Contact-Geometric Description of Dissipative Nonlinear Waves

This paper introduces a contact-geometric framework for dissipative field theories with two main theorems: a Least Constraint Theorem for complex fields and a theorem linking contact geometry to probability measures. Applying this to the 2D Complex Ginzburg-Landau Equation (CGLE), we derive its dissipative contact dynamics, resulting in a Contact Hamilton-Jacobi (CHJ) equation. The theory yields exact travelling-wave solutions and shows a continuous parametric transition from periodic cnoidal waves to localised solitons. Importantly, the geometric probability functional reveals a first-order statistical phase transition, marked by a sharp switch and hysteresis in the probability landscape. Geometrically, probability results from projecting high-dimensional contact information onto the configuration space, where the statistical weight is encoded in the action functional. While demonstrated for CGLE, the principles provide a fundamental framework for analysing pattern selection and phase transitions in nonlinear dissipative systems.