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Contact-Geometric Dynamics for Dissipative Nonlinear Systems

D. Y. Zhong

TL;DR

This work develops a contact-geometric framework for dissipative complex-field dynamics and extends the Least Constraint Theorem to complex fields, linking contact geometry to probabilistic measures. Applied to the 2D Complex Ginzburg-Landau Equation, it derives a dissipative Contact Hamilton-Jacobi (CHJ) equation and, via a canonical transformation to a real field, obtains exact travelling-wave CHJ solutions expressed with Jacobi elliptic functions, including a soliton limit. A geometric probability measure is constructed from the contact structure, revealing a universal switching line and a first-order periodon–soliton transition with hysteresis; the switching point and hysteresis width are derived from geometric invariants without empirical fitting. The conserved contact potential, rather than energy, governs pattern formation, offering a unifying analytical tool for pattern selection and state transitions in dissipative nonlinear systems with potential applicability beyond the CGLE.

Abstract

Dissipative nonlinear waves are ubiquitous in nonequilibrium physical systems, and the Complex Ginzburg-Landau Equation (CGLE) serves as a fundamental model for describing their dynamics. This paper develops a contact-geometric formulation of dissipative field theories, extending the least constraint theorem to complex fields and establishing a link between contact geometry and probability measures. By applying this framework to the 2D CGLE, we derive the dissipative Contact Hamilton-Jacobi (CHJ) equation, which governs the evolution of the action functional. Through canonical transformation and travelling-wave reduction, exact solutions of the CHJ equation are obtained. From a probabilistic perspective, we derive the probability density functional of the 2D CGLE, identify a universal switching line that separates different dynamical regimes, and reveal a first-order periodon--soliton phase transition with a hysteresis loop. The conserved contact potential is identified as the key geometric quantity governing pattern formation in dissipative media, playing a role analogous to energy in conservative systems. This contact-geometric framework provides a unified analytical tool for studying pattern selection and state transitions.

Contact-Geometric Dynamics for Dissipative Nonlinear Systems

TL;DR

This work develops a contact-geometric framework for dissipative complex-field dynamics and extends the Least Constraint Theorem to complex fields, linking contact geometry to probabilistic measures. Applied to the 2D Complex Ginzburg-Landau Equation, it derives a dissipative Contact Hamilton-Jacobi (CHJ) equation and, via a canonical transformation to a real field, obtains exact travelling-wave CHJ solutions expressed with Jacobi elliptic functions, including a soliton limit. A geometric probability measure is constructed from the contact structure, revealing a universal switching line and a first-order periodon–soliton transition with hysteresis; the switching point and hysteresis width are derived from geometric invariants without empirical fitting. The conserved contact potential, rather than energy, governs pattern formation, offering a unifying analytical tool for pattern selection and state transitions in dissipative nonlinear systems with potential applicability beyond the CGLE.

Abstract

Dissipative nonlinear waves are ubiquitous in nonequilibrium physical systems, and the Complex Ginzburg-Landau Equation (CGLE) serves as a fundamental model for describing their dynamics. This paper develops a contact-geometric formulation of dissipative field theories, extending the least constraint theorem to complex fields and establishing a link between contact geometry and probability measures. By applying this framework to the 2D CGLE, we derive the dissipative Contact Hamilton-Jacobi (CHJ) equation, which governs the evolution of the action functional. Through canonical transformation and travelling-wave reduction, exact solutions of the CHJ equation are obtained. From a probabilistic perspective, we derive the probability density functional of the 2D CGLE, identify a universal switching line that separates different dynamical regimes, and reveal a first-order periodon--soliton phase transition with a hysteresis loop. The conserved contact potential is identified as the key geometric quantity governing pattern formation in dissipative media, playing a role analogous to energy in conservative systems. This contact-geometric framework provides a unified analytical tool for studying pattern selection and state transitions.

Paper Structure

This paper contains 30 sections, 4 theorems, 174 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Contact Geometric Formulation for Complex Fields
  3. Least Constraint Theorem and Complex Field Extension
  4. Geometry of Probability Measure
  5. Geometric Dynamics of 2D CGLE
  6. 2D Complex Ginzburg-Landau Equation
  7. Contact Dynamics of CGLE
  8. Conservation Laws and Symmetries
  9. Contact potential $\mathcal{H}$.
  10. Energy: $\mathcal{N} = |W|^2$.
  11. Canonical Transformation
  12. Contact Hamilton--Jacobi Equation
  13. Traveling-Wave Reduction and Analytical Solutions
  14. Travelling-Wave Reduction
  15. Solution of the CHJ Equation
  16. ...and 15 more sections

Key Result

Theorem 2.1

Let $E\subset \mathbb{R}^n$ be an $n$-dimensional vector space and $\mathbb{P}$ be the space of probability measures about ${E}$. The infinite-order stochastic jet bundle $\pi_{E,0}^{\infty}: J^{\infty}(E,\mathbb{P}) \to E$ satisfies the following properties:

Figures (1)

  • Figure 1: Spatial visualisation of solutions to 2D CGLE based on contact geometry. Column 1 (3D Re(W), subplots (a), (e), (i), (m)) shows the real part of $W(x,y)$; Column 2 (3D $|W|^2$, subplots (b), (f), (j), (n)) presents the intensity $|W|^2 = J\Phi^2(y)$; Column 3 (2D Re(W) Contour, subplots (c), (g), (k), (o)) displays the transverse gradient of the real part; Column 4 ($|W|^2$-arg(W) Overlay, subplots (d), (h), (l), (p)) maps intensity $|W|^2$ to scatter size and phase $\arg(W) = k_xx - \omega t$ to scatter color; Bottom subplots: The left one (subplot (q)) compares shape functions $\phi(y)$ across different $m$; the right one (subplot (r)) shows the real/imaginary parts of $W(x,y)$ at the $x=0$ cross-section, illustrating the transition of solitons from periodicity to localization as $m \to 1^-$. In the calculations, we set $\mu=1.0$ and $|k_x|=0.8$.

Theorems & Definitions (7)

  • Theorem 2.1: Least Constraint for Vector Bundles
  • Theorem 2.2: Least Constraint for Complex Fields
  • proof
  • Lemma 2.1: Evolution of Complex Structure under Contact Flow
  • proof
  • Theorem 2.3: Probability Measure of Complex Fields from Contact Geometry
  • proof