Characteristic analysis for integrable soliton models on two-dimensional target spaces

TL;DR

This paper analyzes integrable soliton models with two-dimensional target spaces, revealing their rich solution structures and the ill-posedness of wave propagation due to non-hyperbolic equations.

Contribution

It provides a characteristic analysis of these models, highlighting the topological solutions and the fundamental ill-posedness of wave propagation.

Findings

Models have infinitely many conserved quantities

Existence of exact solutions with nontrivial topology

Wave propagation is ill-posed despite smooth solutions

Abstract

We investigate the evolutionary aspects of some integrable soliton models whose Lagrangians are derived from the pullback of a volume-form to a two-dimensional target space. These models are known to have infinitely many conserved quantities and support various types of exact analytic solutions with nontrivial topology. In particular, we show that, in spite of the fact that they admit nice smooth solutions, wave propagation about these solutions will always be ill-posed. This is related to the fact that the corresponding Euler-Lagrange equations are not of hyperbolic type.