A New Class of Elliptic Finite-Gap Densities of the Polar Operator and Stationary Solutions of the Harry Dym equation

TL;DR

This paper introduces a new class of elliptic densities for the polar operator, constructed via a modified approach, and explores their stationary solutions and transformations within the context of the Harry Dym equation.

Contribution

It presents a novel family of finite-gap elliptic densities for the polar operator and analyzes their stationary solutions and Bäcklund transformations.

Findings

Constructed new one- and two-gap elliptic densities

Developed auto-Bäcklund transformations for stationary solutions

Studied properties of these solutions

Abstract

A new family of one- and two-gap elliptic densities of the polar operator has been constructed by modifying the so-called "higher time approach" to constructing finite-gap solutions of the Harry Dym equation. Auto-B\"{a}cklund transformations of the obtained stationary solutions have been constructed and their properties have been studied.