Linear vs. Nonlinear Effects for Nonlinear Schrödinger Equations with Potential
نویسنده
چکیده
We review some recent results on nonlinear Schrödinger equations with potential, with emphasis on the case where the potential is a second order polynomial, for which the interaction between the linear dynamics caused by the potential, and the nonlinear effects, can be described quite precisely. This includes semi-classical régimes, as well as finite time blow-up and scattering issues. We present the tools used for these problems, as well as their limitations, and outline the arguments of the proofs.
منابع مشابه
Global Existence Results for Nonlinear Schrödinger Equations with Quadratic Potentials
We prove that no finite time blow up can occur for nonlinear Schrödinger equations with quadratic potentials, provided that the potential has a sufficiently strong repulsive component. This is not obvious in general, since the energy associated to the linear equation is not positive. The proof relies essentially on two arguments: global in time Strichartz estimates, and a refined analysis of th...
متن کاملNon-Linear Analysis of Functionally Graded Sector Plates with Simply Supported Radial Edges Under Transverse Loading
In this study, nonlinear bending of functionally graded (FG) circular sector plates with simply supported radial edges subjected to transverse mechanical loading has been investigated. Based on the first-order shear deformation plate theory with von Karman strain-displacement relations, the nonlinear equilibrium equations of sector plates are obtained. Introducing a stress function and a potent...
متن کاملNonlinear Vibration Analysis of the Fluid-Filled Single Walled Carbon Nanotube with the Shell Model Based on the Nonlocal Elacticity Theory
Nonlinear vibration of a fluid-filled single walled carbon nanotube (SWCNT) with simply supported ends is investigated in this paper based on Von-Karman’s geometric nonlinearity and the simplified Donnell’s shell theory. The effects of the small scales are considered by using the nonlocal theory and the Galerkin's procedure is used to discretize partial differential equations of the governing i...
متن کاملRemarks on the Schrödinger Equation
Various origins of linear and nonlinear Schrödinger equations are discussed in connection with diffusion, hydrodynamics, and fractal structure. The treatment is mainly expository, emphasizing the quantum potential, with a few new observations.
متن کاملConservative Numerical Scheme in Complex Arithmetic for Coupled Nonlinear Schrödinger Equations
For the Coupled Nonlinear Schrödinger Equations (CNLSE) we construct a conservative fully implicit numerical scheme using complex arithmetic which allows to reduce the computational time fourfold. We obtain various results numerically and investigate the role of the nonlinear and linear coupling. For nontrivial but moderate nonlinear coupling parameter, we find that the polarization of the syst...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2004