Nonlinear Brinkman-penalised Hamiltonian PDEs

Develop a detailed treatment of nonlinear Hamiltonian partial differential equations within the Brinkman-penalised multi-symplectic class, including nonlinear wave and Schrödinger-type equations with partially damped penalisations and convergence rates as the penalty parameter η tends to zero that preserve the conformal structure.

Background

The paper establishes multi-conformal symplectic structure, structure-preserving splitting schemes, and conformal symplectic neural operators primarily for the linear wave equation and Maxwell’s equations with Brinkman-type penalisation. These examples satisfy the compatibility condition linking the penalisation projection to the symplectic structure matrix.

The authors explicitly identify the extension to nonlinear Hamiltonian PDEs as unresolved. The open direction includes both determining how partial damping interacts with nonlinear Hamiltonian structure and obtaining convergence estimates for the penalised equations as η approaches zero without losing the conformal geometric properties.