Instabilities of standing waves and positivity in traveling waves to a higher-order Schrödinger equation

Abstract

The aim of this paper is to explore a Schrödinger equation that incorporates a higher-order operator. Traditional models for electron dynamics have utilized a second-order diffusion Schrödinger equation, where oscillatory behavior is achieved through complex domain formulations. Incorporating a higher-order operator enables the induction of oscillatory spatial patterns in solutions. Our analysis initiates with a variational formulation within generalized spaces, facilitating the examination of solution boundedness. Subsequently, we delve into the oscillatory characteristics of solutions, drawing upon a series of lemmas originally applied to the Kuramoto–Sivashinsky equation, the Cahn–Hilliard equation, and other equations that employ higher-order operators. Specific solution types, such as standing waves, are numerically investigated to illustrate the oscillatory spatial patterns. The discussion then extends to the theory of traveling waves to establish general conditions for positive solutions. A contribution of this work is the precise evaluation of a critical traveling wave speed, denoted as c* above which the first minimum remains positive. For values of the traveling wave speed significantly greater than c*, the solutions can be entirely positive.