\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Asymptotic expansion of the ground state energy for nonlinear Schrödinger system with three wave interaction

  • * Corresponding author

    * Corresponding author

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

Abstract / Introduction Full Text(HTML) Related Papers Cited by
  • 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^* $.

    Mathematics Subject Classification: Primary: 35Q55, 35B40; Secondary: 35J50.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [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. 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.
    [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.
    [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. TianZ. 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. 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. 
  • 加载中
SHARE

Article Metrics

HTML views(2153) PDF downloads(177) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return