No Arabic abstract
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.
For a class of Kirchhoff functional, we first give a complete classification with respect to the exponent $p$ for its $L^2$-normalized critical points, and show that the minimizer of the functional, if exists, is unique up to translations. Secondly, we search for the mountain pass type critical point for the functional on the $L^2$-normalized manifold, and also prove that this type critical point is unique up to translations. Our proof relies only on some simple energy estimates and avoids using the concentration-compactness principles. These conclusions extend some known results in previous papers.
We study the existence and stability of ground state solutions or solitons to a nonlinear stationary equation on hyperbolic space. The method of concentration compactness applies and shows that the results correlate strongly to those of Euclidean space.
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.
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 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 .