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: |
[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.
![]() |