Ground state solutions for a non-autonomous nonlinear Schrödinger-KdV system

Abstract

We study the Schrödinger-KdV system

\{\begin{array}{ll}-\mathrm{\Delta }u+{\lambda }_{\mathrm{1}}(x)u=u^{\mathrm{3}}+\beta uv,& u\in H^{\mathrm{1}}({ℝ}^N),\\ -\mathrm{\Delta }v+{\lambda }_{\mathrm{2}}(x)v={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{v}^{\mathrm{2}}+{\displaystyle \frac{\beta }{\mathrm{2}}}{u}^{\mathrm{2}},& v\in H^{\mathrm{1}}({ℝ}^N),\end{array}

where N=\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{}{\lambda }_i(x)\in C({ℝ}^N,ℝ),{\lim }_{\left|x\right|\to \infty }{\lambda }_i(x)={\lambda }_i(\infty ), and {\lambda }_i(x)\le {\lambda }_i(\infty ),i= 1,2,a.e. x\in {ℝ}^N.We obtain the existence of nontrivial ground state solutions for the above system by variational methods and the Nehari manifold.

Key words： Schrödinger-KdV system; variational methods; Nehari manifold; ground state solutions

References

1 Albert J, Bhattarai S. Existence and stability of a two-parameter family of solitary waves for an NLS-KdV system. Adv Differential Equations, 2013, 18: 1129–1164

2 Albert J, Pava J A. Existence and stability of ground-states solutions of a Schrödinger-Kdv system. Proc Roy Soc Edinburgh Sect A, 2003, 133(05): 987–1029

3 Ardila A H. Orbital stability of standing waves for a system of nonlinear Schrödinger equations with three wave interaction. Nonlinear Anal, 2018, 167: 1–20

4 Ardila A H. Existence and stability of a two-parameter family of solitary waves for a logarithmic NLS-KdV system. Nonlinear Anal, 2019, 189: 111563

5 Colorado E. Existence of bound and ground states for a system of coupled nonlinear Schrödinger-KdV equations. C R Math Acad Sci Paris, 2015, 353(6): 511–516

6 Colorado E. On the existence of bound and ground states for some coupled nonlinear Schrödinger-Korteweg-de Vries equations. Adv Nonlinear Anal, 2017, 6(4): 407–426

7 Dias J P, Mário Figueira, Oliveira F. Existence of bound states for the coupled Schrödinger-KdV system with cubic nonlinearity. C R Math Acad Sci Paris, 2010, 348(19-20): 1079–1082

8 Funakoshi M, Oikawa M. The resonant interaction between a long internal gravity wave and a surface gravity wave packer packet. J Phys Soc Jpn, 1983, 52(6): 1982–1995

9 Kawahara T, Sugimoto N, Kakutani T. Nonlinear interaction between short and long capillary-gravity waves. J Phys Soc Jpn, 1975, 39(5): 1379–1386

10 Makhankov V G. On stationary solutions of the Schrödinger equation with a selfconsistent potential satisfying Boussinesqs equation. Phys Lett A, 1974, 50(1): 42–44

11 Nishikawa K, Hojo H, Mima H, Ikezi H. Coupled nonlinear electron-plasma and ionacoustic waves. Phys Rev Lett, 1974, 33(3): 148–151

12 Rabinowitz P H. On a class of nonlinear Schrödinger equations. Z Angew Math Phys, 1992, 43(2): 270–291

13 Willem M. Minimax Theorems. Progr Nonlinear Differential Equations Appl, Vol 24. Boston: Birkhäuser, 1996

14 Yajima N, Satsuma J. Soliton solutions in a diatomic lattice system. Prog Theor Phys, 1979, 62(2): 370–378