Dynamics of Dissipative Nonlinear Systems: A Study via 2D CGLE by Contact Geometry

TL;DR

This paper introduces a contact-geometric framework for dissipative nonlinear systems, exemplified by the Complex Ginzburg-Landau Equation, revealing new insights into pattern formation, phase transitions, and the role of geometric quantities.

Contribution

It extends contact geometry to dissipative fields, derives exact solutions, and uncovers a universal switching line and phase transition in the CGL equation.

Findings

Exact Jacobi elliptic solutions for CGL

Identification of a universal switching line

Discovery of a first-order phase transition with hysteresis

Abstract

We develop a contact-geometric framework for dissipative nonlinear field theories by extending the least constraint theorem to complex fields and establishing a rigorous link with probability measures. The Complex Ginzburg-Landau Equation serves as a paradigmatic example, yielding a dissipative Contact Hamilton-Jacobi equation that governs the evolution of the action functional. Through canonical transformation and travelling-wave reduction, exact Jacobi elliptic solutions are obtained, revealing a continuous transition from periodic periodons to localised solitons. Probabilistic analysis identifies a universal switching line separating dynamical regimes and uncovers a first-order periodon-soliton phase transition with a hysteresis loop. The conserved contact potential emerges as the key geometric quantity governing pattern formation in dissipative media, analogous to energy in conservative systems.