Energy asymptotic expansion for a system of nonlinear Schrödinger equations with three wave interaction

Abstract

In this paper we consider the asymptotic expansion of the energy $I_{\alpha }^{\beta }(\gamma ,\mu ,s)$ associated with a nonlinear Schrödinger system with three wave interaction as $\beta \to \infty$ with $\alpha ={\beta }^{\kappa }$ for a given $\kappa \in ℝ$. In particular, we classify the asymptotic expansion formula into five cases for the parameter $\kappa$.