Generalized Darboux Transformation and Rational Solutions for the Nonlocal Nonlinear Schrödinger Equation with the Self-induced Parity-time Symmetric Potential

In this paper, I construct a generalized Darboux transformation for the nonlocal nonlinear Schrö-dinger equation with the self-induced parity-time symmetric potential. The N-order rational solution is derived by the iterative rule and it can be expressed by the determinant form. In particular, I calculate first-order and second-order rational solutions and obtain their figures according to different parameters.


Introduction
Parity-time (PT) symmetry was firstly proposed by Bender and Boettcher in quantum mechanics [1].And it has been widespread concerned in optical solitons [2], non-reciprocal light propagation [3], unidirectional invisibility [4], perfect absorber [5] and so on.Since then, there has been attracted more and more attentions in the non-Hermitian systems with PT symmetry [6]- [13].Generally, the non-Hermitian Hamiltonian ( ) is deemed to be PT symmetric if ( ) ( ) the real part of a PT symmetry complex potential must be an even function whereas the imaginary part should be odd.In optical system, the PT symmetric potential can be realized by controlling the complex refractive index distribution ( ) ( ) ( ), where the refractive index profile ( ) R n x is an even function in the trans- verse direction, the gain or loss component ( ) I n x is an odd one [10]- [12].In the nonlinear optics, the PT symmetric and Kerr nonlinearity linear potentials have been intensively researched in the nonlinear Schrödinger (NLS) equations.For example, [14] has studied the soliton in PT symmetric potential with competing nonlinearity; [15] has studied the dynamical behaviors of 2D nonautonomous solitons in PT symmetric potentials; and [16] has studied stable dark solitons in PT symmetric dual-core waveguides.
In this paper, I will consider a nonlocal NLS equation [17] ( ) which is non-Hermitian but PT symmetric, where ( ) , u x z is a complex valued function of real variables x and z, and 1 σ = ± .The signs ( ) + and ( ) − denote the focusing and defocusing respectively.In Equation ( 1), the nonlinear term brings a self-induced potential of the form ( ) ( ) ( ) The exact moving one-soliton solution of Equation ( 1) has been obtained in [17] via the inverse scattering transform.The dark and antidark soliton interactions have been given in [18] via the classical Darboux transformation (DT) method.However, there are no papers on high-order rational solutions of Equation ( 1) by generalized Darboux transformation (gDT).
The organization of this paper is as follows: In Section 2, a determinant expression of N-order gDT will be constructed based on the Lax pair.In Section 3, I will obtain a general determinant expression of N-order rational solution of Equation (1).In addition, I calculate first-order and second-order rational solutions and obtain their figures according to different parameters.The conclusions will be given in Section 4.

Lax Pair and Generalized Darboux Transformation
The Lax pair of Equation ( 1) can be expressed as follows [17]: , , where ( ) is the vector eigenfunction of Lax pair (2), and T signifies the vector transpose.Matrices U and V have the following forms: where λ is a spectral parameter, the asterisk denotes the complex conjugate.The compatibility condition 0 1) by a direct computation.The classical DT for Equation (1) has been constructed in [18]: where = is an eigenfunction of Lax pair (2) with a seeding solution ( ) = , the above DT procedure can be easily ite- rated.Based on Crum theorem [19], I can obtain a general case for Equation (1) in the form of determinant.
Next, I suppose ( ) ( ) ≤ ≤ then iterate the above DT N times, I obtain the N-fold DT for Equation (1) in the form of a determinant as , In the following, I derive the determinant form of the gDT for Equation (1).Considering N different eigenfunctions , for the Lax pair (2) with ( ) , , .
Thus, on the basis of the work in [20] [21], I can perform the limit on Formula (4), then obtain the following result: where , , , , , , , .

Rational Solutions
To construct the rational solutions of Equation ( 1), I take a plane wave solution 0 e , where a is real constant, and the frequency ω satisfies the nonlinear dispersion relation where 1 c and 2 c are both complex constants.In order to obtain the rational solitonic structure, I must impose s to be real numbers, which is satisfied only when 1 Re 0 λ = .I point out that, a special seed solution and suitable eigenvalue enable us to get higher rational solutions in determinant forms according to Formula (5).In the following discussions, I may set 1 a = to simply our calculation process.Then set ( ) where ( ) ( ) The relevant Taylor expansions are It follows that the N-order rational solution for Equation (1), reads ( ) where Setting N = 1 in Formula (10), then I obtain the first-order rational solution (see Figure 1(a)) with the parameters 1 2 Then with N = 2, the second-order rational solution (see where

Conclusion
In this paper, I have studied the nonlocal nonlinear Schrödinger equation with the self-induced parity-timesymmetric potential.Then I have constructed a gDT for Equation (1) and derived the N-fold rational solutions in determinant forms.In particular, I have calculated first-order and second-order rational solutions from a planewave solution and obtained their figures according to different parameters.

Figure 1 Figure 1 .
Figure 1.(a) Plot of the first-order rational solution; (b) Plot of the second-order rational solution.