A cohomological approach to immersed submanifolds via integrable systems

TL;DR

This paper introduces a cohomological framework for understanding immersed soliton surfaces, providing new tools to analyze their integrability, deformations, and generalizations within the context of Lie algebras and PDE systems.

Contribution

It develops a cohomological approach to soliton surface theory, including Poincaré lemmas and generalizations to soliton submanifolds, enhancing the understanding of integrability conditions.

Findings

Established Poincaré-type lemmas for new cohomologies

Clarified the structure of soliton surface deformations

Generalized soliton surface theory to submanifolds

Abstract

A geometric approach to immersion formulas for soliton surfaces is provided through new cohomologies on spaces of special types of $\mathfrak{g}$-valued differential forms. This leads us to introduce Poincar\'e-type lemmas for these cohomologies, which appropriately describe the integrability conditions of Lax pairs associated with systems of PDEs. Our methods clarify the structure and properties of the deformations and soliton surfaces for the aforesaid Lax pairs. Our findings also allow for the generalization of the theory of soliton surfaces in Lie algebras to general soliton submanifolds. Techniques from the theory of infinite-dimensional jet manifolds and diffieties enable us to justify certain common assumptions of the theory of soliton surfaces. Theoretical results are illustrated through $\mathbb{C}P^{N-1}$ sigma models.