General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions
Bao-Feng Feng Xu-Dan Luo Mark J. Ablowitz Ziad H. Musslimani
nlin.SI 39A10, 35Q58

]General soliton solution to a nonlocal nonlinear Schrödinger
equation with zero and nonzero boundary conditions

^1 School of Mathematical and Statistical Sciences,
The University of Texas Rio Grande Valley, Edinburg, TX 78539
^2 Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260
^3 Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309-0526
^4 Department of Mathematics, Florida State University, Tallahasse, Florida 32306-4510















General soliton solutions to a nonlocal nonlinear Schrödinger (NLS) equation with PT-symmetry for both zero and nonzero boundary conditions are considered via the combination of Hirota's bilinear method and the Kadomtsev-Petviashvili (KP) hierarchy reduction method. First, general N-soliton solutions with zero boundary conditions are constructed. Starting from the tau functions of the two-component KP hierarchy, it is shown that they can be expressed in terms of either Gramian or double Wronskian determinants. On the contrary, from the tau functions of  single component KP hierarchy, general soliton solutions to the nonlocal NLS equation with nonzero boundary conditions are obtained. All possible soliton solutions to nonlocal NLS with Parity (PT)-symmetry for both zero and nonzero boundary conditions are found in the present paper.




02.30.Ik, 05.45.Yv





§ INTRODUCTION

The study of the nonlinear Schrödinger (NLS) equation and its multi-component generalization
lies at the forefront of research in applied mathematics and mathematical physics. This is the case since they have been recognized as generic models for describing the evolution of slowly varying wave packets in general nonlinear wave system <cit.>.
In 1972 Zakharov and Shabat found that the NLS equation possesses a Lax
pair and can be solved/linearized by inverse scattering <cit.>.


Motivated by these results in 1973 Ablowitz, Kaup, Newell and Segur
<cit.> generalized the linear operators used by Zakharov and Shabat; they
showed that the NLS, sine-Gordon<cit.>, modified-KdV, KdV all could be
solved/linearized by inverse scattering. Soon afterwards in 1974, AKNS
<cit.> developed a general framework to find integrable systems solvable by
what they termed the Inverse Scattering Transform (IST). The method
was associated with classes of equations (later called recursion operators)
and was applied to find solutions to the initial value problem with rapidly
decaying data on the line.

The NLS and coupled NLS equations arise in a variety of physical contexts such as nonlinear optics <cit.>
, Bose-Einstein condensates <cit.>, water waves <cit.>
and plasma physics <cit.>, amongst many others.

The idea in <cit.> was to consider the scattering problem


    v _x=Xv = (
    [    -ik q(x,t); r(x,t)     ik ]
    ) ,


    v_t=Tv=(
    [  2ik^2+iqr  -2kq-iq_x;  -2kr+ir_x -2ik^2-iqr ]
    ) .

The compatibility condition of the linear spectral problem (<ref>)–(<ref>)
leads to

    {[  iq_t=q_xx-2rq^2 ,; -ir_t=r_xx-2qr^2 . ]
    .

Under the standard symmetry reduction

    r(x,t)=σ q^∗(x,t) ,   σ = ± 1 ,

the system (<ref>)  gives the classical NLS
equation

    iq_t(x,t)=q_xx(x,t)-2σ q^∗(x,t) q^2(x,t) .

Recently, an integrable nonlocal NLS equation

    iq_t(x,t)=q_xx(x,t)-2σ q^∗(-x,t) q^2(x,t)

was found in <cit.> under a new symmetry reduction

    r(x,t)=σ q^∗(-x,t) ,  σ = ± 1 .

This nonlocal equation is shown to be related to an unconventional magnetic system <cit.>. It is parity-time (PT) symmetric, i.e., it is invariant under
the joint transformations of x → -x, t → t and
complex conjugation, thus is related to a hot research area of contemporary
physics <cit.>.
Due to this potential application, it has been shown that different symmetry reductions from the linear problem of general AKNS hierarchy and other integrable hierarchies can also lead to various types of new nonlocal equations, which have been extensively studied <cit.>–<cit.>.  Typical examples are the
reverse space-time nonlocal NLS equation and the reverse time nonlocal NLS
equation, the complex/real space-time Sine-Gordon equation <cit.>, the complex/real reverse
space-time mKdV equation <cit.>, the
nonlocal derivative NLS equation <cit.>, and the multi-dimensional
nonlocal Davey-Stewartson equation <cit.>. On
the other hand, corresponding to the classical semi-discrete NLS equation <cit.>, the nonlocal NLS
equation also admits its semi-discrete version <cit.> and
multi-component generalizations <cit.>.

It has been known that that the inverse scattering transform (IST) has been successfully developed for the NLS, coupled NLS, as well as their semi-discrete analogues <cit.>–<cit.>.
Moreover, the IST to solve the initial-value problem
with vanishing boundary condition for the nonlocal NLS equation has been
developed in <cit.>.
For pure soliton solutions corresponding to reflectionless potentials, one obtains
one- and two-soliton solutions; these solutions can have singularities. 
In particular, the decaying one-soliton solution is found to be <cit.>

    q=2(η +η̅)e^-2η̅x-4iη̅^2t+
    iθ/1-e^-2(η +η̅)x+4i(η ^2-η̅^2)t+i(θ +θ̅) .

This solution is singular at x=0 when η≠η̅ and  4(η^2-η̅^2)t + (θ +θ̅)= 2n π, n ∈ℤ. When η = η̅, θ +θ̅≠ 2n π, n ∈ℤ the solution is not singular. 
It is also noted that the same solution has been found in <cit.> via Darboux transformations and AKNS reductions, respectively.

The inverse scattering transform (IST) to solve the initial-value problem
with nonzero boundary conditions has also been developed in <cit.>. To be specific, under the following boundary
conditions

    q(x,t)→ q_±(t)=q_0e^i(α t+θ _±),  asx→±∞ ,

where q_0>0, 0≤θ _±<2π, the following cases are studied.

  
  * σ =-1, Δ=θ _+-θ _-=π: a one-soliton solution
is found:

    q=q_0e^i(2q_0^2t+θ _+-π )tanh [q_0x-i
    θ _∗]

where θ _∗=1/2(θ _++θ _1+π ), where θ _1 is related
to the associated scattering data (normalization constant). This
solution is stationary in space and oscillating in time.
  
  * σ =-1, Δ=0: in this case, there is no 'proper
exponentially decaying' pure one soliton solution. The simplest decaying
pure reflectioness potential generates a two-soliton solution.
  
  * σ =1, Δ=0: 
 in this case only an even number of soliton solutions arise; they are related to an even number (2N) of eigenvalues.
  
  * σ =1, Δ=π: in this case, there is no eigenvalues/solitons.

Regardless of the success of the IST for the nonlocal NLS equation, it is still not easy to come up with its general soliton solution in closed form by this method. On the other hand, Hirota's bilinear method is a direct and powerful method in finding multi-soliton solutions to soliton equations <cit.>. In the early 1980's, suggested by Hirota's direct method, Sato discovered that the soliton solutions to the Kadomtsev-Petviashvili (KP) equation and its hierarchy (named tau-functions afterwards) satisfy the so-called Plücker relations, and found that the totally of solutions to the KP hierarchy forms an infinite dimensional Grassmann manifold <cit.>. Later on, Ueno and Takasaki extended Sato's discovery to include two-dimensional Toda lattice (2DTL) hierarchy <cit.>; Date, Jimbo and Miwa further developed Sato's idea with transformation groups theory, and presented Lie algebraic classification of soliton equations <cit.>.


The KP-hierarchy reduction method was  first developed by the Kyoto school
<cit.>, and later was used to obtain soliton solutions in the NLS
equation, the modified KdV equation, the Davey-Stewartson equation and the
coupled higher order NLS equations <cit.>. Recently, this
method has been applied to derive dark-dark soliton solution in two-coupled
NLS equation of the mixed type <cit.>. Unlike the 
the inverse scattering transform method <cit.>, and Hirota's
bilinear method <cit.>, the KP-hierarchy reduction method starts
with the general KP hierarchy including the two-dimensional Toda-hierarchy
<cit.> and derives the general soliton solution in either
determinant or pfaffian form reduced directly the tau functions of the KP
hierarchy.
In spite of many successes of the KP-hierarchy reduction method, constructing general soliton solutions by this method to the nonlocal NLS equation remains an unsolved and challenging problem, especially for the nonzero boundary conditions. The difficulty lies in the simultaneous realization of both the nonlocal and complex conjugate reductions. Therefore, it motivates the present work, which intends to construct general soliton solutions in determinant forms to the nonlocal NLS equation with both the zero and nonzero boundary conditions.




 In the present paper, we find solitons that are both decaying and have nonzero boundary values.
For the decaying boundary condition, we construct the general N-soliton solutions expressed in both the Gram-type and the double-Wronskin determinants. The explicit one-soliton solution agrees with the solution found by the inverse scattering transform <cit.>, which is also listed in (<ref>). For the nonzero boundary condition, the general soliton solutions are constructed for all cases except for the case: σ =1, Δ=π where there is no soliton solution. To be specific, for the case of σ =-1, Δ=π, the solution (<ref>) is the same as above; for the case of σ =1, Δ=0, it is true that only the determinant soliton solution 2N × 2N can be constructed, which is consistent with the fact that only the even number of soliton solutions with an even number of eigenvalues exist. The simplest solution based on the tau-functions found here corresponds to (5.90) in <cit.> but with a simpler form. In the last, for the case of σ =-1, Δ=0, the soliton solution (<ref>) found in this paper is the same as the one found <cit.>.

The outline of the present paper is organized as follows. In section 2, we
construct the bright soliton solution to the nonlocal NLS equation with a zero boundary condition.
Both the Gram-type and double Wronski-type solution start 
from the tau functions in Gram-type or double Wronski-type of two-component KP hierarchy.
In section 3, general multi-soliton solutions for the nonlocal NLS equation of σ=1  and of σ=-1 are constructed, which recover all possible solutions found in <cit.>.
The paper concludes with 
comments and remarks in section 4.


§ GENERAL SOLITON SOLUTION WITH ZERO BOUNDARY CONDITION

In this section, we consider general soliton solutions to the nonlocal NLS equation (<ref>).



under the boundary condition q(x,t) → 0 as x →±∞. This type of soliton solution is usually called bright soliton solution. We first give the bilinear forms for  the nonlocal NLS equation (<ref>).
By introducing the dependent variable transformation

    q(x,t)=g(x,t)/f(x,t),

the nonlocal NLS equation (<ref>)  is converted into the following
bilinear equations

    {[              (iD_t-D_x^2)g(x,t)· f(x,t)=0,; D_x^2f(x,t)· f(x,t)+2σ g(x,t)g^∗(-x,t)=0 , ]
    .

where the Hirota's bilinear operator is defined as

    D_x^nD_t^lf(x,t)· g(x,t)=( ∂/∂ x-
    ∂/∂ x^') ^n( ∂/
    ∂ t-∂/∂ t^')
    ^lf(x,t)g(x^',t^')|_x=x^',t=t^'.


Similar to the classical NLS equation, the bright soliton solutions to the nonlocal NLS can be obtained from the tau functions of two-component KP hierarchy. It is known that the tau functions have two
different forms: one is the Gram-type, the other is the Wronski-type. They are basically equivalent, however, each form has advantage and disadvantage in view of different symmetries. In what follows, we will derive two types of soliton solutions via KP hierarchy reduction method.


 §.§ General bright soliton solution expressed by Gramian determinant

In this subsection, we will construct soliton solution in terms of Gram-type determinant.  To this end, let us start with a Gram-type determinant
expression of the tau functions for two-component KP hierarchy,

    τ _0=| A| ,



    τ _1=|[    A Φ ^T;  -Ψ̅    0 ]
    |,   τ _-1=|[    A Ψ ^T;  -Φ̅    0 ]
    |

where the elements of matrix A are

    a_ij=1/p_i+p̅_je^ξ _i+ξ̅_j+1/q_i+
    q̅_je^η _i+η̅_j

with

    ξ _i=p_ix_1+p_i^2x_2+ξ _i0,  ξ̅_j=p̅
    _jx_1-p̅_j^2x_2+ξ̅_j0,



    η _i=q_iy_1+η _i0,  η̅_j=q̅
    _jy_1+η̅_j0,

and Φ, Φ̅, Ψ, Ψ̅ are row vectors defined by

    Φ =( e^ξ _1,⋯ ,e^ξ _N)  ,  Ψ =(
    e^η _1,⋯ ,e^η _N)  ,


    Φ̅=( e^ξ̅_1,⋯ ,e^ξ̅_N)  ,  Ψ̅=( e^η̅_1,⋯ ,e^η̅_N)  .

It can be shown below that the tau functions given above satisfy the following two bilinear
equations

    {[      (D_x_2-D_x_1^2)τ _1·τ _0=0 ,; D_x_1D_y_1τ _0·τ _0=-2τ _1τ
    _-1 . ]
    .

The proof is given as follows.  Since

    ∂ _x_1τ _0=|[    A Φ ^T;  -Φ̅    0 ]
    |,   ∂ _x_1∂ _y_1τ _0=|[    A Φ ^T Ψ ^T;  -Φ̅    0    0;  -Ψ̅    0    0 ]
    | ,

then based on the Jacobi identify for the determinant, we have

    (2∂ _x_1∂ _y_1)τ _0×τ _0=2(∂
    _x_1τ _0)× (∂ _x_1τ _y)-2τ _1×τ
    _-1 ,

which is exactly the second bilinear equation.

Moreover, we can easily verify the following relations

    ∂ _x_2τ _0=|[        A Φ _x_1^T;      -Φ̅        0 ]
    | -|[        A Φ _x_1^T;  -Φ̅_x_1        0 ]
    | ,



    ∂ _x_1^2τ _0=|[        A Φ _x_1^T;      -Φ̅        0 ]
    | +|[        A Φ _x_1^T;  -Φ̅_x_1        0 ]
    | ,   ∂ _x_1τ _1=|[        A Φ _x_1^T;      -Ψ̅        0 ]
    | ,

and

    ∂ _x_1^2τ _1=|[        A     Φ ^T Φ _x_1^T;      -Φ̅        0        0;      -Ψ̅        0        0 ]
    | +|[           A Φ _x_1x_1^T;         -Ψ̅           0 ]
    | ,



    ∂ _x_2τ _1=-|[        A     Φ ^T Φ _x_1^T;      -Φ̅        0        0;      -Ψ̅        0        0 ]
    | +|[           A Φ _x_1x_1^T;         -Ψ̅           0 ]
    | ,

where Φ _x_1 and Φ _x_1x_1 represent the first and second derivatives of Φ respect to
x_1, respectively. Therefore we have

    (∂ _x_2-∂ _x_1^2)τ _0=2|[        A Φ _x_1^T;      -Ψ̅        0 ]
    | ,



    (∂ _x_2-∂ _x_1^2)τ _1=-2|[        A     Φ ^T Φ _x_1^T;      -Φ̅        0        0;      -Ψ̅        0        0 ]
    | .

From the Jacobi identity for the determinant, we have

    ((∂ _x_2-∂ _x_1^2) τ _1)×τ _0=((∂
    _x_2-∂ _x_1^2)τ _0) ×τ _1-2(∂
    _x_1τ _1)× (∂ _x_1τ _f
    0)

which gives the first bilinear equation. The proof is done.

Next, we show the reduction processes. By row
operations τ _0 can be rewritten as

    τ _0=∏_j=1^Ne^ξ _j+ξ̅_j|
    a_ij^'|_1≤ i,j ≤ N =∏_j=1^Ne^ξ _j+ξ̅
    _j|1/p_i+p̅_j+1/q_i+q̅_j
    e^(η _i-ξ _i)+(η̅_j-ξ̅_j)|

where

    η _i-ξ _i=q_iy_1-p_ix_1+⋯ ,
      η̅_j-ξ̅_j=q̅_jy_1-p̅_jx_1+⋯ .

Thus if we impose constraints on parameters

    q_j=p_j ,  q̅_j=p̅_j ,   j=1,⋯ ,N,

then the following relation holds

    (∂ _x_1+∂ _y_1)a_ij^'=0

which implies

    ∂ _x_1τ _0=-∂ _y_1τ _0.

Consequently, the second bilinear equation in (<ref>) becomes

    D_x_1^2τ _0·τ _0=2τ _1τ _-1 .

Due to the dimension reduction (<ref>),
y_1 becomes a dummy variable, which can be treated as zero.

We then have the following Lemma.

Assume x_1=x is real and x_2=-it is
purely imaginary variable. If p_j and p̅_j (j=1,⋯, N) are real, η_j0, η̅_j0 are puly imaginary, or the subset of (p_j, η_j0) or (p̅_j, η̅_j0) occurs in pair such that p_k=p^*_k', η_k0=η^*_k'0 or p̅_k=p̅^*_k', η̅_k0=η̅^*_k'0, then

    τ _0(x,t)=C τ^∗ _0(-x,t)   τ_-1(x,t)=C τ^∗ _1(-x,t) ,

where C=C_1 C_2, C_1=∏_j=1^N e^(ξ _j+ξ̅_j),
C_2=∏_j=1^N e^(η _j+η̅_j).

  First, we prove the case when p_i and p̅_i are
all real, η_i0, η̅_i0 are all purely imaginary. Under
this case,

    ξ _i=p_ix-ip_i^2t ,
      η _i=iθ_i ,


    ξ̅_j=p̅_jx+ip̅_j^2t ,   η̅_j= iθ̅_j ,

it then follows

    ξ _i^∗(-x,t)=-ξ _i(x,t) ,   ξ̅_i^∗(-x,t)=-ξ̅_i(x,t) .

We note that τ _0(x,t) can be rewritten as

    τ _0(x,t)    =   ∏_j=1^N e^(ξ _j+ξ̅_j)|1/p_i+
    p̅_j(1+e^-(p_i+p̅_j)x+i(p_i^2-p̅_j^2)t+i(θ _i+θ̅_j))| , 
       =   ∏_j=1^N e^i(θ _j+θ̅_j)|1
    /p_i+p̅_j(1+e^(p_i+p̅_j)x-i(p_i^2-
    p̅_j^2)t-i(θ _i+θ̅
    _j))| ,

which implies

    τ _0^∗(-x,t)= ∏_j=1^N e^-(ξ_j+ξ̅_j)|1/p_i+p̅_j
    (1+e^(p_i+p̅_j)x-i(p_i^2-p̅_j^2)t-
    i(θ _i+θ̅_j))| .

Therefore τ _0(x,t)=C τ^∗ _0(-x,t).
On the other hand,

    τ _1(x,t)    =   |[ 1/p_i+p̅_j(e^(p_i+p̅_j)x-i
    (p_i^2-p̅_j^2)t+e^i(θ _i+θ̅_j))                                             e^p_ix-ip_i^2t;                                                   -e^iθ̅_j                                                          0 ]
    | , 
       =   ∏_j=1^N e^i(θ _j+θ̅_j)|[ 1/p_i+p̅_j(1+e^ξ _i+ξ̅_j-i(θ _i+θ̅_j))                          e^ξ _i -iθ _i;                                     -1                                      0 ]
    | , 
       =   ∏_j=1^N e^(ξ _j+ξ̅_j)|[ 1/p_i+p̅_j(1+e^-ξ _i-ξ̅_j+i(θ _i+θ̅_j))                                       1;                         -e^-ξ̅_j+i
    θ̅_j                                       0 ]
    | ,



    τ _-1(x,t)    =   |[ 1/p_i+p̅_j(e^(p_i+p̅_j)x-i
    (p_i^2-p̅_j^2)t+e^i(θ _i+θ̅_j))                                                    e^iθ _i;                                          -e^p̅_jx+ip̅_j^2t                                                          0 ]
    | , 
       =   ∏_j=1^N e^(ξ _j+ξ̅_j)|[ 1/p_i+p̅_j(1+e^-ξ _i-ξ̅_j+i(θ _i+θ̅_j))                          e^-ξ _i +iθ _i;                                      -1                                       0 ]
    | .

Therefore

    τ^∗_1(-x,t)
       =   ∏_j=1^N e^(ξ _j+ξ̅_j)^∗(-x,t) |[ 1/p_i+p̅_j(1+e^ξ _i+ξ̅_j-i(θ _i+θ̅_j))                                      1;                         -e^ξ̅_j-i
    θ̅_j                                      0 ]
    | , 
       =   ∏_j=1^N e^-i(θ _j+θ̅_j)|[ 1/p_i+p̅_j(1+e^-ξ _i-ξ̅_j+i(θ _i+θ̅_j))                            e^-ξ_i+iθ _i;                                      -1                                       0 ]
    | .

Obviously τ_-1(x,t)=C τ^∗ _1(-x,t).
Next we prove the case where there are pairs of wave numbers p_k,p_k^',
p̅_k,p̅_k^' such that p_k=p_k^'^∗,
p̅_k=p̅_k^'^∗. Moreover, η
_k^'0=-η_k0^∗, η̅_k^'0=-η̅_k0^∗.
Note that p_k,p_k^' or p̅_k,p̅_k^' being real; η_k^'0, η_k0 or η̅_k^'0, η̅_k0 being purely imaginary is simply a special case. Under this case, we
also have ξ _k^∗(-x,t)=-ξ _k'(x,t),ξ̅_k^∗(-x,t)=-ξ̅_k'(x,t), thus

    τ _0(x,t)    =   |[ 1/p_k+p̅_k^'(e^ξ _k+ξ̅_k^'+e^η_k0+η̅_k^'0)                                          ⋯;                                          ⋯ 1/p_k^'+p̅_k(e^ξ _k^'+ξ̅_k+e^η_k^'0+η̅_k0) ]
    |
       =   C_2|[ 1/p_k+p̅_k^'(1+e^ξ _k+ξ̅_k^'-
    η_k0-η̅_k^'0)                                           ⋯;                                           ⋯   1/p_k^'+p̅_k(1+e^ξ _k^'+ξ̅_k-η_k^'0+η̅_k0 ]
    |
       =   C_1|[  1/p_k+p̅_k^'(1+e^-(ξ _k+ξ̅_k^'-η_k0-η̅_k^'0))                                              ⋯;                                              ⋯ 1/p_k^'+p̅_k(1+e^-(ξ _k^'+
    ξ̅_k-η_k^'0-η̅_k0)) ]
    |
       =   C_1|[ 1/p_k^'+p̅_k(1+e^-(ξ _k^'+ξ̅
    _k-η_k^'0-η̅_k0))                                              ⋯;                                              ⋯ 1/p_k+p̅_k^'(1+e^-(ξ _k+ξ̅
    _k^'-η_k0-η̅_k^'0)) ]
    | .

Therefore

    τ _0^∗(-x,t)=C_1^-1|[     1/p_k+p̅_k^'(1+e^ξ _k+ξ̅_k^'-
    η_k0-η̅_k0))                                              ⋯;                                              ⋯ 1/p_k^'+p̅_k(1+e^-(ξ _k+ξ̅
    _k^'-η_k0-η̅_k^'0)) ]
    | .

Obviously τ_0(x,t)=C τ^∗ _0(-x,t). On the other hand,

    τ _1(x,t)    =   |[ 1/p_k+p̅_k^'(e^ξ _k+ξ̅_k^'+e^η_k0+η̅_k^'0)                                          ⋯                                     e^ξ _k;                                          ⋯ 1/p_k^'+p̅_k(e^ξ _k^'+ξ̅_k+e^η_k^'0+η̅_k0)                                   e^ξ _k^';                                 -e^η̅_k^'0                                   -e^η̅_k0                                          0 ]
    |
       =   C_2|[ 1/p_k+p̅_k^'(1+e^ξ _k+ξ̅_k^'-
    η_k0-η̅_k^'0)                                           ⋯                                 e^ξ _k-η_k0;                                           ⋯  1/p_k^'+p̅_k(1+e^ξ _k^'+ξ̅_k-η_k^'0-η̅_k0)                            e^ξ _k^'-
    η_k^'0;                                          -1                                          -1                                           0 ]
    |
       =   C_1|[  1/p_k+p̅_k^'(1+e^-(ξ _k+ξ̅_k^'-η_k0-η̅_k^'0))                                              ⋯                                              1;                                              ⋯ 1/p_k^'+p̅_k(1+e^-(ξ _k^'+
    ξ̅_k-η_k^'0-η̅_k0))                                              1;                                     -e^η̅_k^'0                                       -e^η̅_k0                                              0 ]
    |
       =   C_1|[ 1/p_k^'+p̅_k(1+e^-(ξ _k^'+ξ̅
    _k-η_k^'0-η̅_k0))                                              ⋯                                              1;                                              ⋯ 1/p_k+p̅_k^'(1+e^-(ξ _k+ξ̅
    _k^'-η_k0-η̅_k^'0))                                              1;                                       -e^η̅_k0                                     -e^η̅_k^'0                                              0 ]
    | ,



    τ _-1(x,t)    =   |[ 1/p_k+p̅_k^'(e^ξ _k+ξ̅_k^'+e^η_k0+η̅_k^'0)                                          ⋯                                     e^η_k0;                                          ⋯ 1/p_k^'+p̅_k(e^ξ _k^'+ξ̅_k+e^η_k^'0+η̅_k0)                                  e^η
    _k^'0;                                  -e^ξ̅_k^'                                    -e^ξ̅_k                                          0 ]
    |
       =   C_1|[ 1/p_k^'+p̅_k(1+e^-(ξ _k^'+ξ̅
    _k-η_k^'0-η̅_k0))                                              ⋯                               e^-ξ
    _k^'+η_k^'0;                                              ⋯ 1/p_k+p̅_k^'(1+e^-(ξ _k+ξ̅
    _k^'-η_k0-η̅_k^'0))                                  e^-ξ _k+
    η_k0;                                             -1                                             -1                                              0 ]
    | .

Thus

    τ _-1^∗(-x,t)=C_1^-1|[ 1/p_k+p̅_k^'(1+e^ξ _k+ξ̅_k^'-
    η_k0-η̅_k^'0)                                           ⋯                                 e^ξ _k-η_k0;                                           ⋯  1/p_k^'+p̅_k(1+e^ξ _k^'+ξ̅_k-η_k^'0-η̅_k0)                             e^ξ _k^'-η_k^'0;                                          -1                                          -1                                           0 ]
    | .

Consequently, τ _-1(-x,t)=C τ _1^∗(x,t).

Following above Lemma, if we define τ_0(x,t)=√(C)f(x,t), τ_1(x,t)=√(C)g(x,t), τ_-1(x,t)=√(C)g̅(x,t), then we have f(x,t)=f^∗(-x,t), g̅(x,t)=g^∗(-x,t), which lead to the following bilinear equations

    {[             (i D_t-D_x^2)g(x,t) · f(x,t)=0 ,; D^2_xf(x,t) · f(x,t) -2g(x,t) g^∗ (-x,t)=0 . ]
    .

Moreover, if we define


    q(x,t)=g(x,t)/f(x,t),    r(x,t)= -g̅(x,t)/f(x,t) ,

we then have


    r(x,t)=-q^∗(-x,t).

In summary, we have the general N-bright soliton solution (<ref>) to the nonlocal NLS equation
(<ref>) with σ=-1.

    f(x,t)= 1/√(C)|1/p_i+p̅_j(e^(p_i+p̅_j)x-i
    (p_i^2-p̅_j^2)t+e^(η _i+η̅_j))|_N × N,,


    g(x,t) =1/√(C)|[ 1/p_i+p̅_j(e^(p_i+p̅_j)x-i
    (p_i^2-p̅_j^2)t+e^(η _i+η̅_j))                                            e^p_ix-ip_i^2t;                                                   -e^η̅_j                                                         0 ]
    |_(N+1) × (N+1) .

In what follows, we list one- and two-soliton solutions: 

One-soliton solution:

    f(x,t)=1/
    p_1+p̅_1 e^-1/2 (ξ _1+ξ̅_1-iθ _1-iθ̅_1)( 1+e^(p_1+p̅_1)x-i(p_1^2-
    p̅_1^2)t-i(θ _1+θ̅
    _1))



    g(x,t)=e^-1/2 (ξ _1+ξ̅_1+ iθ _1+ iθ̅_1) e^p_1x-i p_1^2t+iθ̅_1 .

So

    q(x,t)=(p_1+p̅_1)e^p_1x- i p_1^2t-iθ_1/1+e^(p_1+p̅_1)x-i(p_1^2-
    p̅_1^2)t-i(θ _1+θ̅
    _1) .

 if we let p_1=-2η̅, p̅_1=-2 η, θ_1=-θ̅+π, θ̅_1=-θ, then above solution exactly recovers the one-soliton solution found in <cit.>, also mentioned in the introduction.

Two-soliton solution:

    f    =   |[  1/p_1+p̅_1(e^ξ _1+ξ̅_1+e^η_10+η̅_10) 1/p_1+p̅_2(e^ξ _1+ξ̅_2+e^
    η_10+η̅_20);  1/p_2+p̅_1(e^ξ _2+ξ̅_1+e^η_20+η̅_10) 1/p_2+p̅_2(e^ξ _2+ξ̅_2+e^
    η_20+η̅_20) ]
    |
       =   D ( 1+e^ξ _1+ξ̅_1+ξ _2+ξ̅_2-η_10-
    η̅_10-η_20-η̅_20+(p_1+p̅
    _2)(p_2+p̅_1)/(p_1-p_2)(p̅_1-p̅_2)(
    e^ξ _1+ξ̅_1-η_10-η̅_10+e^ξ _2+ξ̅
    _2-η_20-η̅_20) . 
          . +(p_1+p̅_1)(p_2+p̅_2)/(p_1-p_2)(p̅_1-
    p̅_2)( e^ξ _1+ξ̅_2-η_10-η̅
    _20+e^ξ _2+ξ̅_1-η_20-η̅_10) ) ,



    g    =   |[  1/p_1+p̅_1(e^ξ _1+ξ̅_1+e^η_10+η̅_10) 1/p_1+p̅_2(e^ξ _1+ξ̅_2+e^
    η_10+η̅_20)                                e^ξ _1;  1/p_2+p̅_1(e^ξ _2+ξ̅_1+e^η_20+η̅_10) 1/p_2+p̅_2(e^ξ _2+ξ̅_2+e^
    η_20+η̅_20)                                e^ξ _2;                              -e^η̅_10                              -e^η̅_20                                     0 ]
    |
       =   D( (p_1+p̅_1)(p_2+p̅_1)/p̅_1-p̅
    _2e^ξ _1+ξ _2+ξ̅_2-η_10-η_20-η̅
    _20+(p_1+p̅_1)(p_1+p̅_2)/p_1-p_2e^ξ _1-
    η_10) 
          -( (p_1+p̅_2)(p_2+p̅_2)/p̅_1-p̅
    _2e^ξ _1+ξ _2+ξ̅_2-η_10-η_20-η̅
    _20+(p_2+p̅_1)(p_2+p̅_2)/p_1-p_2e^ξ _2-
    η_20) ,

where

    D=√(C_2/C_1)(p_1-p_2)(p̅_1-p̅_2)/(p_1+p̅_1)(p_1+
    p̅_2)(p_2+p̅_1)(p_2+p̅_2) .

Two soliton solution could have the following cases:

  
  * p_1,p_2,p̅_1,p̅_2 are all real and η_10,
η_20,η̅_10,η̅_20 are all purely imaginary.
  
  * p_1,p_2,p̅_1,p̅_2, η_10,η_20,η̅_10,η̅_20 are all complex, p_1=p_2^∗,p̅
_1=p̅_2^∗,η_10=-η_20^∗,η̅_10=
η̅_20^∗.
  
  * p_1,p_2, η_10,η_20 are complex number, p_1=p_2^∗,η_10=-η_20^∗; p̅_1,p̅
_2 are real and η̅_10,η̅_20 are all purely imaginary.
  
  * p̅_1,p̅_2, η̅_10,η̅_20 are complex
number, p̅_1=p̅_2^∗,η̅_10=η̅
_20^∗; p_1,p_2 are real and η_10,η_20 are
all purely imaginary.

It is noted that all above four cases regarding the bright soliton solution to the nonlocal NLS equation have been recognized and their dynamics have been detailed discussed in <cit.>.





 §.§ General bright soliton solution expressed by double Wronskian determinant

Alternatively, we can also present the general bright soliton solution to the nonlocal NLS equation (<ref>) in terms of the double Wronskian determinant. To this end, we start with
the tau-functions for two-component KP hierarchy expressed in double Wronskian determinant

    τ _N,M=|[     ϕ _1^(0)            ⋯   ϕ _1^(N-1)     ψ _1^(0)            ⋯   ψ _1^(M-1);     ϕ _2^(0)            ⋯   ϕ _2^(N-1)     ψ _2^(0)            ⋯   ψ _2^(M-1);            ⋮            ⋮            ⋮            ⋮            ⋮            ⋮;   ϕ _N+M^(0)            ⋯ ϕ _N+M^(N-1)   ψ _N+M^(0)            ⋯ ψ _N+M^(M-1) ]
    | _(N+M)× (N+M) ,

here ϕ _i^(n) and ψ _i^(n) take the form

    ϕ _i^(n)=p_i^ne^ξ _i,   ψ _i^(n)=q_i^ne^η
    _i ,

with

    ξ _i=p_ix_1+p_i^2x_2+ξ _0i+⋯ ,   η _i=q_iy_1+η _0i+⋯ .


The above tau functions satisfy the following bilinear equations

    {[               (D_x_2-D_x_1^2)τ _N+1,N-1·τ _N,N=0,;         [5pt]
    (D_x_2-D_x_1^2)τ _N,N·τ
    _N-1,N+1=0,; D_x_1D_y_1τ _N,N·τ _N,N-2τ
    _N+1,N-1τ _N-1,N+1=0 . ]
    .

The proof of above equations can be done by using the determinant
technique <cit.>,which is omitted here. In what follows, we will
perform reduction procedure from bilinear equations (<ref>) to the bilinear equations (<ref>). For the sake of convenience,
we take the following abbraevation

    τ _N,M =| 0,⋯ ,N-1,0^^',⋯
    ,(M-1)^^'| ,

First, we consider the dimension reduction. Imposing the condition N=M and  q_i=p_i (i=1,⋯ ,2N), we then have

    (∂ _x_1+∂ _y_1)τ _N,M=(
    ∏_j=1^2Np_j) τ _N,M .

Under this reduction, y_1 becomes a dummy variable, which can be taken
as zero. Applying variable transformations

    x_1=x,   x_2=-it ,

which implies

    ∂ _x_1=∂ _x,  ∂ _x_2=i∂
    _t .

Then the bilinear equations (<ref>) can be rewritten as

    {[             (iD_t-D_x^2)τ _N+1,N-1·τ _N,N=0,;             (iD_t-D_x^2)τ _N,N·τ
    _N-1,N+1=0,; D_x^2τ _N,N·τ _N,N+2τ _N+1,N-1τ
    _N-1,N+1=0 , ]
    .



Assume x_1=x is real and x_2=-it is purely imaginary variable. Suppose there are K pairs of wave
numbers (p_k, η_k0) and (p̅_k, η̅_k0) for k=1, ⋯, K such that
 p_k=p^*_k', η_k0=η^*_k'0 or p̅_k=p̅^*_k', η̅_k0=η̅^*_k'0; and the rest of wave numbers p_j and p̅_j are real, η_j0, η̅_j0 are purely imaginary. Then

    τ _N,N(x,t) = C' (-1)^N+Kτ^∗ _N,N(-x,t) ,


    τ _N-1,N+1(x,t) = C' (-1)^N+K+1τ^∗ _N+1,N-1(-x,t) ,

where C'=∏^N_i=1 e^ξ_i(x,t)-η _i.








We only give the proof when p_j and p̅_j are real and η_j0, η̅_j0 are purely imaginary. Under this case,
 ξ _i(x,t)=p_ix-ip_i^2t=-ξ _i^∗(-x,t), then it can be shown

    τ _N-1,N+1(x,t)    =   | 0,⋯ ,N-2,0^^'⋯
    ,,N^^'| , 
       =   | e^ξ _i(x,t),⋯ ,p_i^N-2e^^ξ
    _i(x,t),e^-iθ _i,⋯ ,p_i^Ne^-iθ
    _i| , 
       =   ∏^N_i=1 e^ξ _i(x,t)-iθ _i| e^
    iθ _i,⋯ ,p_i^N-2e^iθ _i,e^-ξ
    _i(x,t),⋯ ,p_i^Ne^-ξ _i(x,t)| , 
       =    (-1)^N ∏^N_i=1 e^ξ _i(x,t)-iθ _i| e^-ξ _i(x,t),⋯ ,p_i^Ne^-ξ
    _i(x,t),e^iθ _i,⋯ ,p_i^N-2e^iθ
    _i| , 
       =   (-1)^N ∏^N_i=1 e^ξ _i(x,t)-iθ _i| e^ξ _i^∗(-x,t),⋯
    ,p_i^Ne^ξ _i^∗(-x,t),e^iθ _i,⋯
    ,p_i^N-2e^iθ _i| , 
       =    (-1)^N+1 C' τ^∗ _N+1,N-1(-x,t) .


    τ _N,N(x,t)   =   | 0,⋯ ,N-1,0^^',⋯
    ,(N-1)^^'| ,  
       =   | e^ξ _i(x,t),⋯ ,p_i^N-1e^ξ
    _i(x,t),e^-iθ _i,⋯ ,p_i^N-1e^-i
    θ _i| ,  
       =   ∏^N_i=1 e^ξ_i(x,t)-iθ _i| e^iθ _i,⋯
    ,p_i^N-1e^iθ _i,e^-ξ _i(x,t),⋯
    ,p_i^N-1e^-ξ _i(x,t)| ,  
       =   (-1)^N+1∏^N_i=1 e^ξ_i(x,t)-iθ _i| e^-ξ _i(x,t),⋯ ,p_i^N-1e^-ξ
    _i(x,t),e^iθ _i,⋯ ,p_i^N-1e^iθ
    _i| ,  
       =    (-1)^N∏^N_i=1 e^ξ_i(x,t)-iθ _i| e^ξ
    _i^∗(-x,t),⋯ ,p_i^N-1e^ξ _i^∗(-x,t),e^
    iθ _i,⋯ ,p_i^N-1e^iθ _i| ,  
       =    (-1)^N+1 C' τ^∗ _N,N(-x,t) .

For the case of complex conjuate wavenumbers pairs, it can be approved the same way as the Lemma in previous subsection. So we omit the proof here.

Therefore if we define τ_N,N(x,t)=√(C')f(x,t), τ_N+1,N-1(x,t)=√(C')g(x,t), τ _N-1,N+1(x,t)=-√(C')g̅(x,t), then the following bilinear equations follow

    {[             (i D_t-D_x^2)g(x,t) · f(x,t)=0 ,; D^2_xf(x,t) · f(x,t) -2g(x,t) g^∗ (-x,t)=0 . ]
    .

Moreover, if we define


    q(x,t)=g(x,t)/f(x,t),    r(x,t)= g̅(x,t)/f(x,t) ,

we then have


    r(x,t)=-q^∗(-x,t).

To summarize, we have the general N-bright soliton solution
expressed by 
    q(x,t)=g'(x,t)/f'(x,t),

 where

    f'(x,t)=|[    ϕ̃_1^(0)           ⋯  ϕ̃_1^(N-1)    ψ̃_1^(0)           ⋯  ψ̃_1^(N-1);    ϕ̃_2^(0)           ⋯  ϕ̃_2^(N-1)    ψ̃_2^(0)           ⋯  ψ̃_2^(N-1);           ⋮           ⋮           ⋮           ⋮           ⋮           ⋮;   ϕ̃_2N^(0)           ⋯ ϕ̃_2N^(N-1)   ψ̃_2N^(0)           ⋯ ψ̃_2N^(N-1) ]
    | _2N× 2N ,


    g'(x,t)=|[    ϕ̃_1^(0)           ⋯    ϕ̃_1^(N)    ψ̃_1^(0)           ⋯  ψ̃_1^(N-2);    ϕ̃_2^(0)           ⋯    ϕ̃_2^(N)    ψ̃_2^(0)           ⋯  ψ̃_2^(N-2);           ⋮           ⋮           ⋮           ⋮           ⋮           ⋮;   ϕ̃_2N^(0)           ⋯   ϕ̃_2N^(N)   ψ̃_2N^(0)           ⋯ ψ̃_2N^(N-2) ]
    | _2N× 2N ,

here ϕ̃_i^(n) and ψ̃_i^(n) take the form

    ϕ̃_i^(n)=p_i^ne^p_ix - i p_i^2 t,   ψ̃_i^(n)=p_i^ne^-iθ_i .

In what follows, we will illustrate
one- and two-soliton solutions and make some comments. By taking N=1, we
get the tau functions for one-soliton solution,

    f'(x,t)    =   |[   e^ξ _1 e^-iθ _1;   e^ξ _2 e^-iθ _2 ]
    |  =e^ξ _1-iθ _2(1-e^iθ _2-
    iθ _1e^ξ _2-ξ _1)



    g'(x,t)    =   |[    e^ξ _1 p_1e^ξ _1;    e^ξ _2 p_2e^ξ _2 ]
    |
    =(p_2-p_1)e^ξ _2+ξ _1 .

The above tau functions lead to the one-soliton as follows

    q(x,t)    =   (p_2-p_1)e^ξ _2+ξ _1/e^ξ _1-iθ
    _2(1-e^iθ _2-iθ _1e^ξ _2-ξ _1)
     =(p_2-p_1)e^ξ _2+iθ _2/(1-e^i
    θ _2-iθ _1e^ξ _2-ξ _1)

 which can be shown to be exactly the same as the one-soliton solution obtained in previous subsection, thus the same solution found in <cit.>.



 


§ SOLITON SOLUTION TO THE NONLOCAL NLS EQUAOTION WITH NONZERO BOUNDARY CONDITION




In this section, we consider the general soliton solution to the nonlocal NLS equation (<ref>) with  the same nonzero boundary condition as considered in <cit.>

    q(x,t)→ρ e^iθ_± ,  (ρ>0)   as  x→±∞ ,

where Δθ =θ_+-θ_- is either 0 or π.

Similar to the classical NLS equation, to construct soliton solutions of the nonlocal NLS equation with NZBCs,
we need to start with the tau functions for single-component KP hierarchy expressed in Gram-type determinants

    τ _k=| m_ij(k)| _1≤ i,j≤ N

where

    m_ij(k)=c_ij+1/p_i+p̅_j( -p_i/
    p̅_j) ^ke^ξ _i+ξ̅_j


    ξ _i   =   p_i^-1x_-1+p_ix_1+p_i^2x_2+ξ _0i+⋯
    ξ̅_i   =   p̅_i^-1x_-1+p̅_ix_1-
    p̅_i^2x_2+ξ̅_0i+⋯

Based on the Sato theory, the above tau functions satisfies the following
bilinear equations

    {[              (D_x_2-D_x_1^2)τ _k+1·τ _k=0,; (1/2D_x_1D_x_-1-1)τ _k·τ
    _k=-τ _k+1τ _k-1. ]
    .

As stated in the subsequent sections, above two set of bilinear equations will be the key in constructing soliton solutions to the nonlocal NLS equation.  We also assume c_ij=c_iδ _ij hereafter.


 §.§ General soliton solution to the nonlocal NLS equation of σ=1, Δθ=0 with nonzero boundary condition

The nonlocal NLS equation (<ref>) of σ=1 is converted into a set of bilinear equations

    {[                         (iD_t-D_x^2)g· f=0,; (D_x^2-2 ρ ^2)f· f=-2 ρ
    ^2g(x,t)g^∗(-x,t) , ]
    .

via a variable transformation

    q(x,t)=ρg(x,t)/f(x,t)e^i2 ρ ^2t

under the condition f(x,t)=f^∗ (-x,t).
In what follows, we will show how to reduce the bilinear equations (<ref>) to the bilinear equations
(<ref>) by the KP hierarchy reduction method.

Firstly, we perform the dimension reduction. Note that, by row operations, τ _k can be rewritten as

    τ _k=∏_j=1^Ne^ξ _j+ξ̅_j| m_ij^'| =∏_j=1^Ne^ξ _j+ξ̅_j|
    c_iδ _ije^-(ξ _i+ξ̅_j)+1/p_i+
    p̅_j( -p_i/p̅_j)
    ^k| .

Since

    (∂ _x_1- ρ ^2∂ _x_-1)m_ij^'=(p_i+p̅_j)( 1-ρ ^2/p_ip̅
    _j) c_iδ _ije^-(ξ _i+ξ̅_j),

thus, if we impose the constraints

    p_jp̅_j=ρ ^2,   j=1,⋯ ,N ,

then

    ∂ _x_1m_ij^'=ρ ^2∂
    _x_-1m_ij^',

which leads to

    ∂ _x_1τ _k=ρ ^2∂ _x_-1τ _k.

Therefore, the bilinear equations (<ref>) become

    {[                (D_x_2-D_x_1^2)τ _k+1·τ _k=0,; (D^2_x_1-2 ρ^2)τ _k·τ
    _k=-2 ρ^2τ _k+1τ _k-1. ]
    .

Furthermore, we assume x_1=x, x_2=-it and define

    τ _0(x,t)=Cf(x,t),τ _1(x,t)=Cg(x,t),τ _-1(x,t)=Cg̅(x,t),

with C=∏_j=1^Ne^ξ _j+ξ̅_j, then the bilinear equations become

    {[                     (iD_t-D_x^2)g(x,t)· f(x,t)=0,; (D_x^2-2ρ ^2)f(x,t)· f(x,t)=-2ρ ^2g(x,t)g̅(x,t) . ]
    .


Consider 2N × 2N matrices for f(x,t), g(x,t) and g̅(x,t). If p̅_j=p_j^∗ to be the complex conjugate of p_j, let c_N+j=-c_j^∗,

p_N+j=-p_j to be complex, for j=1,⋯, N, and ξ _0,N+j=ξ _0j to be real, then we have

    f(x,t)=f^∗(-x,t),   g̅(x,t)=g^∗(-x,t).



Since

    (ξ _i+ξ̅_i)(x,t)=(p_i+p_i^∗)x-(p_i^2-(p_i^∗)^2)it+ξ _0i ,

and

    (ξ _N+i+ξ̅_N+i)(x,t)    =   (p_N+i+p_N+i^∗)x-(p_N+i^2-(p_N+i^∗)^2)it+ξ _0,N+i
       =   -(p_i+p_i^∗)x-(p_i^2-(p_i^∗)^2)it+ξ
    _0i ,

so we have

    (ξ _N+i+ξ̅_N+i)(x,t)=(ξ _i+ξ̅_i)^∗(-x,t) ,    (ξ _i+ξ̅_i)(x,t)=(ξ _N+i+ξ̅_N+i)^∗(-x,t).

Note that

    f(x,t)    =   |[                  c_iδ _ije^-(ξ_i+ξ̅_j)+1/p_i+p_j^∗                                      1/p_i+p_N+j^∗;                                      1/p_N+i+p_j^∗ c_N+iδ _N+i,N+je^-(ξ _N+i+
    ξ̅_N+j)+1/p_N+i+p_N+j^∗ ]
    |_1≤ i,j ≤ N
       =   |[ c_N+iδ _N+i,N+je^-(ξ _N+i+ξ̅_N+j)+1/
    p_N+i+p_N+j^∗                                      1/p_N+i+p_j^∗;                                      1/p_i+p_N+j^∗                 c_iδ _ije^-(ξ_i+ξ̅
    _j)+1/p_i+p_j^∗ ]
    |
       =   |[   -c_i^∗δ _ije^-(ξ _N+i+ξ̅_N+j)-1/
    p_i+p_j^∗                               -1/p_i+p_N+j^∗;                               -1/p_N+i+p_j^∗ -c_N+i^∗δ _i,je^-(ξ_i+
    ξ̅_j)-1/p_N+i+p_N+j^∗ ]
    |
       =   |[   c_i^∗δ _ije^-(ξ _N+i+ξ̅_N+j)+1/
    p_i+p_j^∗                               1/p_i+p_N+j^∗;                               1/p_N+i+p_j^∗ c_N+i^∗δ _i,je^-(ξ_i+
    ξ̅_j)+1/p_N+i+p_N+j^∗ ]
    |

On the other hand,

    f(x,t)    =   |[                  c_iδ _ije^-(ξ_i+ξ̅_j)+1/p_i+p_j^∗                                      1/p_i+p_N+j^∗;                                      1/p_N+i+p_j^∗ c_N+iδ _N+i,N+je^-(ξ _N+i+
    ξ̅_N+j)+1/p_N+i+p_N+j^∗ ]
    |
       =   |[                 c_jδ _jie^-(ξ _j+ξ̅_i)+1/p_j+p_i^∗                                      1/p_j^∗+p_N+j;                                      1/p_N+i^∗+p_j c_N+jδ _N+j,N+ie^-(ξ _N+j+
    ξ̅_N+i)+1/p_N+i^∗+p_N+j ]
    | ,

thus

    f^∗(-x,t)=|[                c_jδ _jie^-(ξ _i+ξ̅_j)^∗(-x,t)+1/
    p_i+p_j^∗                                              1/p_i+p_N+j^∗;                                              1/p_N+i+p_j^∗ c_N+jδ _N+j,N+ie^-(ξ _N+i+
    ξ̅_N+j)^∗(-x,t)+1/p_N+i+p_N+j^∗ ]
    | .

Obviously f(x,t)=f^∗(-x,t). Next, let us proceed to prove g(x,t)=g^∗(-x,t). Note g(x,t)
can be written as

    g(x,t)    =   |[                     c_iδ _ij( -p_i/p_j^∗) e^-(ξ _i+
    ξ̅_j)+1/p_i+p_j^∗                                                       1/p_i+p_N+j^∗ ,;                                                         1/p_N+i+p_j^∗ c_N+iδ _N+i,N+j( -
    p_N+i/p_N+j^∗) e^-(ξ _N+i+ξ̅_N+j)+1/
    p_N+i+p_N+j^∗ ]
    |, 
       =   |[ c_N+iδ _N+i,N+j( -p_N+i/p_N+j^∗)
    e^-(ξ _N+i+ξ̅_N+j)+1/p_N+i+p_N+j^∗                                                    1
    /p_N+i+p_j^∗ ,;                                                       1/p_i+p_N+j^∗                  c_iδ _ij( -p_i/
    p_j^∗) e^-(ξ _i+ξ̅_j)+1/
    p_i+p_j^∗ ]
    |, 
       =   |[              -c_i^∗δ _ij( -p_i/p_j^∗)
    e^-(ξ _N+i+ξ̅_N+j)-1/p_i+p_j^∗                                                     -1/
    p_i+p_N+j^∗,;                                                       -1/p_N+i+p_j^∗ -c_N+i^∗( -p_N+i/
    p_N+j^∗) δ _N+i,N+je^-(ξ _i+ξ̅_j)-
    1/p_N+i+p_N+j^∗ ]
    |, 
       =   |[              c_i^∗δ _ij( -p_i/p_j^∗) e^-(ξ
    _N+i+ξ̅_N+j)+1/p_i+p_j^∗                                                      1/
    p_i+p_N+j^∗;                                                       1/p_N+i+p_j^∗ c_N+i^∗( -p_N+i/
    p_N+j^∗) δ _N+i,N+je^-(ξ _i+ξ̅_j)+
    1/p_N+i+p_N+j^∗ ]
    |,

thus

    g^∗(-x,t)=|[        c_iδ _ij( -p_i^∗/p_j) e^-(ξ _N+i+
    ξ̅_N+j)^∗(-x,t)+1/p_i^∗+p_j                                                       1/
    p_i^∗+p_N+j;                                                        1/p_N+i^∗+p_j c_N+i( -p_N+i^∗/
    p_N+j) δ _ije^-(ξ _i+ξ̅_j)^∗(-x,t)+
    1/p_N+i^∗+p_N+j ]
    | .

On the other hand,

    g̅(x,t)    =   |[                      c_iδ _ij( -p_i/p_j^∗) ^-1e^-(ξ
    _i+ξ̅_j)+1/p_i+p_j^∗                                                           1/
    p_i+p_N+j^∗;                                                            1/p_N+i+p_j^∗ c_N+iδ _N+j,N+i( -
    p_N+i/p_N+j^∗) ^-1e^-(ξ _N+i+ξ̅_N+j)+
    1/p_N+i+p_N+j^∗ ]
    |, 
       =   |[                      c_jδ _ji( -p_j/p_i^∗) ^-1e^-(ξ
    _j+ξ̅_i)+1/p_j+p_i^∗                                                           1/
    p_N+j+p_i^∗;                                                            1/p_j+p_N+i^∗ c_N+jδ _N+j,N+i( -
    p_N+j/p_N+i^∗) ^-1e^-(ξ _N+j+ξ̅_N+i)+
    1/p_N+i^∗+p_N+j ]
    | .

Obviously g̅(x,t)=g^∗(-x,t).

Based on above Lemma, two sets of bilinear equations (<ref>) and (<ref>) coincide.






Therefore, we have the following theorem regarding the general soliton solution.

The nonlocal NLS equation admits the solution

    q(x,t)=ρg(x,t)/f(x,t)e^i2ρ ^2t ,

where

    f(x,t)=|
    c_iδ _ije^-(ξ _i+ξ̅_j)+1/p_i+p_j^∗
    
    
    | _0≤ i,j≤ 2N ,


    g(x,t)=|
    c_iδ _ij( -p_i/p_j^∗) e^-(ξ _i+
    ξ̅_j)+1/p_i+p_j^∗
    
    
    
    | _0≤ i,j≤ 2N ,

with c_N+i=-c_i^∗, p_N+i=-p_i, ξ_0,N+i=ξ _0i for i=1,⋯,N.

In what follows, we  give explicit form of the solution for N=1 and detailed analysis.
By taking N=1 in (<ref>) and (<ref>) and p_1=ρ e^-iθ
we get the tau functions

    f    =   |[     c_1e^-(ξ _1+ξ̅_1)+1/p_1+p_1^∗                      1/
    p_1+p_2^∗;                       1/p_2+p_1^∗ -c_1^∗e^-(ξ _2+ξ̅_2)+
    1/p_2+p_2^∗ ]
    |
       =   -ρ ^-2(|c_1|^2ρ ^2e^4ρ ^2sin 2θ t+
    1/2ρ c_1θ e^-2ρcosθ x+2ρ ^2sin 2θ t
          +1/2ρ c_1^∗θ e^2ρcosθ x+2ρ
    ^2sin 2θ t+ ^22θ) ,



    g    =   |[    c_1e^-(ξ _1+ξ̅_1)+1/p_1+p_1^∗( -p_1/p_1^∗
    )                      1/p_1+p_2^∗( -
    p_1/p_2^∗);                       1/p_2+p_1^∗( -p_2/p_1^∗) -c_1^∗e^-(ξ _2+ξ̅_2)+1/p_2+p_2^∗
    ( -p_2/p_2^∗) ]
    |
       =   -e^2iθρ ^-2(|c_1|^2ρ ^2e^4ρ ^2sin 2θ
    t-2iθ-1/2ρ c_1θ e^-2ρcosθ x+ 2ρ ^2sin 2θ t
    
          -1/2ρ c_1^∗θ e^2ρcosθ x+2ρ ^2sin
    2θ t+ ^22θ e^2iθ) .

 It can be easily shown that the resulting solution q(x,t)=ρg(x,t)/f(x,t)e^i2 ρ ^2t corresponds to the solution (5.90) in <cit.> with σ=1 and Δθ=0.
Next, we investigate the asymptotic behavior of above two-soliton solution. To
this end, we assume 0< θ < π/2 without loss of generality, and κ=2ρcosθ >0. We define the right-moving soliton along the line
η_1=x-2ρsinθ t as soliton 1, and the left-moving soliton along the line η_2=x+2ρsinθ t as soliton 2.
For the above choice of parameters, we have
(i) η_1 ≈ 0, η_2 →∓∞ as t→∓∞ for soliton 1 and (ii) η_2 ≈ 0, η_1 →±∞ as t→∓∞ for soliton
2. This leads to the following asymptotic forms for the two-soliton solution.
(i) Before collision (t→ -∞)

Soliton 1 (η _1≈ 0, η _2→ -∞):

    q     →   ρ ^2θ-1/2ρ c_1θ e^-κη_1 -2iθ/ ^2θ+1/2ρ c_1θ e^-κη_1 e^i2ρ ^2t+4iθ , 
       =   ρ1- 2ρ |c_1|sin^2 θcosθ e^-κη_1 -i(2θ-ϕ)/1+2ρ |c_1|sin^2 θcosθ e^-κη_1+iϕ e^i2ρ ^2t+4iθ ,


    |q|^2  →ρ^2 cosh (η_1-η^-_10)-cos (2θ-ϕ)/cosh (η_1-η^-_10)+cosϕ .


Soliton 2 (η_2≈ 0, η_1→∞):

    q  →ρ ^22θ-1/2ρ c^∗_1θ e^-κη_2 -2iθ/ ^22θ+1/2ρ c^∗_1θ e^-κη_2 e^i2ρ ^2t+4iθ , 
       =   ρ1- 2ρ |c_1|sin^2 θcosθ e^-κη_1 -i(2θ+ϕ)/1+2ρ |c_1|sin^2 θcosθ e^-κη_1-iϕ e^i2ρ ^2t+4iθ ,



    |q|^2  →ρ^2 cosh (η_1-η^-_10)-cos (2θ+ϕ)/cosh (η_1-η^-_10)+cosϕ ,

where e^η^-_10=2ρ |c_1| sin^2 θcosθ.

(ii) After collision (t →∞) 

Soliton 1 (η_1≈ 0, η_2→∞):

    q     →   ρ|c_1|^2ρ ^2-1/2ρ c^∗_1θ e^κη_1 +2iθ/|c_1|^2ρ ^2+1/2ρ c^∗_1θ e^κη_1  e^i2ρ ^2t , 
       =   ρ2ρ |c_1| cosθ e^-κη_1 -i(2θ-ϕ)-1/2ρ |c_1| cosθ e^-κη_1 +iϕ+1
    e^i2ρ ^2t+4iθ


    |q|^2  →ρ^2 cosh (η_1-η^+_10)-cos (2θ-ϕ)/cosh (η_1-η^+_10)+cosϕ .

Soliton 2 (η _2≈ 0, η _1→ -∞):

    q     →   ρ|c_1|^2ρ ^2-1/2ρ c_1θ e^κη_2 +2iθ/|c_1|^2ρ ^2+1/2ρ c_1 θ e^κη_2  e^i2ρ ^2t , 
       =   ρ2ρ |c_1| cosθ e^-κη_1 -i(2θ+ϕ)-1/2ρ |c_1| cosθ e^-κη_1 -iϕ+1
    e^i2ρ ^2t+4iθ


    |q|^2  →ρ^2 cosh (η_1-η^+_10)-cos (2θ+ϕ)/cosh (η_1-η^+_10)+cosϕ ,

where e^η^+_10=2ρ |c_1|  cosθ.











 §.§ Soliton solution with nonzero boundary condition to the nonlocal NLS equation with σ =-1, Δθ=π.


In this subsection, we consider the soliton solutions to the nonlocal NLS equation with σ=-1

    iq_t(x,t)=q_xx(x,t)+2q^∗(-x,t) q^2(x,t)

under boundary condition q(x,t)→ρ as x→±∞. To this end, we introduce a variable transformation

    q(x,t)=ρg(x,t)/f(x,t)e^ 2iϵρ ^2t,

with the condition f(x,t)=± f^∗(-x,t), then the nonlocal NLS equation is converted into the following bilinear
equations
 

    {[                          (iD_t-D_x^2)g(x,t)· f(x,t)=0,; (D_x^2-2ϵρ ^2)f(x,t)· f(x,t)= ± 2 ρ^2g(x,t)g^∗(-x,t) , ]
    .


We recall the tau functions of single component KP hierarchy τ_k and the
bilinear equations (<ref>) the tau functions satisfy in previous section.
We take the simplest case p_1=p̅_1=ρ, then the second bilinear equation in
(<ref>) reduces to the second equation in (<ref>).
Furthermore, if we let c_1=1/2ρ, and define
 

    f(x,t)=τ_0=1/2ρ( 1+e^ξ _1+ξ̅_1) ,


    g(x,t)=τ_1=1/2ρ( 1-e^ξ _1+ξ̅_1) ,


    g̅(x,t)=τ_-1=1/2ρ( 1-e^ξ _1+ξ̅_1) ,

where e^ξ _1+ξ̅_1=e^2 ρ x. It then follows

    f(x,t)=e^(ξ _1+ξ̅_1) f^∗(-x,t) ,   g̅(x,t)=-e^(ξ _1+ξ̅_1) g^∗(-x,t) .

Furthermore, if we let x_1=x, x_2=- i t and ϵ=1, the bilinear equation (<ref>) reduces to the bilinear equation (<ref>).
  Under this case, if we define

    q(x,t)=g(x,t)/f(x,t)e^i2ρ ^2t ,    r(x,t)=g̅(x,t)/f(x,t)e^-i2ρ ^2t ,

it is obvious that

    r(x,t)=-q^∗(-x,t),

which leads to the 1-soliton solution
 

    q(x,t)=ρ 1-e^ξ _1+ξ̅_1/ 1+e^ξ _1+ξ̅_1 e^2 iρ ^2t
      =ρ e^2 iρ ^2ttanh (ρ x)  .

This solution coincides with the solution (3.200) and (3.201) found in <cit.> for the case: σ =-1, Δθ=π.



 §.§ Soliton solution with nonzero boundary condition to the nonlocal NLS equation with σ =-1, Δθ=0.


Again for the tau functions of single component KP hierarchy τ_k and the
bilinear equations the tau functions satisfy in previous section. If we impose the condition
p_jp̅_j=-ρ ^2, j=1,⋯ ,N ,
then similarly, we have

    ∂_x_1τ _k=-ρ^2∂ _x_-1τ _k .

Furthermore, if we assume x_1=x, x_2=-it,








then the bilinear equation (<ref>) reduces to

    {[                               (iD_t-D_x^2) τ _1(x,t) ·τ _0(x,t)=0,; (D_x^2+2ρ ^2)τ _0(x,t)  ·τ _0(x,t)=  2 ρ ^2τ _1(x,t)  τ _-1(x,t) . ]
    .


Consider a 2 × 2 matrix for τ _0(x,t), τ _1(x,t) and τ _-1(x,t), let p_1=v_1, p̅_1=-ρ ^2/v_1
where v_1>ρ is real, p_2=p̅_1, p̅_2=p_1, c_2=-c_1 and

    c_1^2=1/
    (p_1+p̅_1)^2-1/4p_1p̅_1
. 



Since

    (ξ _1+ξ̅_1)(x,t)=(v_1-ρ ^2/v_1)x+(v_1^2-ρ
    ^4/v_1^2)it ,


    (ξ _2+ξ̅_2)(x,t)=(v_1-ρ ^2/v_1)x-(v_1^2-ρ
    ^4/v_1^2)it ,






    τ _0(x,t)    =   |[  c_1+1/p_1+p̅_1e^ξ _1+ξ̅_1     1/p_1+
    p̅_2e^ξ _1+ξ̅_2;      1/p_2+p̅_1e^ξ _2+ξ̅_1 c_2+1/p_2+
    p̅_2e^ξ _2+ξ̅_2 ]
    |
       =   -c_1^2+c_1/p_2+p̅_2e^ξ _2+ξ̅_2-
    c_1/p_1+p̅_1e^ξ _1+ξ̅_1+c_1^2e^ξ
    _1+ξ̅_1+ξ _2+ξ̅_2 
    
    
       =   2c_1 e^(v_1-ρ^2/v_1)x{c_1sinh[(v_1-ρ^2/v_1)x ] -i/p_1+p̅_1sin[(v_1^2-ρ^4/v_1^2)t ]} .

Therefore

    τ _0^∗(-x,t)    =   -e^-2(v_1-ρ^2/v_1)xτ _0(x,t) .

Moreover

    τ _1(x,t)    =   |[  c_1+1/p_1+p̅_1( -p_1/p̅_1)
    e^ξ _1+ξ̅_1     1/p_1+p̅_2( -p_1/
    p̅_2) e^ξ _1+ξ̅_2;      1/p_2+p̅_1( -p_2/p̅_1) e^ξ
    _2+ξ̅_1 c_2+1/p_2+p̅_2( -p_2/
    p̅_2) e^ξ _2+ξ̅_2 ]
    |
       =   -c_1^2+c_1/p_2+p̅_2( -p_2/
    p̅_2) e^ξ _2+ξ̅_2-c_1/p_1+p̅_1
    ( -p_1/p̅_1) e^ξ _1+ξ̅
    _1+c_1^2e^ξ _1+ξ̅_1+ξ _2+ξ̅_2 
       =   c_1 e^(v_1-ρ^2/v_1)x{2 c_1sinh[(v_1-ρ^2/v_1)x ] +1/p_1+p̅_1[ p_1/p̅_1e^(v_1^2-ρ^4/v^2_1)it-
    p̅_1/p_1 e^-(v_1^2-ρ^4/v^2_1)it]} .


    τ _-1(x,t)    =   |[  c_1+1/p_1+p̅_1( -p̅_1/p_1)
    e^ξ _1+ξ̅_1     1/p_1+p̅_2( -p̅
    _2/p_1) e^ξ _1+ξ̅_2;      1/p_2+p̅_1( -p̅_1/p_2) e^ξ
    _2+ξ̅_1 c_2+1/p_2+p̅_2( -p̅
    _2/p_2) e^ξ _2+ξ̅_2 ]
    |
       =    -c_1^2+c_1/p_2+p̅_2( -p̅
    _2/p_2) e^ξ _2+ξ̅_2-c_1/p_1+p̅_1
    ( -p̅_1/p_1) e^ξ _1+ξ̅
    _1+c_1^2e^ξ _1+ξ̅_1+ξ _2+ξ̅_2 
       =   c_1 e^(v_1-ρ^2/v_1)x{2 c_1sinh[(v_1-ρ^2/v_1)x ] +1/p_1+p̅_1[ p̅_1/p_1 e^(v_1^2-ρ^4/v^2_1)it-
     p_1/p̅_1 e^-(v_1^2-ρ^4/v^2_1)it]} .

Therefore















    τ _1^∗(-x,t)    =   -e^-2(v_1-ρ^2/v_1)xτ _-1(x,t) .

Consequently, if we define

    τ _0(x,t)=Cf(x,t),τ _1(x,t)=Cg(x,t),τ _-1(x,t)=Cg̅(x,t),

with C=e^(v_1-ρ^2/v_1)x, then we can easily show that

    f(x,t)=-f^∗(-x,t),   g̅(x,t)=-g^∗(-x,t),

which reduces the bilinear equation (<ref>) to

    {[                       (iD_t-D_x^2) g(x,t)· f(x,t)=0,; (D_x^2+2ρ ^2)f(x,t) · f(x,t)=2ρ ^2g(x,t) g^*(-x,t) . ]
    .

Above bilinear equation coincides with the bilinear equation (<ref>) with ϵ=-1 under the case
f(x,t)=-f^*(-x,t). Moreover, for

    q(x,t)=g(x,t)/f(x,t)e^-i2ρ ^2t ,    r(x,t)=-g̅(x,t)/f(x,t)e^-i2ρ ^2t ,

it is obvious that

    r(x,t)=-q^∗(-x,t),

which leads to the 1-soliton solution

    q(x,t)=ρ (v_1^2+ρ^2) sinh[(v_1-ρ^2/v_1)x ] + 2ρ^3/v_1e^-(v_1^2-ρ^4/v^2_1)it-
    2v_1^3/ρ e^(v_1^2-ρ^4/v^2_1)it/(v_1^2+ρ^2) sinh[(v_1-ρ^2/v_1)x ] -4 iρ v_1 sin[(v_1^2-ρ^4/v_1^2)t ]  e^-2 iρ ^2t .

This solution coincides with the solution (4.130) in <cit.> for the case: σ =-1, Δθ=0.



§ SUMMARY AND CONCLUDING REMARKS

In this paper, via the combination of Hirota's bilinear method and the KP hierarchy reduction method, we have constructed various general soliton solutions in the form of determinants for the nonlocal NLS equation with both zero and nonzero boundary conditions. The solutions obtained here recover the solutions found in the nonlocal NLS equation so far and reformulate them in compact form by using tau functions of the KP hierarchy.

A natural extension of the present work will be the construction of the general soliton solutions to the coupled nonlocal NLS equation

    {[   iq_1,t(x,t)=q_1,xx(x,t)-2(σ_1 q_1^2(x,t)+σ_2 q_2^2(x,t))q^∗_1(-x,t) ,; iq_2,t(x,t)=q_2,xx(x,t)-2(σ_1 q_1^2(x,t)+σ_2 q_2^2(x,t))q^∗_2(-x,t) , . ]
    .

It is expected that the soliton solution to the above coupled nonlocal NLS equation is even richer and more complicated than the single component nonlocal NLS equation. We will explore this interesting topic and report the results elsewhere in the near future.


§ ACKNOWLEDGEMENTS

MJA was partially supported by NSF under Grant No. DMS-1712793. BF was partially supported by NSF Grant under Grant No. (DMS-1715991) and the COS Research Enhancement Seed Grants Program at UTRGV.


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