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Quadratic invariants for discrete clusters of weakly interacting waves
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Harper, Katie L., Bustamante, Miguel D. and Nazarenko, Sergey (2013) Quadratic invariants for discrete clusters of weakly interacting waves. Journal of Physics A: Mathematical and Theoretical, Volume 46 (Number 24). Article number 245501. doi:10.1088/1751-8113/46/24/245501 ISSN 1751-8113.
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Official URL: http://dx.doi.org/10.1088/1751-8113/46/24/245501
Abstract
We consider discrete clusters of quasi-resonant triads arising from a Hamiltonian three-wave equation. A cluster consists of N modes forming a total of M connected triads. We investigate the problem of constructing a functionally independent set of quadratic constants of motion. We show that this problem is equivalent to an underlying basic linear problem, consisting of finding the null space of a rectangular M × N matrix with entries 1, −1 and 0. In particular, we prove that the number of independent quadratic invariants is equal to J ≡ N − M* ≥ N − M, where M* is the number of linearly independent rows in Thus, the problem of finding all independent quadratic invariants is reduced to a linear algebra problem in the Hamiltonian case. We establish that the properties of the quadratic invariants (e.g., locality) are related to the topological properties of the clusters (e.g., types of linkage). To do so, we formulate an algorithm for decomposing large clusters into smaller ones and show how various invariants are related to certain parts of a cluster, including the basic structures leading to M* < M. We illustrate our findings by presenting examples from the Charney–Hasegawa–Mima wave model, and by showing a classification of small (up to three-triad) clusters.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Algebras, Linear, Hamiltonian systems, Mathematical analysis, Nonlinear theories, Nonlinear waves, Waves | ||||
Journal or Publication Title: | Journal of Physics A: Mathematical and Theoretical | ||||
Publisher: | IOP Publishing Ltd | ||||
ISSN: | 1751-8113 | ||||
Official Date: | 30 May 2013 | ||||
Dates: |
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Volume: | Volume 46 | ||||
Number: | Number 24 | ||||
Page Range: | Article number 245501 | ||||
DOI: | 10.1088/1751-8113/46/24/245501 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Open Access (Creative Commons) | ||||
Date of first compliant deposit: | 25 December 2015 | ||||
Date of first compliant Open Access: | 25 December 2015 | ||||
Funder: | University College Dublin (UCD), Russia (Federation), Engineering and Physical Sciences Research Council (EPSRC) | ||||
Grant number: | SF564, SF652 (UCD) ; 12.740.11.1430 (RF) ; |
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