Generalized ∂¯ -dressing method for coupled nonlocal NLS equation

Abstract

Abstract In this paper, we investigate a general integrable coupled nonlocal nonlinear Schrödinger (cnNLS) equation with reverse space and reverse time. First, we introduce a 4-component nonlocal nonlinear Schrödinger (nNLS) equation. Based on two 3 × 3 matrices ∂ ¯ -problems, we obtain the N-soliton solutions of the 4-component nNLS equation by constructing two spectral transformation matrices R and R ˆ with some specific scattering data { k i , α i , β i } i = 1 N and { λ j , ξ j , η j } j = 1 N ˜ . We have the symmetry condition for R and R ˆ . We express the determinant in the N-soliton solution in the form of sums with the help of Cauchy matrix properties to facilitate follow-up reduction. The general nonlocal reduction of the 4-component nNLS equation to the cnNLS equation is discussed in detail. After obtaining the 1-soliton solution of cnNLS equation, we analyse the nonsingular region of the single soliton solution, with spectral parameters k 1 and λ 1 are conjugate. We also draw some typical images of 1-soliton solution. The 2-soliton solutions for cnNLS equation are derived and their asymptotic behaviours are discussed. After that we discuss the dynamic behaviour of the two waves in 2-soliton solution under the condition of different spectral parameters values.