Abstract
In this article, we consider the classification of the systems of two ordinary differential equations with a complex-valued unknown and gauge-invariant cubic nonlinearities of polynomial type. The class of systems of ODEs arises in the study of the large time behavior of a system of cubic nonlinear Schrödinger equations or that of cubic nonlinear Klein-Gordon equations. We give the complete classification of the systems which possess a conserved quantity of quadratic type. Our argument is based on the representation of a system in terms of a pair of a matrix and a vector.
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