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Register a new userOn a Fractional Schrodinger equation in the presence of Harmonic potential
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In this paper, we establish the existence of ground state solutions for a fractional Schrodinger equation in the presence of a harmonic trapping potential. We also address the orbital stability of standing waves. Additionally, we provide interesting numerical results about the dynamics and compare them with other types of Schrodinger equations. Our results explain the effect of each term of the Schrodinger equation : The fractional power, the power of the nonlinearity and the harmonic potential.
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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 article we prove a reducibility result for the linear Schrodinger equation on a Zoll manifold with quasi-periodic in time pseudo-differential perturbation of order less or equal than $1/2$. As far as we know, this is the first reducibility results for an unbounded perturbation of a linear system which is not integrable.
In this paper, we show the scattering of the solution for the focusing inhomogenous nonlinear Schrodinger equation with a potential begin{align*} ipartial_t u+Delta u- Vu=-|x|^{-b}|u|^{p-1}u end{align*} in the energy space $H^1(mathbb R^3)$. We prove a scattering criterion, and then we use it together with Morawetz estimate to show the scattering theory.
This paper is devoted to the asymptotic analysis of a fractional version of the Ginzburg-Landau equation in bounded domains, where the Laplacian is replaced by an integro-differential operator related to the square root Laplacian as defined in Fourier space. In the singular Ginzburg-Landau limit, we show that solutions with uniformly bounded energy converge weakly to sphere valued 1/2-harmonic maps, i.e., the fractional analogues of the usual harmonic maps. In addition, the convergence holds in smooth functions spaces away from a (n-1)-rectifiable closed set of finite (n-1)-Hausdorff measure. The proof relies on the representation of the square root Laplacian as a Dirichlet-to-Neumann operator in one more dimension, and on the analysis of a boundary version of the Ginzburg-Landau equation. Besides the analysis of the fractional Ginzburg-Landau equation, we also give a general partial regularity result for stationary 1/2-harmonic maps in arbitrary dimension.
We are concerned with the direct and inverse scattering problems associated with a time-harmonic random Schrodinger equation with unknown source and potential terms. The well-posedness of the direct scattering problem is first established. Three uniqueness results are then obtained for the corresponding inverse problems in determining the variance of the source, the potential and the expectation of the source, respectively, by the associated far-field measurements. First, a single realization of the passive scattering measurement can uniquely recover the variance of the source without the a priori knowledge of the other unknowns. Second, if active scattering measurement can be further obtained, a single realization can uniquely recover the potential function without knowing the source. Finally, both the potential and the first two statistic moments of the random source can be uniquely recovered with full measurement data. The major novelty of our study is that on the one hand, both the random source and the potential are unknown, and on the other hand, both passive and active scattering measurements are used for the recovery in different scenarios.
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