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
References:
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Y. Osada, Energy asymptotic expansion for a system of nonlinear Schrödinger equations with three wave interaction, submitted. Google Scholar

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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.  Google Scholar

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show all references

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.  Google Scholar

[2]

M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interactions, Differ. Integral Equ., 17 (2004), 297-330.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. ColinT. 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.  Google Scholar

[6]

M. ColinT. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

Y. Osada, Energy asymptotic expansion for a system of nonlinear Schrödinger equations with three wave interaction, submitted. Google Scholar

[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.  Google Scholar

[11]

R. TianZ. Q. Wang and L. Zhao, Schrödinger systems with quadratic interactions, Commun. Contemp. Math., 21 (2019), 1850077.  doi: 10.1142/S0219199718500773.  Google Scholar

[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.  Google Scholar

[13]

L. ZhaoF. 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.   Google Scholar

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