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Register a new userStability and instability of standing waves for a generalized Choquard equation with potential
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We are going to study the standing waves for a generalized Choquard equation with potential: $$ -ipartial_t u-Delta u+V(x)u=(|x|^{-mu}ast|u|^p)|u|^{p-2}u, hbox{in} mathbb{R}timesmathbb{R}^3, $$ where $V(x)$ is a real function, $0<mu<3$, $2-mu/3<p<6-mu$ and $ast$ stands for convolution. Under suitable assumptions on the potential and appropriate frequency $omega$ , the stability and instability of the standing waves $u=e^{i omega t}varphi(x)$ are investigated .
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We study the instability of standing-wave solutions $e^{iomega t}phi_{omega}(x)$ to the inhomogeneous nonlinear Schr{o}dinger equation $$iphi_t=-trianglephi+|x|^2phi-|x|^b|phi|^{p-1}phi, qquad inmathbb{R}^N, $$ where $ b > 0 $ and $ phi_{omega} $ is a ground-state solution. The results of the instability of standing-wave solutions reveal a balance between the frequency $omega $ of wave and the power of nonlinearity $p $ for any fixed $ b > 0. $
In this paper we establish the orbital stability of standing wave solutions associated to the one-dimensional Schrodinger-Kirchhoff equation. The presence of a mixed term gives us more dispersion, and consequently, a different scenario for the stability of solitary waves in contrast with the corresponding nonlinear Schrodinger equation. For periodic waves, we exhibit two explicit solutions and prove the orbital stability in the energy space.
The orbital instability of standing waves for the Klein-Gordon-Zakharov system has been established in two and three space dimensions under radially symmetric condition, see Ohta-Todorova (SIAM J. Math. Anal. 2007). In the one space dimensional case, for the non-degenerate situation, we first check that the Klein-Gordon-Zakharov system satisfies Grillakis-Shatah-Strauss assumptions on the stability and instability theorems for abstract Hamiltonian systems, see Grillakis-Shatah-Strauss (J. Funct. Anal. 1987). As to the degenerate case that the frequency $|omega|=1/sqrt{2}$, we follow Wu (ArXiv: 1705.04216, 2017) to describe the instability of the standing waves for the Klein-Gordon-Zakharov system, by using the modulation argument combining with the virial identity. For this purpose, we establish a modified virial identity to overcome several troublesome terms left in the traditional virial identity.
In this paper we study the existence and the instability of standing waves with prescribed $L^2$-norm for a class of Schrodinger-Poisson-Slater equations in $R^{3}$ %orbitally stable standing waves with arbitray charge for the following Schrodinger-Poisson type equation label{evolution1} ipsi_{t}+ Delta psi - (|x|^{-1}*|psi|^{2}) psi+|psi|^{p-2}psi=0 % text{in} R^{3}, when $p in (10/3,6)$. To obtain such solutions we look to critical points of the energy functional $$F(u)=1/2| triangledown u|_{L^{2}(mathbb{R}^3)}^2+1/4int_{mathbb{R}^3}int_{mathbb{R}^3}frac{|u(x)|^2| u(y)|^2}{|x-y|}dxdy-frac{1}{p}int_{mathbb{R}^3}|u|^pdx $$ on the constraints given by $$S(c)= {u in H^1(mathbb{R}^3) :|u|_{L^2(R^3)}^2=c, c>0}.$$ For the values $p in (10/3, 6)$ considered, the functional $F$ is unbounded from below on $S(c)$ and the existence of critical points is obtained by a mountain pass argument developed on $S(c)$. We show that critical points exist provided that $c>0$ is sufficiently small and that when $c>0$ is not small a non-existence result is expected. Concerning the dynamics we show for initial condition $u_0in H^1(R^3)$ of the associated Cauchy problem with $|u_0|_{2}^2=c$ that the mountain pass energy level $gamma(c)$ gives a threshold for global existence. Also the strong instability of standing waves at the mountain pass energy level is proved. Finally we draw a comparison between the Schrodinger-Poisson-Slater equation and the classical nonlinear Schrodinger equation.
We consider a system of two coupled non-linear Klein-Gordon equations. We show the existence of standing waves solutions and the existence of a Lyapunov function for the ground state.
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