# Generating mechanism and dynamic of the smooth positons for the derivative nonlinear Schr\"odinger equation

## TL;DR

This paper constructs and analyzes smooth positon solutions for the derivative nonlinear Schrödinger equation using Darboux transformations, revealing their complex dynamics and interactions through detailed theoretical and graphical analysis.

## Contribution

It introduces a method to generate higher-order smooth positon solutions and explores their dynamic behaviors and interactions, which were not previously detailed.

## Key findings

- Generated explicit n-order smooth positon solutions.
- Analyzed the dynamic behaviors and phase shifts of positons.
- Presented hybrid solutions with complex interaction patterns.

## Abstract

Based on the degenerate Darboux transformation, the $n$-order smooth positon solutions for the derivative nonlinear Schr\"{o}dinger equation are generated by means of the general determinant expression of the $N$-soliton solution, and interesting dynamic behaviors of the smooth positons are shown by the corresponding three dimensional plots in this paper. Furthermore, the decomposition process, bent trajectory and the change of the phase shift for the positon solutions are discussed in detail. Additional, three kinds of mixed solutions, namely (1) the hybrid of one-positon and two-positon solutions, (2) the hybrid of two-positon and two-positon solutions, and (3) the hybrid of one-soliton and three-positon solutions are presented and their rather complicated dynamics are revealed.