Bosonization, Painleve property, exact solutions for N=1 supersymmetric mKdV equation

TL;DR

This paper applies bosonization and symmetry methods to analyze the N=1 supersymmetric mKdV equation, deriving exact solutions and transformations that extend understanding of its integrability and solution space.

Contribution

It introduces a bosonization approach for the supersymmetric mKdV, deriving new exact solutions, Bäcklund transformations, and symmetry reductions not previously reported.

Findings

The bosonized system passes the Painlevé test, indicating integrability.

New exact traveling wave solutions are constructed using the mapping and deformation method.

A nonauto-Bäcklund theorem is established, leading to novel explicit solutions.

Abstract

The N=1 supersymmetric modified Korteweg-de Vries (SmKdV) system is transformed to a system of coupled bosonic equations with the bosonization approach. The bosonized SmKdV (BSmKdV) passes the Painlev\'{e} test and allows a set of B\"{a}cklund transformation (BT) by truncating the series expansions of the solutions about the singularity manifold. The traveling wave solutions of the BSmKdV system are obtained using the mapping and deformation method. Some special types of exact solutions for the BSmKdV system are found with the solutions and symmetries of the usual mKdV equation. In the meanwhile, the similarity reduction solutions of the system are investigated by using the Lie point symmetry theory. The generalized tanh function expansion method for the BSmKdV system leads to a nonauto-BT theorem. Using the nonauto-BT theorem, the novel exact explicit solutions of the BSmKdV system can be obtained. All these solutions obtained via the bosonization procedure are different from those obtained via other methods.