Integrable Systems of Partial Differential Equations Determined by Structure Equations and Lax pair

TL;DR

This paper demonstrates how evolution equations for submanifolds can be derived from structure equations and Lax pairs, revealing their equivalence under certain constraints and allowing more flexible coefficient choices.

Contribution

It introduces a method to generate integrable PDE systems from geometric structure equations and Lax pairs, expanding the class of solvable models with variable coefficients.

Findings

Systems from structure equations and Lax pairs are equivalent under constraints

Coefficients of second fundamental form can be chosen more generally

Provides a framework for generating integrable PDEs from geometry

Abstract

It is shown how a system of evolution equations can be developed both from the structure equations of a submanifold embedded in three-space as well as from a matrix SO(6) Lax pair. The two systems obtained this way correspond exactly when a constraint equation is selected and imposed on the system of equations. This allows the coefficients of the second fundamental form to be selected in a more general way so they need not be constants.