A Guide to Applications of $k$-Contact Geometry in Dissipative Field Equations

TL;DR

This paper explores the use of $k$-contact geometry as a framework for modeling dissipative field equations, providing explicit Hamiltonian descriptions for various nonlinear PDEs with dissipative terms.

Contribution

It develops new geometric tools and criteria within $k$-contact formalism to analyze dissipative PDEs, including explicit Hamiltonian models for several nonlinear equations.

Findings

Derived explicit Hamiltonian descriptions for nonlinear dissipative PDEs.

Established criteria for hyperbolicity and ellipticity in $k$-contact systems.

Developed tools for analyzing regularity and dissipation in geometric PDE frameworks.

Abstract

We study the practical scope of the $k$-contact Hamilton--De Donder--Weyl formalism as a geometric framework for dissipative field equations. In particular, our work focuses on canonical $k$-contact manifolds on $\bigoplus^k {\rm T}^*Q\times\mathbb{R}^k$ and $k$-contactifications of exact $k$-symplectic phase spaces. A special two-contactification of exact two-symplectic structures on cotangent bundles is defined and analysed. We also develop several tools for applications, including splitting results for the Hamilton--De Donder--Weyl equations on $k$-contactifications, regularity conditions for such spaces, criteria for the ultrahyperbolicity, hyperbolicity, or ellipticity of PDEs associated with Hamiltonian $k$-contact systems, dissipation laws associated with infinitesimal dynamical symmetries, relevance and applications of quadratic dissipative terms in the Hamiltonian, etc. Our methods yield explicit Hamiltonian descriptions for several nonlinear nonconservative PDEs with polynomial dissipative terms, including damped Klein--Gordon, Allen--Cahn, generalized Burgers, porous medium equations with linear absorption, complex Ginzburg--Landau, damped nonlinear Schr\"odinger, Fisher--KPP, damped $\phi^4$, damped sine--Gordon, and FitzHugh--Nagumo equations, and many others. Our work also stresses the many further practical applications of this framework.