-
Previous Article
Stability of standing waves for a nonlinear Schrödinger equation wdelta potentialith a repulsive Dirac
- DCDS Home
- This Issue
-
Next Article
Superposition of selfdual functionals in non-homogeneous boundary value problems and differential systems
Schrödinger equations with a spatially decaying nonlinearity: Existence and stability of standing waves
| 1. | Station 8, IACS-FSB, EPFL, CH-1015, Lausanne, Switzerland, Switzerland |
$i\partial_{t}w+\Delta_{x}w+V(x) |w| ^{p-1}w=0\text{ where }w=w(t,x):\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{C}$
with a potential $V$ that decays at infinity like $| x|^{-b}$ for some $b\in (0,2)$. A standing wave is a solution of the form
$w(t,x)=e^{i\lambda t}u(x)\text{ where }\lambda>0\text{ and }u:\mathbb{R}^{N}\rightarrow\mathbb{R}.$
For $ 1 < p < 1+(4-2b)/(N-2)$, we establish the existence of a $C^1$-branch of standing waves parametrized by frequencies $\lambda $ in a right neighbourhood of $0$. We also prove that these standing waves are orbitally stable if $ 1 < p < 1+(4-2b)/N$ and unstable if $1+(4-2b)/N < p < 1+(4-2b)/(N-2)$.
| [1] |
Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure & Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843 |
| [2] |
Hiroaki Kikuchi. Remarks on the orbital instability of standing waves for the wave-Schrödinger system in higher dimensions. Communications on Pure & Applied Analysis, 2010, 9 (2) : 351-364. doi: 10.3934/cpaa.2010.9.351 |
| [3] |
Fábio Natali, Ademir Pastor. Orbital stability of periodic waves for the Klein-Gordon-Schrödinger system. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 221-238. doi: 10.3934/dcds.2011.31.221 |
| [4] |
Sevdzhan Hakkaev. Orbital stability of solitary waves of the Schrödinger-Boussinesq equation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1043-1050. doi: 10.3934/cpaa.2007.6.1043 |
| [5] |
Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359 |
| [6] |
Marco Ghimenti, Stefan Le Coz, Marco Squassina. On the stability of standing waves of Klein-Gordon equations in a semiclassical regime. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2389-2401. doi: 10.3934/dcds.2013.33.2389 |
| [7] |
Reika Fukuizumi. Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 525-544. doi: 10.3934/dcds.2001.7.525 |
| [8] |
Salvador Cruz-García, Catherine García-Reimbert. On the spectral stability of standing waves of the one-dimensional $M^5$-model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1079-1099. doi: 10.3934/dcdsb.2016.21.1079 |
| [9] |
François Genoud. Existence and stability of high frequency standing waves for a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1229-1247. doi: 10.3934/dcds.2009.25.1229 |
| [10] |
Fábio Natali, Ademir Pastor. Stability properties of periodic standing waves for the Klein-Gordon-Schrödinger system. Communications on Pure & Applied Analysis, 2010, 9 (2) : 413-430. doi: 10.3934/cpaa.2010.9.413 |
| [11] |
Alex H. Ardila. Stability of standing waves for a nonlinear SchrÖdinger equation under an external magnetic field. Communications on Pure & Applied Analysis, 2018, 17 (1) : 163-175. doi: 10.3934/cpaa.2018010 |
| [12] |
Reika Fukuizumi, Louis Jeanjean. Stability of standing waves for a nonlinear Schrödinger equation wdelta potentialith a repulsive Dirac. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 121-136. doi: 10.3934/dcds.2008.21.121 |
| [13] |
Riccardo Adami, Diego Noja, Cecilia Ortoleva. Asymptotic stability for standing waves of a NLS equation with subcritical concentrated nonlinearity in dimension three: Neutral modes. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5837-5879. doi: 10.3934/dcds.2016057 |
| [14] |
Aslihan Demirkaya, Milena Stanislavova. Numerical results on existence and stability of standing and traveling waves for the fourth order beam equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 197-209. doi: 10.3934/dcdsb.2018097 |
| [15] |
Yonggeun Cho, Hichem Hajaiej, Gyeongha Hwang, Tohru Ozawa. On the orbital stability of fractional Schrödinger equations. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1267-1282. doi: 10.3934/cpaa.2014.13.1267 |
| [16] |
Michael Herrmann. Homoclinic standing waves in focusing DNLS equations. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 737-752. doi: 10.3934/dcds.2011.31.737 |
| [17] |
Michiel Bertsch, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. Standing and travelling waves in a parabolic-hyperbolic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5603-5635. doi: 10.3934/dcds.2019246 |
| [18] |
Masahito Ohta. Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1671-1680. doi: 10.3934/cpaa.2018080 |
| [19] |
Xiaoyu Zeng. Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1749-1762. doi: 10.3934/dcds.2017073 |
| [20] |
Yue Liu. Existence of unstable standing waves for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 193-209. doi: 10.3934/cpaa.2008.7.193 |
2018 Impact Factor: 1.143
Tools
Metrics
Other articles
by authors
[Back to Top]