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In this paper, we study the large-time behavior of small solutions to thestandard form of the systems of 1D cubic nonlinear Schr\"odinger equationsconsisting of two components and possessing a coercive mass-like conservedquantity. The cubic nonlinearity is known to be critical in one space dimensionin view of the large-time behavior. By employing the result by Katayama andSakoda, one can obtain the large-time behavior of the solution if we canintegrate the corresponding ODE system. We introduce an integration schemesuited to the system. The key idea is to rewrite the ODE system, which iscubic, as a quadratic system of quadratic quantities of the original unknown.By using this technique, we described the large-time behavior of solutions interms of elementary functions and the Jacobi elliptic functions for severalexamples of standard systems.

Partial classification of the large-time behavior of solutions to cubic nonlinear Schr\"odinger systems