Exact quasi-periodic solutions to the MKdV equation

TL;DR

This paper derives exact quasi-periodic solutions for the modified Korteweg-de Vries (mKdV) equation using algebraic geometry, involving hyperelliptic spectral curves and $ ext{wp}$-functions, with applications to wave phenomena.

Contribution

It introduces a hierarchy of solutions to the mKdV equation via algebraic geometry, specifically using hyperelliptic curves and $ ext{wp}$-functions, extending previous finite-gap solutions.

Findings

Explicit finite-gap solutions in terms of $ ext{wp}$-functions.

Complete characterization of reality conditions for solutions.

Visual illustrations of solutions in small genera.

Abstract

In the present paper, a hierarchy of the mKdV equation is integrated by the methods of algebraic geometry. The mKdV hierarchy in question arises on coadjoint orbits in the loop algebra of $\mathfrak{sl}(2)$, and employs a family of hyperelliptic curves as spectral curves. A generic form of the finite-gap solution in any genus is obtained in terms of the $\wp$-functions, which generalize the Weierstrass $\wp$-function. Reality conditions for quasi-periodic wave solutions are completely specified. The obtained solutions are illustrated by plots in small genera.