Soliton solutions to a (2+1)-dimensional nonlocal NLS equation

In this paper, applying the Hirota’s bilinear method and the KP hierarchy reduction meth-od, we obtain the general soliton solutions in the forms of N× N Gram-type determinants to a (2+1)-dimensional nonlocal nonlinear Schrödinger equation with time reversal under zero and nonzero boundary conditions. The general bright soliton solutions with zero boundary condition are derived via the tau functions of two-component KP hierarchy. Under nonzero boundary condition, we first construct general soliton solutions on periodic background, when N is odd. Furthermore, we discuss typical dynamics of solutions analytically, and graphically.


Introduction
In 2013, Ablowitz and Mussilimani [1] considered a nonlocal nonlinear Schrödinger (NLS) equation iu t (x,t) + u xx (x,t) + 2σ u 2 (x,t)u * (−x,t) = 0, where σ = ±1, and first derived its soliton solutions making use of the inverse scattering method (IST). Actually, Eq. (1) comes from a symmetry reduction of the AKNS hierarchy, and it is a parity-time (PT ) symmetry introduced by Bender and Boettcher [2,3] in 1998, as the equation (1) is invariant under the parity-time (PT ) operator, that is, the variable transformation x → −x, t → −t and the complex conjugate. After that, soliton solutions of the nonlocal NLS equations and various new nonlocal equations appearing from different symmetry reductions of the AKNS hierarchy were paid attention by many authors [4][5][6][7][8][9][10][11]. In addition, people also considered some other nonlocal equations with self-induced PT symmetric potential [12][13][14]. For example, in 2016, Fokas [5] provided soliton solutions of the nonlocal Davey-Stewartson (DS) equation iA t = A xx + γ 2 A yy + (εV − 2Q)A, Q xx − γ 2 Q yy = (εAA * (−x, −y,t)) xx , where ε = ±1, by using the same method (IST) in the work of Ablowitz and Mussilimani [1]. Later, the general breather and rogue wave solutions were obtained by Rao et al. [15] applying the Hirota bilinear method. An extension of the usual DS II euqation involving a PT symmetric potential was considered by Liu et al [16], and the families of n-order rational solutions were obtained. In [5], Fokas also considered soliton solutions of a (2+1)-dimensional NLS equation Cao et al. [17] kept on investigating this multidimensional equation (3) and gave families of rational and semi-rational solutions. Its general soliton solutions with zero and nonzero boundary conditions were studied by Liu and Li [18], they derived the soliton solutions expressing by N × N Gram-type determinants with even N, however, the odd case is not involved. It is known that there are a variety of useful and powerful tools to deal with the nonlocal equations, namely, the inverse scattering transformation method, the Hirota's bilinear method, the KP hierarchy reduction method, and so on. Combining the Hirota's bilinear method and the KP reduction hierarchy method, very recently, Li et al. [19] discussed the nonlocal Mel'nikov equation and obtained its general soliton solution and (semi-)rational solutions on the periodic background. By employing the Hirota bilinear method, recently, Ma and Zhu [6] constructed the N-soliton solution for an integrable nonlocal discrete focusing nonlinear Schrödinger(dNLS+) equation, and gave the asymptotic analysis of two-soliton solution. By using the KP hierarchy reduction method, Rao et al. [20] found new families of semi-rational solutions termed as lump-soliton solutions to the nonlocal DS I equation.
In this paper, inspired by Liu and Li [18] and Li et al. [19], we study the (2+1)-dimensional nonlocal nonlinear Schrdinger equation with time reversal instead of space reversal, We also present the general soliton solutions of (4) in terms of N × N Gram-type determinants with zero and nonzero boundary conditions. Under the nonzero boundary condition, when N is odd, the general soliton solutions on the periodic background are constructed. The rest of this article is organized as follows. In Sec.2 and 3, we present our main theorem for soliton solutions of the multi-dimensional nonlocal NLS equation with zero and nonzero boundary conditions, respectively. The construction of explicit soliton solutions appears in Sec.3.1. In Sec.4, we illustrate the dynamics of the two-, four-and multi-soliton solutions on different backgrounds. The conclusion is given in Sec.5.
v(x, y,t) = 2(ln f (x, y,t)) xx , where f (x, y,t) = A 0,i j + A 1,i j N×N , with the parameters p i , q i ,p i ,q i satisfying following constrains:

Derivation of Solution with zero boundary condition
Under the transformation (4) can be converted into the following bilinear forms where D is the Hirota's bilinear differential operator defined by We start with the Gram-type determinant expression of the tau functions, and the Φ,Φ, Ψ ,Ψ are row vectors defined by According to the Sato theory, it is clear that the tau functions given above satisfy the following two bilinear equations Indeed, assume that Then the tau function τ 0 (x, y,t) can be rewritten as follow: Obviously, we obtain τ 0 (x, y,t) = τ 0 (x, y, −t). Besides, we can also claim that τ 1 (x, y,t) = τ −1 (x, y, −t). Actually, it is not hard to see that and then (12) can be converted to (8), therefore, we complete the proof of Theorem 1.

Bilinearization
In order to construct soliton solutions, we first introduce the dependent variable transformations where f and g are functions with respect to three variables x, y and t. Similar to the case in zero boundary condition, plugging Eq.(26) into Eq.(4), we have

Construction of solutions
In this section, we start with tau functions of single component KP heirarchy with where p i , q j , c i , ξ i0 and η j0 are arbitrary complex constants, δ i j = 1 when i = j and δ i j = 0 elsewhere. Due to the Sato theory, we know that the tau functions τ n satisfy the bilinear equations, By taking the variables transformation, x 1 = x, x 2 = it and x −1 = −y, we can rewrite τ n as with Furthermore, if we set where C = ∏ N j=1 e ξ j +η j , then the bilinear equations (29) can be as the following, D 2 x + iD t g(x, y,t) · f (x, y,t) = 0, (D x D y + 2) f (x, y,t) · f (x, y,t) = 2g(x, y,t)h(x, y,t).

(32)
After simple algebra calculation, it follows that By substituting the parameters constrains (24) and (25), we can obtain Thus, we have and then Eq. (32) reduces into Eq. (27). Therefore, the proof of Theorem 2 is completed.

Dynamics of the soliton solutions
In this section, we would like to present the concrete form of the one-soliton solution with the periodic background and the explicit form of the two-soliton solutions on constant and periodic backgrounds. Furthermore, we also give the asymptotic behaviour of the two-soliton solution.

The periodic background
When N = 1, Theorem 2 yields the following solution If we set p 1 = m + ni, c 1 = r + di, where m, n, r, d are real numbers, then (41) is expressed as and α = 2ny m 2 +n 2 + 2nx, β = −2mx + 2m m 2 +n 2 y + ξ 01 + η 01 . Clearly, the solutions are periodic in both x and y with period 2n and 2n m 2 +n 2 respectively when p 1 is purely imaginary. In this paper, regular solutions (4.1) provide the periodic background for higher-order soliton solution (see Fig. 3).
Next, two-soliton solutions on the periodic background can be derived when N = 3 in Theorem 2. By selecting the different parameters, the same three types of two-soliton appear on the periodic background (see Figs. 4 (d)-(f)).

Four-soliton solutions on both constant and periodic backgrounds
Likewise, we construct the four-soliton solutions on the constant and periodic background when N = 4 and N = 5 respectively. Four-soliton solutions describe the superposition of two-soliton. As illustrated in Fig. 4, twosoliton solutions are presented as three patterns, so foursoliton solutions have more different types. By adjusting different parameter values, five types of solutions can be obtained, that is, 4-dark soliton, 3-dark-1-antidark soliton, 2-dark-2-anti-dark soliton, 1-dark-3-antidark soliton, and 4-anti-dark soliton (see Fig. 5). Similarly, when N = 5, we also have five types of solutions (see Fig. 6). Next, we give the dynamical behavior of multi-soliton solutions in general cases.
i) When N = 2M, N + 1 types of N-soliton solutions on the constant background can be obtained, which contains k-dark-(N − k)-anti-dark solutions, where 0 ≤ k ≤ N. ii) When N = 2M + 1, N + 1 types of N-soliton solutions on the periodic background can be obtained.

Conclusion
In this paper, under the zero and nonzero boundary conditions, we derive the soliton solutions of the (2+1)dimensional nonlocal nonlinear Schrödinger equation with time reversal. By applying the reduction method of KP hierarchy and the Hirota's bilinear method, we describe the concrete solutions in terms of N ×N Gramtype determinants. With some parameter restrictions, we illustrate the regularity of solutions, two-and foursoliton solutions on constant and periodic backgrounds. For example, when N = 1, Fig. 3 presents the regular periodic solutions. We also consider the classical dynamical behaviors of two-soliton solutions and the elasticity of the collisions of two-solitons. Typical twoand four-soliton solutions on different backgrounds can be referred to as Figs. 4 -6. Finally, we conclude two dynamical behaviours of multi-soliton solutions. One is N +1 types of N-soliton solutions on the constant background with even N; another is the N-soliton solutions on the periodic background with odd one.