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

Abstract

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.

Solutions of Nonlocal Equations Reduced from the AKNS Hierarchy

“…Remark 1. By observation of equations in the hierarchy (11) one can find the following. On the level of equations, (11) with (σ, δ) = (±1, 1) and ( 11) with (σ, δ) = (±1, −1) can be transformed from each other by simply taking q → iq.…”

“…By observation of equations in the hierarchy (11) one can find the following. On the level of equations, (11) with (σ, δ) = (±1, 1) and ( 11) with (σ, δ) = (±1, −1) can be transformed from each other by simply taking q → iq. Such a transformation can be also seen on the level of solutions that we present through (40) with choices listed in Table 1.…”

“…Such a transformation can be also seen on the level of solutions that we present through (40) with choices listed in Table 1. First, for q (σ,δ)=(1,−1) , due to arbitrariness of vector C + , if we replace C + by Diag(I n+1 , iI n+1 )C + , then, q (σ,δ)=(1,−1) defined by (40) will be changed to iq (σ,δ)=(1,−1) , which gives a solution to (11) with (σ, δ) = (1, 1). Second, for q given in the form (40), from the line (σ, δ) = (−1, −1) and line (σ, δ) = (−1, 1)…”

“…Meanwhile, solving techniques are developed to deal with nonlocalness resulting from −x. Among the main techniques are the Inverse Scattering Transform [1,6,9], Darboux transformations [10][11][12][13][14][15][16][17], and the bilinear method [18][19][20]. Actually, it is hard to apply bilinearisation directly to Eq.…”