Model reduction of a higher-order KdV equation for shallow water waves

Tassos Bountis; Ko Van Der Weele; Giorgos Kanellopoulos; Kostis Andriopoulos

doi:10.1007/978-3-642-14941-2_15

Model reduction of a higher-order KdV equation for shallow water waves

Tassos Bountis, Ko Van Der Weele, Giorgos Kanellopoulos, Kostis Andriopoulos

Department of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

1 Citation (Scopus)

Abstract

We present novel results on a non-integrable generalized KdV equation proposed by Fokas [A.S. Fokas, Physica D87, 145 (1995)], aiming to describe unidirectional solitary water waves with greater accuracy than the standard KdV equation. The profile of the solitary wave solutions is determined via a reduction of the partial differential equation (PDE) to a set of ordinary differential equations (ODEs). Subsequently, we study the stability of the wave using this profile as initial condition for the PDE. In the case of the standard KdV equation it is well-known that the solitary wave solutions are always stable, irrespective of their height. However, in the case of our higher-order KdV equation we find that the stability of the solutions breaks down beyond a certain critical height, just like solitary waves in real water experiments.

Original language	English
Title of host publication	Coping with Complexity
Subtitle of host publication	Model Reduction and Data Analysis
Pages	287-298
Number of pages	12
DOIs	https://doi.org/10.1007/978-3-642-14941-2_15
Publication status	Published - Jan 1 2011
Event	International Research Workshop: Coping with Complexity: Model Reduction and Data Analysis - Ambleside, United Kingdom Duration: Aug 31 2009 → Sep 4 2009

Publication series

Name	Lecture Notes in Computational Science and Engineering
Volume	75 LNCSE
ISSN (Print)	1439-7358

Other

Other	International Research Workshop: Coping with Complexity: Model Reduction and Data Analysis
Country	United Kingdom
City	Ambleside
Period	8/31/09 → 9/4/09

ASJC Scopus subject areas

Modelling and Simulation
Engineering(all)
Discrete Mathematics and Combinatorics
Control and Optimization
Computational Mathematics

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Model reduction of a higher-order KdV equation for shallow water waves. / Bountis, Tassos; Van Der Weele, Ko; Kanellopoulos, Giorgos; Andriopoulos, Kostis.

Coping with Complexity: Model Reduction and Data Analysis. 2011. p. 287-298 (Lecture Notes in Computational Science and Engineering; Vol. 75 LNCSE).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Bountis, T, Van Der Weele, K, Kanellopoulos, G & Andriopoulos, K 2011, Model reduction of a higher-order KdV equation for shallow water waves. in Coping with Complexity: Model Reduction and Data Analysis. Lecture Notes in Computational Science and Engineering, vol. 75 LNCSE, pp. 287-298, International Research Workshop: Coping with Complexity: Model Reduction and Data Analysis, Ambleside, United Kingdom, 8/31/09. https://doi.org/10.1007/978-3-642-14941-2_15

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abstract = "We present novel results on a non-integrable generalized KdV equation proposed by Fokas [A.S. Fokas, Physica D87, 145 (1995)], aiming to describe unidirectional solitary water waves with greater accuracy than the standard KdV equation. The profile of the solitary wave solutions is determined via a reduction of the partial differential equation (PDE) to a set of ordinary differential equations (ODEs). Subsequently, we study the stability of the wave using this profile as initial condition for the PDE. In the case of the standard KdV equation it is well-known that the solitary wave solutions are always stable, irrespective of their height. However, in the case of our higher-order KdV equation we find that the stability of the solutions breaks down beyond a certain critical height, just like solitary waves in real water experiments.",

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