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

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.

💡 Research Summary

This paper presents a groundbreaking contact-geometric framework for analyzing dissipative nonlinear field theories, with the two-dimensional Complex Ginzburg-Landau Equation (CGLE) serving as the primary paradigm. The core theoretical advancement is the extension of the Least Constraint Theorem—previously established for real vector bundles—to complex-valued fields. This extension rigorously defines a complex contact manifold equipped with a contact 1-form Θ_C = Hdt - Π*_j dW_j, where W is the complex field (e.g., amplitude in CGLE) and Π* is its canonically conjugate local connection. From this structure, the authors derive a dissipative Contact Hamilton-Jacobi (CHJ) equation, ∂S/∂t + H