A novel hybrid dispersive–dissipative model: A Schrödinger-type Cahn–Hilliard system with memory and elastic coupling (SCHMEC)

We introduce and analyze the Schrödinger-type Cahn–Hilliard–memory–elasticity coupling (SCHMEC) model, a hybrid framework designed to capture the interplay between dispersive wave dynamics, conserved phase separation, memory-dependent chemical evolution, and weak elastic responses. The system blends features of the nonlinear Schrödinger and Gross–Pitaevskii equations with Cahn–Hilliard coarsening and integro-differential memory effects, thereby unifying conservative and dissipative mechanisms within a single partial differential equation structure. Mass conservation in the phase field, convolutional memory in the concentration dynamics, and wave-driven interface motion are embedded in the formulation, ensuring consistency with physical conservation laws. We derive the governing equations, clarify their mathematical structure, and position the model within the broader landscape of quantum, optical, and soft-matter systems. Numerical and analytical results demonstrate how linear instability bands translate into rich nonlinear dynamics, including labyrinthine morphologies, oscillatory ripples, and frozen oscillatory states. The SCHMEC framework thus provides a versatile platform for studying pattern formation, bifurcation, and multiscale interactions in systems where memory, dispersion, and elasticity act in concert.