Soliton surfaces associated with symmetries of ODEs written in Lax representation

TL;DR

This paper extends the Fokas-Gel'fand method for constructing soliton surfaces from PDEs to integrable ODEs with Lax representations, providing explicit formulas and examples involving elliptic functions.

Contribution

It adapts the soliton surface construction procedure to integrable ODEs with Lax pairs, including explicit formulas and applications to elliptic function solutions.

Findings

Derived explicit immersion functions for integrable ODEs with Lax representations.

Applied the method to static φ^4 equations leading to elliptic function surfaces.

Produced diverse soliton surfaces for different Jacobian elliptic functions.

Abstract

The main aim of this paper is to discuss recent results on the adaptation of the Fokas-Gel'fand procedure for constructing soliton surfaces in Lie algebras, which was originally derived for PDEs [Grundland, Post 2011], to the case of integrable ODEs admitting Lax representations. We give explicit forms of the $\g$-valued immersion functions based on conformal symmetries involving the spectral parameter, a gauge transformation of the wave function and generalized symmetries of the linear spectral problem. The procedure is applied to a symmetry reduction of the static $\phi^4$-field equations leading to the Jacobian elliptic equation. As examples, we obtain diverse types of surfaces for different choices of Jacobian elliptic functions for a range of values of parameters.