A $\overline{\partial }$-Dressing Method for the Kundu-Nonlinear Schrödinger Equation

In this paper, we employed the ∂¯-dressing method to investigate the Kundu-nonlinear Schrödinger equation based on the local 2 × 2 matrix ∂¯ problem. The Lax spectrum problem is used to derive a singular spectral problem of time and space associated with a Kundu-NLS equation. The N-solitions of the Kundu-NLS equation were obtained based on the ∂¯ equation by choosing a special spectral transformation matrix, and a gradual analysis of the long-duration behavior of the equation was acquired. Subsequently, the one- and two-soliton solutions of Kundu-NLS equations were obtained explicitly. In optical fiber, due to the wide application of telecommunication and flow control routing systems, people are very interested in the propagation of femtosecond optical pulses, and a high-order, nonlinear Schrödinger equation is needed to build a model. In plasma physics, the soliton equation can predict the modulation instability of light waves in different media.