time independent. Moreover, 4.61 [ ] [ ] biv 1, t = e iϕ1 e 2i q v 2 1 q4 v 1 2 t, b i q2, t = e iϕ2 e 2i q v 2 1 q4 v 2 t 1. v 1 Putting all the above into the formula we had for the reflectionless potential, we obtain the following 2-soliton solution, which is a oscillating in time or breathing

31 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS 31 soliton solution q 2 +v2 1tq 2 +v2 1 iv 1 x qx, t = e i 2 v 2 1 +ϕ 1+ϕ 2 2q 4 e 2itv2 1 + q 2 v +v 1 x+iθ+ϕ 1 1 2v 4 1 e it q q v 1 q 2 + v v4 1 v e v 1x + q v 1 q 2 + v1 2 e i q4 t 2iq2 v 1 x+v2 1 tv2 1 +2θ+ϕ 1 +ϕ 2 v 1 2 / 2iq 4 t v 2 1 q + 2 v +v 1 x+iθ+ϕ 1 2 [ ] [ 1 q 4 2v 1 2iq v 1 sin 2t q 2 v1 2 v1 2 ϕ 1 + ϕ 2 + q 2 + v1 2 2 sinh x + 2v ] 1x + i2θ + ϕ 1 + ϕ 2. v 1 2 A typical breather solution is plotted in figure 7 below. Figure 7. The amplitude of qx, t with θ = π 2, ϕ 1 =, ϕ 2 =, v 1 = 3 and q = The case of σ = 1 with θ = θ + θ = Next we will study the nonlocal nonlinear Schrödinger NLS equation 1.1 with σ = iq t x, t = q xx x, t 2q 2 x, tq x, t with nonero boundary conditions NZBCs 5.2 qx, t q ± t := qt = q e 2iq2 t+iθ, as x ±, where q >, θ < 2π. Note we take θ + = θ = θ Direct Scattering. With this condition, equation 2.13 conveniently reduces to 2 v j 5.3 x 2 = k2 qv 2 j, j = 1, 2. Each of the two equations has two linearly independent solutions e iλx and e iλx as x, where λ = k 2 q 2. The variable k is then thought of as belonging to a Riemann surface K consisting of two sheets C 1 and C 2 with both coinciding with the complex plane cut along, q ] [q, + with its edges glued in such a way that λk is continuous through the cut. The Riemann surface and the definition of λ are the same as discussed above in section 3 σ = 1, θ = π.

32 32 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI 5.2. Eigenfunctions. Following the approach discussed in Section 3 we introduce the eigenfunctions defined by the following boundary conditions 5.4 φx, k we iλx, φx, k we iλx as x, 5.5 ψx, k ve iλx, ψx, k ve iλx as x +, where 5.6 w = λ + k i q i q, w = λ + k i q, v = λ + k λ + k, v = i q satisfy the boundary conditions; they are not unique. As in Section 3, we consider functions with constant boundary conditions and define the same bounded eigenfunctions Mx, k, Nx, k, Mx, k, Nx, k except we use the boundary conditions defined above in 5.6. The bounded eigenfunctions can be represented by means of integral equations, the formulae are the same as except for the constant terms, which are defined above in Symmetry reductions. Taking into account boundary conditions, we can obtain ψx, k = φ x, k 1, ψx, k = φ x, k. 1 1 The scattering relations are the same as except for the relation between b and b, i.e., b k = bk. Using the notation λ = λ + = λ, we have the following symmetry: 5.8 φ x, k = λ + k i q φ± x, k, ψ x, k = λ + k i q ψ ± x, k for k, q ] [q, +. Moreover, 5.9 a ± k = a k, b ± k = q q b k for k, q ] [q, Uniformiation coordinates. As in Section 3 we introduce a uniformiation variable, defined by the conformal mapping: = k = k + λk, and the inverse mapping is given by k = k = q2. Then λ = 1 2 q Symmetries via uniformiation coordinates. We have seen that 1 when, then k, λ k, λ ; 2 when q2, then k, λ k, λ. Hence, ψx, = φ x, 1, ψx, = φ x,, φ x, q2 q 2 = φx,, i q ψ Moreover, x, q2 = i q ψx,, I < a = a, a = a, b = b, q a q 2 = a, I < ; b = b. q q

33 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS Asymptotic behavior of eigenfunctions and scattering data. In order to solve the inverse problem, one has to determine the asymptotic behavior of eigenfunctions and scattering data both as and as. We have 5.14 Nx, iq x, ; Nx, qx q i q,, 5.15 a = { 1,, 1,, a = { 1,, 1,, 5.16 lim b =, lim b 2 = Left and right scattering problems. Using similar methods as in Section 3, we find Nx,, Nx,, Mx, and Mx,. They are almost the same as 3.64, 3.65, 3.75, The only differences are the constant terms, which are shown in Recovery of the potentials. Note that N 1x, 5.17 N 1 x, 1+ J b j e i j q2 j x 2 j a j as, therefore, 5.18 J q = q 1 + j 2a j b j e i j q2 j x qx q as, and N 1 x, j ρξ 2πi ξ 2 ei ξ q2 xn ξ 1 x, ξdξ N 1 x, j ρξ 2πi ξ 2 e i ξ q2 ξ x N 1 x, ξdξ. q Closing the system. We find J = J from a = a. To close the system, by the symmetry relations between the eigenfunctions, we have 5.19 N1 x, = N 2 x, j + J l=1 i q + J 2πi l=1 + ξ + J l=1 i q + J l=1 i q + J b j e i j q2 j x j j a j jb l e i l q2 x l j l l a l N 1 x, l + j + 2πi jb l e i l q2 l x j l l a l N 2 x, l + j 2πi ρξ ξξ e i ξ q2 x ξ + ξb l e i l q2 x l ξ l l a l N 1 x, l + ξ + 2πi ξb l e i l q2 x l ξ l l a l N 2 x, l + ξ + 2πi ρξ ξξ j ei ξ q2 x ξ N 1 x, ξdξ ρξ ξξ j ei ξ q2 x ξ N 2 x, ξdξ ρη ηη ξ ei η q2 x η N 1 x, ηdη ρη ηη ξ ei η q2 x η N 2 x, ηdη dξ.

34 34 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI We note that from eq 5.18 qx is given in only terms of the component N 1. We can use only the first component of eq 5.19 to find this function and hence qx. Hence to complete the inverse scattering we reduce the problem to solving an integral equation in terms of the component N 1 only Trace formula. Using b = b and following the analysis in Section 3, we can get 5.2 J 1 log a = log j q2 j 5.21 J 1 log a = log + j + q2 j q2 + q2 j j j + j J 2 i i=1 q2 i J 2 i=1 where R i =, R j and 2J 1 + J 2 = J. q2 i i + 1 2πi 1 2πi + + log1 + bξb ξ dξ, I >, ξ log1 + bξb ξ dξ, I <, ξ Discrete scattering data and their symmetries. As discussed earlier, the data a j, ā j are calculated from the trace formulae. The symmetries on the data b j, b j, j = 1, 2,...J, can be calculated from their associated eigenfunctions. Using the method in Section 3.12, we can deduce b j b j = 1. If R j = we are led to a contradiction. Thus we must have that R j ; consequently the eigenvalues come in complex conjugate pairs and J must be even Reflectioness potentials and soliton solutions. Reflectioness potentials correspond to ero reflection coefficients, i.e., ρξ = and ρξ = for all real ξ. We also note, from the symmetry relation b = b it follows that the reflection coefficients ρ = b/a, ρ = b/ā will both vanish when b = for on the real axis. By substituting = l in 5.19, the system 5.19 reduces to an algebraic equation that determine the functional form of these special potentials. When time dependence is added the reflectionless potentials correspond to soliton solutions. The reduced equation takes the form 5.22 N1 x, l N 2 x, l i q = l + J l b j e i j q2 j x l j j a j j + J l=1 i q + J l=1 jb l e i l q2 l x j l l a l N 1 x, l The above equations are an algebraic system to solve for Nx, l, l = 1, 2...J. The potential are reconstructed from equation 5.18 with ρξ = ; i.e., 5.23 q = q 1 + J b j e i j q2 j x 2 j a j N 1 x, j. jb l e i l q2 l x j l l a l N 2 x, l Since a 1 as, by the trace formula when bξ = in the real axis, we have the constraint J/2 = 1. j 4 q 4.

35 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS Reflectionless potential solution: 2-eigenvalue. In this subsection, we construct an explicit form for the 2-reflectionless solution this is the 2 soliton solution when time dependence is added by setting J = 2 and 1 = ξ 1 + iη 1, where ξ 1, η 1 > and ξ1 2 + η1 2 = q. 2 Then aξ 1 + iη 1 = a ξ 1 + iη 1 = and aξ 1 iη 1 = a ξ 1 iη 1 =. Thus, 5.24 a = ξ 1 + iη 1 ξ 1 + iη 1 ξ 1 iη 1 ξ 1 iη 1, a = ξ 1 iη 1 ξ 1 iη 1 ξ 1 + iη 1 ξ 1 + iη 1 and we then find 5.25 a ξ 1 + iη 1 = iξ 1ξ 1 iη 1 2q 2 η a ξ 1 iη 1 = iξ 1ξ 1 + iη 1 2q 2 η 1, a ξ 1 + iη 1 = iξ 1ξ 1 + iη 1 2q 2η, 1, a ξ 1 iη 1 = iξ 1ξ 1 iη 1 2q 2η. 1 Moreover, we write bξ 1 + iη 1 = c 1 e iθ1, by b j b j = 1, we have b ξ 1 + iη 1 = 1 c 1 e iθ1 q 2. From b = q q b, we get 5.27 bξ 1 iη 1 = e 2iθbξ 1 +iη 1 = c 1 e i2θ+θ1, b ξ 1 iη 1 = e 2iθb ξ 1 +iη 1 = 1 c 1 e i2θ+θ1. Hence, we have 5.28 N 1 x, ξ 1 + iη 1 = e 2η1x+iθ ξ 1 e 2η1x+2iθ+θ1 q η 1 + c 2 1η 1 iξ 1 ic 2 1q 1 ξ 1 e 6η1x + c 1 ξ 1 ξ 1 + iη 1 e 3θ+θ1i + c 1 η 1 iξ 1 η 1 + c 2 1η 1 + ic 2 1ξ 1 e 4η1x+iθ+θ1 / c 2 1ξ2e 2 8η1x + c 2 1ξ1e 2 4θ+θ1i + e 4η1x+2iθ+θ1 1 + c η c 4 1ξ1 2, 5.29 N 1 x, ξ 1 + iη 1 = c 1 ξ 1 e 2η1x+iθ e 4η1x+iθ+θ1 η 1 + c 2 1η 1 iξ 1 η 1 + iξ 1 ic 1 q ξ 1 e 6η1x + c 2 1ξ 1 ξ 1 iη 1 e 3iθ+θ1 c 1 q η 1 + c 2 1η 1 + ic 2 1ξ 1 e 2η1x+2iθ+θ1 / c 2 1ξ1e 2 8η1x + c 2 1ξ1e 2 4iθ+θ1 + e 4η1x+2iθ+θ1 1 + c η c 4 1ξ1 2 By 5.18, we can obtain 5.3 qx = q [ 1 + c 1e iθ1 e 2η1x iξ 1ξ 1+iη 1 2η 1 N 1 x, ξ 1 + iη c 1 e iθ1 e 2η1x iξ 1ξ 1 iη 1 2η 1 N 1 x, ξ 1 + iη Time evolution. As in previous sections we find that the scattering data satisfies at at bt 5.31 =, =, = 2iq 2 + 2λkbt. t t t Hence, at and at are time independent. Moreover, 5.32 bξ 1 + iη 1, t = c 1 e iθ1 e 2iq2 +2ξ1η1it, b ξ 1 + iη 1, t = 1 c 1 e iθ1 e 2iq2 2ξ1η1it. ]

36 36 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI Putting all the above into the formula we had for the reflectionless potential, we obtain the following 2-soliton solution 5.33 qx, t = e i2q2 t+θ c 2 1q e 8tη1 q 2 η 2 1 2η 2 1 e 4η1x+2iθ+θ1 + q 2 η 2 1e 8η1x + e 4iθ+θ1 + c 4 1q e 4η1x+4t q 2 η2 1 +2iθ+θ1 q 2 2η1 2 2iη 1 q 2 η c1 η 1 e 2η1x+4tη1 q 2 η1 2+iθ+θ1 e 4η1x + e 2iθ+θ1 q 2 + η 1 η 1 i q 2 η2 1 2c3 1η 1 e 2η1x+6t q 2 η2 1 +iθ+θ1 e 4η1x + e 2iθ+θ1 q 2 + η 1 η 1 i q 2 η2 1 + q e 4η1x+2iθ+θ1 q 2 + 2iη 1 iη 1 + q 2 η2 1 / qe 2 4η1x+2iθ+θ1 + c 4 1qe 2 4η1x+4t q 2 η2 1 +2iθ+θ1 c 2 1e 8tη1 q 2 η1 2 2η 2 1 e 4η1x+2iθ+θ1 + η 2 1 q 2 e 8η1x+4iθ+θ1. Typical 2 soliton solutions, which travel and interact, including the one in [28], are shown in Figure 8 below. In terms of magnitude, Fig. 8 a shows the interaction of two waves of elevation, Fig. 8 b shows the interaction of two depressive waves i.e. two dips and Fig. 8 c illustrates the interaction of one elevated wave and one depressive wave. If we choose q = 2, η 1 = 3, θ =, c 1 = p2 p 1 p 2+p 1 get the general formula in [28]. and e 2iθ = p3 ip4 p 3 ip 4, we can a b c Figure 8. a The amplitude of qx, t with θ =, θ 1 = 3π 4, c 1 = 2, η 1 = 3 and q = 2. b The amplitude of qx, t with θ =, θ 1 = 3π 4, c 1 = 2, η 1 = 3 and q = 2. c The amplitude of qx, t with θ =, θ 1 = π, c 1 = 2, η 1 = 3 and q = The case of σ = 1 with θ + θ = π The final case we will study is the nonlocal nonlinear Schrödinger NLS equation 1.1 with σ = iq t x, t = q xx x, t 2q 2 x, tq x, t with nonero boundary conditions NZBCs 6.2 qx, t q ± t = q e 2iq2 t+iθ±, as x ±, where q >, θ ± < 2π, θ + θ = π.

37 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS 37 The analysis of this case is similar to the NZBCs studied in section 4. As discussed in section 4, it is natural to introduce the eigenfunctions defined by the following boundary conditions 6.3 φx, k we iλx, φx, k we iλx as x, 6.4 ψx, k ve iλx, ψx, k ve iλx as x +, where λ + k 6.5 w = iq + iq, w = λ + k iq+, v = λ + k λ + k, v = iq satisfy the boundary conditions. As in Section 3, we consider functions with constant boundary conditions and define the same bounded eigenfunctions Mx, k, Nx, k, Mx, k, Nx, k except we use the boundary conditions defined above in 6.5. The bounded eigenfunctions can be represented by means of integral equations, the formulae are the same as except for the constant terms, which are defined above in Symmetry reductions. Taking into account boundary conditions, we can obtain 5.7. The scattering relations are the same as eq except for the relation between b and b, i.e., b k = bk. Similarly, we can define the eigenfunctions and scattering data on the left/right edge of the cut for k iq, iq. Then 6.6 φ x, k = λ + k φ ± x, k, ψ x, k = λ + k iq i q ψ ± x, k for k iq, iq. Moreover, we have 6.7 a ± k = a k, b ± k = q2 q + q b k for k iq, iq Uniformiation coordinates. Before discussing the properties of scattering data and solving the inverse problem, we introduce a uniformiation variable, defined by the conformal mapping: = k = k + λk, and the inverse mapping is given by k = k = 1 2 q2. Then λ = q Symmetries via uniformiation coordinates. It is known that 1 when, then k, λ k, λ ; 2 when q2, then k, λ k, λ. Hence, we can deduce the same formulae as 5.1 and 5.12 although has different representation here. Moreover, 6.8 φ x, q2 q 2 = φx,, ψ x, q2 = iq + ψx,, D. iq 6.9 a q2 = a, D, b q2 = q2 b, q + q where D is the white regions of Figure 5.

38 38 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI 6.4. Asymptotic behavior of eigenfunctions and scattering data. In order to solve the inverse problem, one has to determine the asymptotic behavior of eigenfunctions and scattering data both as in K 1 and as in K 2. We have 6.1 Nx, iqx iq+, ; Nx, q x q,, 6.11 a = { 1,, 1,, a = { 1,, 1,, 6.12 lim b =, lim b 2 = Left and right scattering problems. Using the methods in the second case, we can get 6.13 Nx, = 6.14 Nx, = 6.15 Mx, = iq iq+ iq 6.16 Mx, = iq + + J b j e i j+ q2 j x Nx, j j j a + ρξ j 2πi Σ ξξ ei ξ+ q2 x ξ Nx, ξdξ, J J J b j e i j+ q2 j x Nx, j j j a j b j Mx, j e i j+ q2 j x j j a j b j Mx, j e i j+ q2 j x j j a j 2πi Σ ρξ ξξ e i ξ+ q2 x ξ Nx, ξdξ, + ρ ξ 2πi Σ ξξ e i ξ+ q2 x ξ Mx, ξdξ, ρ ξ 2πi Σ ξξ ei ξ+ q2 x ξ Mx, ξdξ where we recall from the second case: Σ :=, q q, + q, q {q e iθ, π θ 2π} clockwise,upper circle {q e iθ, π θ } anticlockwise,lower circle Recovery of the potentials. Note that N 1 x, iqx as, and 6.17 J b j e i j+ q2 j x N 1 x, j N 1 x, iq + + j a + 1 ρξ j 2πi Σ ξ e i ξ+ q2 xn ξ 1 x, ξdξ, we have 6.18 qx = q + +i J b j e i j+ q2 j x N 1 x, j j a j + 1 2π Σ ρξ e i ξ+ q2 xn ξ 1 x, ξdξ. ξ

39 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS Trace formula. As in the prior section we can show that b j b j = 1, which implies that R j and J is even. Hence, we can get the Trace formula as follows J/2 log a = log j + q2 j 6.2 J/2 log a = log + j q2 j + 1 2πi + q2 q2 j j j + j 1 2πi Σ Σ log1 + bξb ξ dξ, D +, ξ log1 + bξb ξ dξ, D, ξ where R j. We claim that this case does not admit soliton solutions. Indeed, by the asymptotic of a, i.e., a 1 as and the fact R j. Moreover, the Trace formula yields 6.21 a = J/2 j + q2 j + j q2 j 1 exp 2πi Σ Since the exponential term is real for all ξ Σ, it implies that J/2 j 4 1 q 4 exp 2πi Σ log1 + bξb ξ dξ. ξ log1 + bξb ξ dξ = 1, ξ which is a contradiction. Thus, in this case there are no solitons Closing the system. This time the equation will not have a discrete spectrum contribution, to close the system, by the symmetry relations between the eigenfunctions, we have N 1 x, = N 2 x, iq + ρξ 2πi Σ ξξ ei ξ+ q2 x ξ iq + ξ 6.22 ρη 2πi Σ ηη ξ e i η+ q2 x η N 1 x, ηdη ξ ξ dξ. ρη 2πi ηη ξ e i η+ q2 x η N 2 x, ηdη Σ 6.9. Time evolution. Similar to the case in section 4, we can get 6.23 at at bt =, =, = 2iq 2 bt 2λkbt, = 2iq 2 2λkbt. t t t t Then 6.24 ρ, t = ρ, e 2i[q2 1 2 ξ2 q 4 ξ 2 ]t, ρ, t = ρ, e 2i[q2 1 2 ξ2 q4 ξ 2 ]t. It yields 6.25 qx, t = q e 2iq2 t+iθ π Σ ρξ, e 2i[q2 1 2 ξ2 q 4 ξ 2 ]t e iξ+ q 2 ξ x N 1 x, ξ, tdξ. ξ In principle we can formulate integral equations for M2 and via the symmetry between M and N to obtain qx as done in section 4.

40 4 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI 7. Modulational instability In this section we study modulation instability of a constant in space amplitude solution to the nonlocal NLS equation 1.1. More precisely, we study linear stability properties of the family of solutions defined by 7.1 qx, t = Ae iωt, where A is a constant complex amplitude and ω is a real frequency. In the limit x + we define A A + and when x we let A A. If we let A ± = q e iθ± then in order for Eq. 7.1 to be a solution to 1.1 we require that θ + θ = or π. Thus, we distinguish here between four different cases: θ + θ =, σ = ±1 and θ + θ = π, σ = ± Modulational instability I: θ + θ =. In this case we have A + = A A and ω = 2σ A 2. To study modulational instability we write solutions to 1.1 in the form 7.2 qx, t = A + ε qx, te 2iσ A 2t. Substituting 7.2 into 1.1 and retaining terms up to order ε we obtain 7.3 i q t x, t = q xx x, t 2σ A 2 qx, t 2σA 2 q x, t. Equation 7.3 governs the dynamical evolution of the perturbation qx, t subject to the boundary conditions qx, t decays very fast at plus and minus infinity. We next decompose the perturbation in terms of its Fourier integral representation qx, t = q 1 ke iλt + q 2 ke iλ t e ikx dk. It is evident that the constant solution is modulationally unstable if the eigenvalue λ is complex. After some algebra we find 7.5 λ 2 = k 2 k 2 + 4σ A 2. Thus we conclude that when σ = +1 the solution 7.1 is modulationaly stable whereas it is modulationaly unstable for σ = 1. The unstable case corresponds to having two eigenvalues on the imaginary axis, or two solitons that are stationary and oscillate in time, as discussed in case Modulational instability II: θ + θ = π. In this second case we find A + = A A thus ω = 2σ A 2. Under this situation, the linear stability analysis proceeds as before by writing a perturbative solutions to 1.1 in the form 7.2 with the exception that q x, t = A +ε q x, te 2iσ A 2t. Substituting 7.2 into 1.1 and retaining terms up to order ε we now obtain 7.6 i q t x, t = q xx x, t + 2σ A 2 qx, t 2σA 2 q x, t. Decomposing the perturbation in terms of its Fourier integral representation 7.4 we then find 7.7 λ 2 = k 2 2σ A A 2. It is thus evident that the constant amplitude solution is modulationally stable for both signs of σ.

41 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS Conclusion In this paper the nonlocal NLS equation 1.1 with nonero boundary values at infinity is considered. Depending on the signs of nonlinearity: σ = ±1 and phase difference of the boundary values from plus infinity to minus infinity: θ = θ + θ, there are four distinct cases to consider when the amplitude at infinity is constant. The direct scattering problem and corresponding analytic properties and symmetries of the eigenfunctions and scattering data are obtained. The inverse problem is constructed via a Riemann-Hilbert problem formulated from both the right and left in terms of a convenient uniformiation variable. The simplest pure soliton solutions are obtained. When σ = 1, θ = π a pure one soliton solution is found; when σ = 1, θ = a pure stationary two soliton solution is obtained; and with σ = 1, θ = a traveling bidirectional interacting two soliton solution is found. In the final case σ = 1, θ = π, there are no soliton solutions allowed. A number of box potentials are analyed; their associated eigenvalues are found and in all case shown to be consistent with the prior analytical results. Modulational instability of a plane wave is also discussed. 9. Acknowledgements The authors are pleased to acknowledge Justin T. Cole and Yi-Ping Ma for considerable assistance in the figures of this paper and in Mathematica respectively. MJA was partially supported by NSF under Grant No. DMS Appendix 1.1. Proof of Theorem 3.2. For k, q q, +, the matrices P iλ k ± and P ± iλ are defined as follows: 1.1 P ± iλ k = 1 λ + k iq± 2λ iq, P ± λ k iλ k = 1 λ k iq± 2λ iq. λ + k We have P ± iλ 2 = P ± iλ, P ± iλ 2 = P ± iλ, P ± iλ + P ± iλ = I 2, P ± iλ P ± iλ = P ± iλ P ± iλ =. Moreover, 1.2 ikj + Q ± P ± iλ k = iλp ± iλ k and 1.3 ikj + Q ± P ± iλ k = iλp ± iλ k. Then we can rewrite the Green s functions in terms of the projectors to find x λ + k 1.4 Mx, k = iq+ + [P iλ + e2iλx x P iλ ]Q Q Mx, kdx, 1.5 Mx, k = 1.6 Nx, k = iq λ + k iq+ λ + k λ + k 1.7 Nx, k = iq x + [e 2iλx x P iλ + P iλ ]Q Q Mx, kdx, x + x [e 2iλx x P + iλ + P + iλ ]Q Q +Mx, kdx, [P + iλ + e2iλx x P + iλ ]Q Q +Mx, kdx.

42 42 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI The projections P ± iλ k and P ± iλ k admit a natural continuation to k K \ {±q }, i.e., λ C \ {}. Taking into account that these projections are singular matrices, we can show that their l 2 -norm are given by 1.8 P ± iλ k 2= P ± iλ k λ k 2= 2 + λ + k 2 + 2q 2 2 λ k 4 λ 2 = 2 + 2q 2. 2 λ In particular, if k, q ] [q, +, then 1.9 P ± iλ k 2= k λ. We consider the Neumann series 1.1 Mx, k = where 1.11 M x, k = 1.12 M n+1 x, k = Then 1.13 M n+1 x, k x M n x, k, n= λ + k iq e 2iq2 t iθ+, [P iλ + e2iλx x P iλ ]Q Q M n x, kdx. λ + k 2 + q k x Qx Q M n x, k dx λ if x x and λ U +,, +. Since the entries of Q Q belong to L 1 R, using the identities [ 1 x ] j ξ fξ fξ dξ dξ j! 1.14 = = 1 x d j + 1! dξ [ x 1 j + 1! [ ξ ] j+1 fξ dξ dξ j+1 fξ dξ], we can get the Neumann series ]mjawhich is uniformly convergent for k C 1 \{±q }. It follows that Mx, k is analytic for k C 1 because a uniformly convergent series of analytic functions converges to an analytic function. Similarly, Mx, k is continuous for k C 1 \ {±q }. To extend the continuity properties at k = ±q, we rewrite P iλ + e2iλx x P iλ as I 2 + [e 2iλx x 1]P iλ and use the estimation 1.15 I 2 + [e 2iλx x 1]P iλ x x k max{1, 2 k }1 + x 1 + x max{1, 2 k }1 + x 2

43 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS 43 and the fact that f L 1,2 R L 1 R, the proof then proceeds as before under the condition that the entries of Q Q belong to L 1,2 R. Similarly, the proof for Nx, k, Mx, k and Nx, k can be done by using the above method Proof of Theorem 3.4. The integral equation of Mx, k can be written in the alternative form 1.16 and M 1 x, k = λ + k + x y 1.17 M 2 x, k = iq + + where Mx, k = M1 x, k M 2 x, k. qy q q + q +e 2iλy M 1, kddy x e 2iλx y q y + q +M 1 y, kdy, Since the entries of Q Q ± do not grow faster than e ax2, it is easy to see that Mx, k is analytic in the Riemann surface K. Similarly, we can prove Nx, k, Mx, k and Nx, k are also analytic in K Asymptotic behavior of eigenfunctions for the case of σ = 1 with θ + θ = π. iq, x 1.18 Mx, qx q + iq,, iqx, 1.19 Nx, iq+ q x,, q iqx, 1.2 Mx, iq + q x,. q 1.4. Time evolution for the case of σ = 1 with θ + θ = π. Since 2ik 1.21 v t = 2 iqx, tq x, t 2kqx, t iq x x, t 2kq x, t + iqx x, t 2ik 2 + iqx, tq v, x, t then 1.22 and 1.23 Note that v 1 t = 2ik2 iqx, tq x, tv 1 + 2kqx, t iq x x, tv 2 v 2 t = 2kq x, t + iq x x, tv 1 + 2ik 2 + iqx, tq x, tv qx, t q ± = q e 2iq2 t+iθ±, as x ±,

44 44 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI where q >, θ ± < 2π, θ + θ = π. Thus, 1.25 and 1.26 as x + ; v 1 t i2k2 + q 2 v 1 2kq e 2iq 2 t+iθ+ v 2 v 2 t 2kq e 2iq 2 t iθ v 1 i2k 2 + q 2 v 2 v t i2k2 + q 2 v 1 2kq e 2iq 2 t+iθ v 2 and v t 2kq e 2iq 2 t iθ+ v 1 i2k 2 + q 2 v 2 as x. As x ±, the eigenfunctions of the scattering problem asymptotically satisfy v1 ik q 1.29 = e 2iq2 t+iθ± v1, v 2 q e 2iq2 t iθ ik v 2 we can get x 1.3 q e 2iq2 t+iθ± v 2 v 1 x + ikv 1 and 1.31 q e 2iq2 t iθ v 1 v 2 x + ikv 2 as x ±. Hence, v t iq2 v 1 2k v 1 x and v t iq2 v 2 2k v 2 x as x ±. Note that the eigenfunctions themselves, whose boundary values at space infinities, are not compatible with this time evolution. Therefore, one introduces time-dependent eigenfunctions 1.34 Φx, t = e ia t φx, t, Φx, t = e ib t φx, t, 1.35 Ψx, t = e ic t ψx, t, Ψx, t = e id t ψx, t to be solutions of We have 1.36 Φ 1 x, t = ia Φ 1 x, t+e ia t φ 1x, t, t t 1.37 Φ 1 x, t = ib Φ 1 x, t+e ib t φ 1x, t, t t 1.38 Ψ 1 x, t t = ic Ψ 1 x, t+e ic t ψ 1x, t, t Φ 2 x, t t Φ 2 x, t t Ψ 2 x, t t = ia Φ 2 x, t+e ia t φ 2x, t, t = ib Φ 2 x, t+e ib t φ 2x, t, t = ic Ψ 2 x, t+e ic t ψ 2x, t, t

45 IST FOR NONLOCAL NLS WITH NONZERO BOUNDARY CONDITIONS Ψ 1 x, t t Note that 1.4 φx, t as x. From 1.41 Φ 1x, t t = id Ψ 1 x, t+e id t ψ 1x, t, t λ + k iq e 2iq2 t iθ+ as x, we can deduce e iλx, Ψ 2 x, t t φx, t t iq 2 Φ 1 x, t 2k Φ 1x, t x 1.42 A = q 2 + 2λk. Similarly, we have 1.43 B = A = q 2 2λk, = id Ψ 2 x, t+e id t ψ 2x, t. t 2q 3 e 2iq2 t iθ+ e iλx = ia Φ 1 x, t + e ia t φ 1x, t, t 1.44 C = A = q 2 2λk, 1.45 D = A = q 2 + 2λk. Then 1.46 φ 2ik t = 2 iqx, tq x, t iq 2 + 2λk 2kqx, t iq x x, t 2kq x, t + iqx x, t 2ik 2 + iqx, tq x, t iq 2 + 2λk 1.47 φ 2ik t = 2 iqx, tq x, t + iq 2 + 2λk 2kqx, t iq x x, t 2kq x, t + iqx x, t 2ik 2 + iqx, tq x, t + iq 2 + 2λk 1.48 ψ 2ik t = 2 iqx, tq x, t + iq 2 + 2λk 2kqx, t iq x x, t 2kq x, t + iqx x, t 2ik 2 + iqx, tq x, t + iq 2 + 2λk 1.49 ψ 2ik t = 2 iqx, tq x, t iq 2 + 2λk 2kqx, t iq x x, t 2kq x, t + iqx x, t 2ik 2 + iqx, tq x, t iq 2 + 2λk φ, φ, ψ, ψ. Note that 1.5 φx, t = btψx, t + atψx, t and 1.51 φx, t = atψx, t + btψx, t,

46 46 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI then ik 2 iqx, tq x, t iq 2 + 2λk 2kqx, t iq x x, t 2kq x, t + iqx x, t 2ik 2 + iqx, tq x, t iq 2 + 2λk btψ1 x, t + atψ 1 x, t = bt ψ1 x, t btψ 2 x, t + atψ 2 x, t t ψ 2 x, t 2ik + bt iqx, tq x, t + iq 2 + 2λk 2kqx, t iq x x, t 2kq x, t + iqx x, t 2ik 2 + iqx, tq x, t + iq 2 + 2λk + at ψ1 x, t + at t ψ 2 x, t 2ik 2 iqx, tq x, t iq 2 + 2λk 2kqx, t iq x x, t 2kq x, t + iqx x, t 2ik 2 + iqx, tq x, t iq 2 + 2λk i.e., 1.53 bt ψ1 x, t t ψ 2 x, t 2iA ψ1 x, t +bt 2iA ψ 2 x, t + at ψ1 x, t t ψ 2 x, t Taking x +, since ψx, t and ψx, t are linearly independent, so 1.54 Similarly, we can get at t =, bt t = 2iA bt. at bt 1.55 =, = 2iA bt. t t Therefore, both at and at are time independent, and 1.56 biq, t = biq, e 2iq2 +2λkt = e iθ1 e 2iq2 t and 1.57 b iq, t = b iq, e 2iq2 +2λkt = e iθ1 e 2iq2 t = e iθ1+2θ+ e 2iq2 t. References =. 1. Ablowit M J and Musslimani Z H, Integrable nonlocal nonlinear Schrödinger equation, Phys. Rev. Lett Ablowit M J and Musslimani Z H, Integrable discrete PT symmetric model, Phys. Rev. E Ablowit M J and Musslimani Z H, Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation, Nonlinearity, Ablowit M J and Musslimani Z H, Integrable nonlocal nonlinear equations, Studies in Applied Mathematics, 139, Issue 1 Pages Ablowit M J and Clarkson P A, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, Ablowit M J, Kaup D J, Newell A C, and Segur H, The Inverse Scattering Transform-Fourier Analysis for Nonlinear Problems, Stud. Appl. Math. 53, Ablowit M J and Segur H, Solitons and Inverse Scattering Transform, SIAM Studies in Applied Mathematics Vol. 4, SIAM, Philadelphia, PA, Ablowit M J Nonlinear Dispersive Waves, Asymptotic Analysis and Solitons, Cambridge University Press, Cambridge, Ablowit M J, Prinari B and Trubatch A D, Discrete and Continuous Nonlinear Schrödinger Systems, Cambridge University Press, Cambridge, 24 ψ1 x, t ψ 2 x, t ψ1 x, t ψ 2 x, t,

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48 48 MARK J. ABLOWITZ, XU-DAN LUO, AND ZIAD H. MUSSLIMANI 38. Prinari B, Ablowit M J, and Biondini G, Inverse scattering transform for the vector nonlinear Schrödinger equation with nonvanishing boundary conditions, J. Math. Phys., Vol , Regensburger A, Bersch C, Mir M-A, Onishchukov G, Christodoulides D N and Peschel U, Paritytime synthetic photonic lattices, Nature, Regensburger A, Miri M A, Bersch C, Näger J, Onishchukov G, Christodoulides D N, and Peschel U, Observation of Defect States in PT-Symmetric Optical Lattices, Phys. Rev. Lett. 11, Ruter C E, Makris K G, El-Ganainy R, Christodoulides D N, Segev M and Kip D, Observation of parity time symmetry in optics, Nature Phys. 6, Segev M, Optical spatial solitons, Opt. Quantum Electron. 3, Zakharov V E and Shabat A B, Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP 34, Zakharov VE and Shabat AB, Interaction between solitons in a stable medium, Sov. Phys. JETP 37, Zverovich E I, Boundary value problems in the theory of analytic functions in Hölder classes on Riemann surfaces, Russ. Math. Surv. 261, Department of Applied Mathematics, University of Colorado, Campus Box 526, Boulder, Colorado address: mark.ablowit@colorado.edu Department of Mathematics, State University of New York at Buffalo, Buffalo, NY address: xudanluo@buffalo.edu Department of Mathematics, Florida State University, Tallahassee, FL address: muslimani@math.fsu.edu

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