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 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. © 1990.