Parameter-free higher-order Schrödinger systems with weak dissipation and forcing

Keeler, J. S., Humphries, B. S., Alberello, A. and Parau, E. (2025) Parameter-free higher-order Schrödinger systems with weak dissipation and forcing. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 481 (2317). ISSN 1364-5021

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Abstract

The higher-order nonlinear Schrödinger equation (NLS) (Dysthe’s equation in the context of water waves) models the time evolution of the slowly modulated amplitude of a wave packet in physical systems described by dispersive partial differential equations (PDEs). These systems, of which water waves are a canonical example, require the presence of a small-valued ordering parameter so that a multi-scale expansion can be performed. However, often the resulting system itself contains this parameter. Thus, these models are difficult to interpret from a formal asymptotics perspective. This article describes a procedure to derive a parameter-free, higher-order evolution equation for a generic infinite-dimensional dispersive PDE with weak linear damping and/or forcing. This is achieved by placing the PDE in an infinite-dimensional Hilbert space and Taylor expanding with Fréchet derivatives. An attractive feature of this procedure is that it can be used in many different physical settings, including water waves, nonlinear optics and any dispersive system with weak dissipation or forcing and does not assume any additional structure to the governing PDE, for example its Hamiltonian nature. To complement this, two specific examples with accompanying symbolic algebra code are demonstrated that can be used as a template for other physical systems.

Item Type:	Article
Additional Information:	Data accessibility: The data are provided in the electronic supplementary material. Funding information: J.S.K. acknowledges support from the Leverhulme trust grant no. (ECF-2021-017) and A.A and E.P acknowledge funding from EPSRC grant no. (EP/Y02012X/1).
Faculty \ School:	Faculty of Science > School of Engineering, Mathematics and Physics
UEA Research Groups:	Faculty of Science > Research Groups > Fluids & Structures Faculty of Science > Research Groups > Quantum Matter
Related URLs:	https://royalsocietypublishing.org/doi/a...
Depositing User:	LivePure Connector
Date Deposited:	09 Jul 2025 09:30
Last Modified:	09 Jul 2025 12:30
URI:	https://ueaeprints.uea.ac.uk/id/eprint/99870
DOI:	10.1098/rspa.2024.0967

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