Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSemiclassical states for weakly coupled nonlinear Schrodinger systems
200
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We consider systems of weakly coupled Schrodinger equations with nonconstant potentials and we investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.
rate research
Read More
We prove the existence of infinitely many non square-integrable stationary solutions for a family of massless Dirac equations in 2D. They appear as effective equations in two dimensional honeycomb structures. We give a direct existence proof thanks to a particular radial ansatz, which also allows to provide the exact asymptotic behavior of spinor components. Moreover, those solutions admit a variational characterization. We also indicate how the content of the present paper allows to extend our previous results for the massive case [5] to more general nonlinearities.
In this paper, a class of Schr{o}dinger-Poisson system involving multiple competing potentials and critical Sobolev exponent is considered. Such a problem cannot be studied with the same argument of the nonlinear term with only a positive potential, because the weight potentials set ${Q_i(x)|1le i le m}$ contains nonpositive, sign-changing, and nonnegative elements. By introducing the ground energy function and subtle analysis, we first prove the existence of ground state solution $v_varepsilon$ in the semiclassical limit via the Nehari manifold and concentration-compactness principle. Then we show that $v_varepsilon$ converges to the ground state solution of the associated limiting problem and concentrates at a concrete set characterized by the potentials. At the same time, some properties for the ground state solution are also studied. Moreover, a sufficient condition for the nonexistence of the ground state solution is obtained.
We consider the propagation of wave packets for the nonlinear Schrodinger equation, in the semi-classical limit. We establish the existence of a critical size for the initial data, in terms of the Planck constant: if the initial data are too small, the nonlinearity is negligible up to the Ehrenfest time. If the initial data have the critical size, then at leading order the wave function propagates like a coherent state whose envelope is given by a nonlinear equation, up to a time of the same order as the Ehrenfest time. We also prove a nonlinear superposition principle for these nonlinear wave packets.
This paper provides rigorous definitions and analysis of the dynamics of weakly-coupled systems and gives sufficient conditions for an infinite dimensional quantum control system to be weakly-coupled. As an illustration we provide examples chosen among common physical systems.
We prove the existence of ground state solutions for a class of nonlinear elliptic equations, arising in the production of standing wave solutions to an associated family of nonlinear Schrodinger equations. We examine two constrained minimization problems, which give rise to such solutions. One yields what we call $F_lambda$-minimizers, the other energy minimizers. We produce such ground state solutions on a class of Riemannian manifolds called weakly homogeneous spaces, and establish smoothness, positivity, and decay properties. We also identify classes of Riemannian manifolds with no such minimizers, and classes for which essential uniqueness of positive solutions to the associated elliptic PDE fails.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
