The Effect of Nonlinearities With Arbitrary-Order Derivatives on Dynamic Transitions

Abstract

The primary objective of this paper is to classify the first transitions of a general class of one spatial dimensional nonlinear partial differential equations on a bounded interval. The linear part of the equation is assumed to have a real discrete spectrum with a complete set of eigenfunctions, which are of the form $\sin kx$ or $\cos kx,k\in \mathbb{N}$. The nonlinear operator consists of arbitrary finite products and sums of the unknown function and its derivatives of arbitrary order. The equations allow for a trivial steady-state solution that becomes unstable when a control parameter exceeds a certain threshold. Unlike most of the previous research in this direction that considers specific equations, this general framework is suitable for extension in several directions such as the higher spatial dimensions and different basis vectors. Under a set of assumptions that are often valid in many interesting applications, we derive two numbers called the transition number and the critical index which completely describe the first dynamic transition. We make detailed numerical computations that reveal the properties of the transition numbers. To show the applicability of our theoretical results, we determine the first transitions of several well-known equations including the Cahn–Hilliard, thin film, Harry Dym, Kawamoto, and Rosenau–Hyman equations.