Modeling Wave Packet Dynamics and Exploring Applications: A Comprehensive Guide to the Nonlinear Schrödinger Equation

Natanael Karjanto ()
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Natanael Karjanto: Department of Mathematics, University College, Natural Science Campus, Sungkyunkwan University, Suwon 16419, Republic of Korea

Mathematics, 2024, vol. 12, issue 5, 1-32

Abstract: The nonlinear Schrödinger (NLS) equation stands as a cornerstone model for exploring the intricate behavior of weakly nonlinear, quasi-monochromatic wave packets in dispersive media. Its reach extends across diverse physical domains, from surface gravity waves to the captivating realm of Bose–Einstein condensates. This article delves into the dual facets of the NLS equation: its capacity for modeling wave packet dynamics and its remarkable breadth of applications. We illuminate the derivation of the NLS equation through both heuristic and multiple-scale approaches, underscoring how distinct interpretations of physical variables and governing equations give rise to varied wave packet dynamics and tailored values for dispersive and nonlinear coefficients. To showcase its versatility, we present an overview of the NLS equation’s compelling applications in four research frontiers: nonlinear optics, surface gravity waves, superconductivity, and Bose–Einstein condensates. This exploration reveals the NLS equation as a powerful tool for unifying and understanding a vast spectrum of physical phenomena.

Keywords: nonlinear Schrödinger equation; Korteweg–de Vries equation; sine-Gordon equation; nonlinear Klein–Gordon equation; Gross–Pitaevskii equation; multiple-scale method; nonlinear optics; gravity waves; superconductivity; Bose–Einstein condensation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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