# $N$-Bright-Dark Soliton Solution to a Semi-Discrete Vector Nonlinear Schr\"odinger Equation

## TL;DR

This paper constructs explicit multi-soliton solutions for a semi-discrete vector nonlinear Schrödinger equation using Pfaffian forms and analyzes their asymptotic behavior, advancing understanding of integrable discrete systems.

## Contribution

It introduces a Pfaffian-based method to derive multi-soliton solutions for semi-discrete vector NLS equations, including explicit forms for multi-component cases.

## Key findings

- Explicit one- and two-soliton solutions are derived.
- Asymptotic analysis of two-soliton interactions is performed.
- The solutions demonstrate integrability and rich soliton dynamics.

## Abstract

In this paper, a general bright-dark soliton solution in the form of Pfaffian is constructed for an integrable semi-discrete vector NLS equation via Hirota's bilinear method. One- and two-bright-dark soliton solutions are explicitly presented for two-component semi-discrete NLS equation; two-bright-one-dark, and one-bright-two-dark soliton solutions are also given explicitly for three-component semi-discrete NLS equation. The asymptotic behavior is analysed for two-soliton solutions.