The Construction of the mKdV Cyclic Symmetric $N$-soliton Solution by the B\"{a}cklund Transformation

TL;DR

This paper explores the algebraic construction of cyclic symmetric N-soliton solutions for the mKdV equation using Bäcklund transformations, highlighting the role of M"obius symmetry and addition formulas.

Contribution

It introduces a novel algebraic method to construct mKdV N-soliton solutions via Bäcklund transformations and M"obius symmetry, revealing underlying group structures.

Findings

Bäcklund transformations generate cyclic symmetric N-soliton solutions.

M"obius symmetry plays a key role in the solution construction.

Addition formulas are observed in the algebraic process.

Abstract

We study group theoretical structures of the mKdV equation. The Schwarzian type mKdV equation has the global M\"{o}bius group symmetry. The Miura transformation makes a connection between the mKdV equation and the KdV equation. We find the special local M\"{o}bius transformation on the mKdV one-soliton solution which can be regarded as the commutative KdV B\"{a}cklund transformation can generate the mKdV cyclic symmetric $N$-soliton solution. In this algebraic construction to obtain multi-soliton solutions, we could observe the addition formula.