On real hyperelliptic solutions of focusing modified KdV equation

TL;DR

This paper constructs and analyzes real hyperelliptic solutions of the focusing MKdV equation of genus three, revealing their properties and numerical behavior through a novel approach involving the gauged MKdV equation.

Contribution

It introduces a new method to construct real hyperelliptic solutions of the focusing MKdV equation via the gauged MKdV framework, especially for genus three.

Findings

Constructed real hyperelliptic solutions for genus three

Established connection with gauged MKdV equation

Numerical analysis of gauge field behavior

Abstract

We study the real hyperelliptic solutions of the focusing modified KdV (MKdV) equation of the genus three. Since the complex hyperelliptic solutions of the focusing MKdV equation over $\mathbb{C}$ are associated with the real gauged MKdV equation, we present a novel construction of the real hyperelliptic solutions of the gauged MKdV equation. When the gauge field is constant, it can be regarded as the real solution of the focusing MKdV equation, and thus we also discuss the behavior of the gauge field numerically.