Partitioning of ensembles of weakly interacting dispersing waves in resonators into disjoint classes

Weakly non-linear interactions of a finite number of waves in a bounded domain are considered. It is evident that the synchronism conditions in this case turn into equations in integers due to boundedness of the domain. A theorem about the partitioning of the set of vectors satisfying synchronism co...

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Bibliographic Details
Published in:Physica. D 1990-10, Vol.46 (1), p.43-56
Main Author: Kartashova, Elena A.
Format: Article
Language:English
Subjects:
Earth, ocean, space
Exact sciences and technology
External geophysics
Physics of the oceans
Surface waves, tides and sea level. Seiches
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Summary:Weakly non-linear interactions of a finite number of waves in a bounded domain are considered. It is evident that the synchronism conditions in this case turn into equations in integers due to boundedness of the domain. A theorem about the partitioning of the set of vectors satisfying synchronism conditions into non-intersecting classes is proved. Cases are found in which for a fixed vector k with integer coordinates there exist no vectors constituting a solution together with it. Examples are given of situations in which the said classes are infinite, finite, or empty. Methods of algebraic number theory have been used.
ISSN:0167-2789
1872-8022
DOI:10.1016/0167-2789(90)90112-3