Pole Dynamics for Elliptic Solutions of the Korteweg-deVries Equation

TL;DR

This paper investigates the complex pole dynamics of real nonsingular elliptic solutions to the Korteweg-deVries equation, revealing a solvable constrained dynamical system and exploring solutions with soliton limits.

Contribution

It introduces a detailed analysis of pole dynamics for elliptic solutions of the KdV equation, including solvability and special soliton limit cases.

Findings

Pole dynamics governed by a constrained dynamical system

Solvability of the constraint for any finite pole set

Identification of elliptic solutions with real nonsingular soliton limits

Abstract

The real, nonsingular elliptic solutions of the Korteweg-deVries equation are studied through the time dynamics of their poles in the complex plane. The dynamics of these poles is governed by a dynamical system with a constraint. This constraint is shown to be solvable for any finite number of poles located in the fundamental domain of the elliptic function, often in many different ways. Special consideration is given to those elliptic solutions that have a real nonsingular soliton limit.