A $$\overline{\partial }$$-dressing method for the mixed Chen–Lee–Liu derivative nonlinear Schrödinger equation

Abstract

In this work, we apply the $$\overline{\partial }$$ ∂ ¯ -dressing method to study the mixed Chen–Lee–Liu derivative nonlinear Schrödinger equation (CLL–NLS) with non-normalization boundary conditions. The spatial and time spectral problems associate with CLL–NLS equation which are derived from local $$2 \times 2$$ 2 × 2 matrix. A CLL–NLS hierachy with source is proposed by using recursive operator. Based on the $$\overline{\partial }$$ ∂ ¯ -equation, the N-solitons of the CLL–NLS equation are constructed by choosing a special spectral transformation matrix. Further more, the explicit two-soliton is obtained.

Higher-order soliton solutions for the Sasa–Satsuma equation revisited via $$\bar{\partial }$$ method

“…Here soliton solutions correspond to the spectral transform matrix R(k) located at the discrete spectrum points of the complex plane where a solution Ψ(k) of the ∂ problem has simple poles. Firstly we consider the discrete spectrum set Z (17), by means of the Eq. ( 10) and the symmetry conditions (15), we can choose where (k) denotes the delta function and c 1j , c 2j , c 3j , c 4j (j = 1, 2, … , N) are given by the following proposition.…”

“…Much research has been conducted for it, we will not dwell on a detailed exposition of various results. The aim of this paper is to study higher-order soliton solutions for the Sasa-Satsuma equation by means of ∂ method [3,11,17,24]. Soliton solutions corresponding to multiple poles have been investigated in the literature before.…”

A ∂¯-Dressing Method for the Kundu-Nonlinear Schrödinger Equation

“…The ∂-dressing method suggested by Zakharov and Shabat [17,18] was later developed by Beals, Coifman, Manakov, Ablowitz, Fokas, and others [19][20][21][22][23]. A wide variety of equations have been successfully investigated using the ∂-dressing method [24][25][26][27][28][29][30]. However, the equations with a normalization boundary condition, ψ → I, at infinity are usually considered.…”

Application of the Dbar-method to a nonlocal coupled modified NLS equation and nonlocal reduction

“…For the past few years, there have been many researches on the application of Dbar method for local equations [19][20][21][22][23][24][25][26]. While few studies have been done about using Dbar method to nonlocal equations.…”