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In this paper, I construct a generalized Darboux transformation for the nonlocal nonlinear Schrodinger 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.

Parity-time (PT) symmetry was firstly proposed by Bender and Boettcher in quantum mechanics [

In the nonlinear optics, the PT symmetric and Kerr nonlinearity linear potentials have been intensively researched in the nonlinear Schrödinger (NLS) equations. For example, [

In this paper, I will consider a nonlocal NLS equation [

which is non-Hermitian but PT symmetric, where

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.

The Lax pair of Equation (1) can be expressed as follows [

where

where

The classical DT for Equation (1) has been constructed in [

where

Here

choose different eigenfunctions

Next, I suppose

where

In the following, I derive the determinant form of the gDT for Equation (1). Considering N different eigenfunctions

where

Thus, on the basis of the work in [

where

To construct the rational solutions of Equation (1), I take a plane wave solution

where a is real constant, and the frequency

Then inserting Equation (7) into the Lax pair (2) and taking

with

where

where

The relevant Taylor expansions are

where

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

Then with N = 2, the second-order rational solution (see

where

In this paper, I have studied the nonlocal nonlinear Schrödinger equation with the self-induced parity-time- symmetric 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.

This work is supported by the Shanghai Leading Academic Discipline Project under Grant No. XTKX2012, by the Natural Science Foundation of Shanghai under Grant No. 12ZR1446800, Science and Technology Commission of Shanghai municipality, and by the National Natural Science Foundation of China under Grant Nos. 11201302 and11171220.