2018 journal article
Nonlinear Schrodinger equations and the universal description of dispersive shock wave structure
STUDIES IN APPLIED MATHEMATICS, 142(3), 241–268.
author keywords: asymptotic analysis; nonlinear waves; partial differential equations
open access via onlinelibrary.wiley.com (publisher PDF)
UN Sustainable Development Goal Categories
14. Life Below Water
(Web of Science)
Source: Web Of Science
Added: April 2, 2019
AbstractThe nonlinear Schrödinger (NLS) equation and the Whitham modulation equations both describe slowly varying, locally periodic nonlinear wavetrains, albeit in differing amplitude‐frequency domains. In this paper, we take advantage of the overlapping asymptotic regime that applies to both the NLS and Whitham modulation descriptions in order to develop a universal analytical description of dispersive shock waves (DSWs) generated in Riemann problems for a broad class of integrable and nonintegrable nonlinear dispersive equations. The proposed method extends DSW fitting theory that prescribes the motion of a DSW's edges into the DSW's interior, that is, this work reveals the DSW structure. Our approach also provides a natural framework in which to analyze DSW stability. We consider several representative, physically relevant examples that illustrate the efficacy of the developed general theory. Comparisons with direct numerical simulations show that inclusion of higher order terms in the NLS equation enables a remarkably accurate description of the DSW structure in a broad region that extends from the harmonic, small amplitude edge.