Memory Driven Pattern Formation

TL;DR

This paper extends the diffusion equation with spatial-temporal memory, leading to non-trivial stationary patterns driven by a balance of currents, with exact solutions and criteria for pattern formation.

Contribution

It introduces a memory-augmented diffusion model that maintains concentration conservation and provides analytical solutions for stationary patterns in various dimensions.

Findings

Memory term induces non-trivial stationary solutions.

Exact solutions are derived and validated numerically.

Pattern formation depends on the memory kernel's properties.

Abstract

The diffusion equation is extended by including spatial-temporal memory in such a manner that the conservation of the concentration is maintained. The additional memory term gives rise to the formation of non-trivial stationary solutions. The steady state pattern in an infinite domain is driven by a competition between conventional particle current and a feedback current. We give a general criteria for the existence of a non-trivial stationary state. The applicability of the model is tested in case of a strongly localized, time independent memory kernel. The resulting evolution equation is exactly solvable in arbitrary dimensions and the analytical solutions are compared with numerical simulations. When the memory term offers an spatially decaying behavior, we find also the exact stationary solution in form of a screened potential.