# American Institute of Mathematical Sciences

December  2021, 20(12): 4239-4251. doi: 10.3934/cpaa.2021157

## 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 Published  December 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 and Applied Analysis, 2021, 20 (12) : 4239-4251. doi: 10.3934/cpaa.2021157
##### References:
 [1] A. H. Ardila, Orbital stability of standing waves for a system of nonlinear Schrödinger equations with three wave interaction, Nonlinear Anal., 167 (2018), 1-20.  doi: 10.1016/j.na.2017.10.013. [2] M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interactions, Differ. Integral Equ., 17 (2004), 297-330. [3] M. Colin and T. Colin, A numerical model for the Raman amplification for laser-plasma interaction, J. Comput. Appl. Math., 193 (2006), 535-562.  doi: 10.1016/j.cam.2005.05.031. [4] M. Colin and M. Ohta, Bifurcation from semitrivial standing waves and ground states for a system of nonlinear Schrödinger equations, SIAM J. Math. Anal., 44 (2012), 206-223.  doi: 10.1137/110823808. [5] M. Colin, T. Colin and M. Ohta, Instability of standing waves for a system of nonlinear Schrödinger equations with three-wave interaction, Funkcial. Ekvac., 52 (2009), 371-380.  doi: 10.1619/fesi.52.371. [6] M. Colin, T. Colin and M. Ohta, Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2211-2226.  doi: 10.1016/j.anihpc.2009.01.011. [7] H. Kikuchi and M. Ohta, Instability of standing waves for the Klein-Gordon-Schrödinger system, Hokkaido Math. J., 37 (2008), 735-748.  doi: 10.14492/hokmj/1249046366. [8] K. Kurata and Y. Osada, Variational problems associated with a system of nonlinear Schrödinger equations with three wave interaction, Discrete Contin. Dyn. Syst. Ser. B, (2021), 37 pp. doi: 10.3934/dcdsb.2021100. [9] Y. Osada, Energy asymptotic expansion for a system of nonlinear Schrödinger equations with three wave interaction, submitted. [10] A. Pomponio, Ground states for a system of nonlinear Schrödinger equations with three wave interaction, J. Math. Phys., 51 (2010), 093513.  doi: 10.1063/1.3486069. [11] R. Tian, Z. Q. Wang and L. Zhao, Schrödinger systems with quadratic interactions, Commun. Contemp. Math., 21 (2019), 1850077.  doi: 10.1142/S0219199718500773. [12] J. Wang, Solitary waves for coupled nonlinear elliptic system with nonhomogeneous nonlinearities, Calc. Var. Partial Differ. Equ., 56 (2017), 1-38.  doi: 10.1007/s00526-017-1147-3. [13] L. Zhao, F. Zhao and J. Shi, Higher dimensional solitary waves generated by second-harmonic generation in quadratic media, Calc. Var. Partial Differ. Equ., 54 (2015), 2657-2691.

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##### References:
 [1] A. H. Ardila, Orbital stability of standing waves for a system of nonlinear Schrödinger equations with three wave interaction, Nonlinear Anal., 167 (2018), 1-20.  doi: 10.1016/j.na.2017.10.013. [2] M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interactions, Differ. Integral Equ., 17 (2004), 297-330. [3] M. Colin and T. Colin, A numerical model for the Raman amplification for laser-plasma interaction, J. Comput. Appl. Math., 193 (2006), 535-562.  doi: 10.1016/j.cam.2005.05.031. [4] M. Colin and M. Ohta, Bifurcation from semitrivial standing waves and ground states for a system of nonlinear Schrödinger equations, SIAM J. Math. Anal., 44 (2012), 206-223.  doi: 10.1137/110823808. [5] M. Colin, T. Colin and M. Ohta, Instability of standing waves for a system of nonlinear Schrödinger equations with three-wave interaction, Funkcial. Ekvac., 52 (2009), 371-380.  doi: 10.1619/fesi.52.371. [6] M. Colin, T. Colin and M. Ohta, Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2211-2226.  doi: 10.1016/j.anihpc.2009.01.011. [7] H. Kikuchi and M. Ohta, Instability of standing waves for the Klein-Gordon-Schrödinger system, Hokkaido Math. J., 37 (2008), 735-748.  doi: 10.14492/hokmj/1249046366. [8] K. Kurata and Y. Osada, Variational problems associated with a system of nonlinear Schrödinger equations with three wave interaction, Discrete Contin. Dyn. Syst. Ser. B, (2021), 37 pp. doi: 10.3934/dcdsb.2021100. [9] Y. Osada, Energy asymptotic expansion for a system of nonlinear Schrödinger equations with three wave interaction, submitted. [10] A. Pomponio, Ground states for a system of nonlinear Schrödinger equations with three wave interaction, J. Math. Phys., 51 (2010), 093513.  doi: 10.1063/1.3486069. [11] R. Tian, Z. Q. Wang and L. Zhao, Schrödinger systems with quadratic interactions, Commun. Contemp. Math., 21 (2019), 1850077.  doi: 10.1142/S0219199718500773. [12] J. Wang, Solitary waves for coupled nonlinear elliptic system with nonhomogeneous nonlinearities, Calc. Var. Partial Differ. Equ., 56 (2017), 1-38.  doi: 10.1007/s00526-017-1147-3. [13] L. Zhao, F. Zhao and J. Shi, Higher dimensional solitary waves generated by second-harmonic generation in quadratic media, Calc. Var. Partial Differ. Equ., 54 (2015), 2657-2691.
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