# Bright-dark mixed $N$-soliton solutions of the multi-component Mel'nikov system

## TL;DR

This paper constructs general bright-dark mixed N-soliton solutions for the multi-component Mel'nikov system using KP hierarchy reduction, analyzing collision behaviors and energy exchanges among solitons in various components.

## Contribution

It provides a unified formula for all-bright, all-dark, and mixed N-soliton solutions in multi-component Mel'nikov systems, including detailed collision analysis.

## Key findings

- Inelastic collisions with energy exchange occur in systems with three or more components.
- Dark solitons and long-wave bright solitons always undergo elastic collisions.
- The solutions encompass all possible combinations of bright and dark solitons across components.

## Abstract

By virtue of the KP hierarchy reduction technique, we construct the general bright-dark mixed $N$-soliton solution to the multi-component Mel'nikov system comprised of multiple (say $M$) short-wave components and one long-wave component with all possible combinations of nonlinearities including all-positive, all-negative and mixed types. Firstly, the two-bright-one-dark (2-b-1-d) and one-bright-two-dark (1-b-2-d) mixed $N$-soliton solutions in short-wave components of the three-component Mel'nikov system are derived in detail. Then we extend our analysis to the $M$-component Mel'nikov system to obtain its general mixed $N$-soliton solution. The formula obtained unifies the all-bright, all-dark and bright-dark mixed $N$-soliton solutions. For the collision of two solitons, the asymptotic analysis shows that for a $M$-component Mel'nikov system with $M \geq 3$, inelastic collision takes place, resulting in energy exchange among the short-wave components supporting bright solitons only if the bright solitons appear at least in two short-wave components. Whereas, the dark solitons in the short-wave components and the bright solitons in the long-wave component always undergo elastic collision which just accompanied by a position shift.