# Smooth positon solutions of the focusing modified Korteweg-de Vries equation

## TL;DR

This paper derives explicit determinant-based formulas for smooth positon solutions of the focusing mKdV equation, including their decomposition into solitons, trajectories, and phase shifts, advancing understanding of these nonlinear wave solutions.

## Contribution

It introduces a determinant representation for n-positon solutions of the focusing mKdV equation and analyzes their structure and dynamics.

## Key findings

- Explicit determinant formulas for n-positon solutions
- Decomposition of positons into single solitons
- Analysis of trajectories and phase shifts

## Abstract

The $n$-fold Darboux transformation $T_{n}$ of the focusing real mo\-di\-fied Kor\-te\-weg-de Vries (mKdV) equation is expressed in terms of the determinant representation. Using this representation, the $n$-soliton solutions of the mKdV equation are also expressed by determinants whose elements consist of the eigenvalues $\lambda_{j}$ and the corresponding eigenfunctions of the associated Lax equation. The nonsingular $n$-positon solutions of the focusing mKdV equation are obtained in the special limit $\lambda_{j}\rightarrow\lambda_{1}$, from the corresponding $n$-soliton solutions and by using the associated higher-order Taylor expansion. Furthermore, the decomposition method of the $n$-positon solution into $n$ single-soliton solutions, the trajectories, and the corresponding "phase shifts" of the multi-positons are also investigated.