# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021157
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## Asymptotic expansion of the ground state energy for nonlinear Schrödinger system with three wave interaction

 Department of Mathematical Sciences, Tokyo Metropolitan University, 1-1 Minami Osawa, Hachioji, Tokyo 192-0397, Japan

* Corresponding author

Received  March 2021 Revised  July 2021 Early access September 2021

Fund Project: The first author was supported by JSPS KAKENHI Grant Numbers 17H01092, 19K03587

In this paper, we consider the asymptotic behavior of the ground state and its energy for the nonlinear Schrödinger system with three wave interaction on the parameter $\gamma$ as $\gamma \to \infty$. In addition we prove the existence of the positive threshold $\gamma^*$ such that the ground state is a scalar solution for $0 \le \gamma < \gamma^*$ and is a vector solution for $\gamma > \gamma^*$.

Citation: Kazuhiro Kurata, Yuki Osada. Asymptotic expansion of the ground state energy for nonlinear Schrödinger system with three wave interaction. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021157
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