The WKB method and geometric instability for non linear Schrödinger equations on surfaces
نویسنده
چکیده
Let (M, g) be a Riemannian surface (i.e. a Riemannian manifold of dimension 2), orientable or not. We assume that M is either compact or a compact perturbation of the euclidian space, so that the Sobolev embeddings are true. Consider ∆ = ∆ g the Laplace-Beltrami operator. In this paper we are interested in constructing WKB approximations for the non linear cubic Schrödinger equation i∂ t u(t, x) + ∆u(t, x) = ε|u| 2 u(t, x), ε = ±1 u(0, x) = u 0 (x) ∈ H σ (M) (1.1)
منابع مشابه
Instability for Semi-classical Schrödinger Equations
Using WKB methods for very small times, we prove some instability phenomena for semi-classical (linear or) nonlinear Schrödinger equations. Like in several recent papers concerned with instability or ill-posedness issues, the main step of the analysis consists in reducing the problem to an ordinary differential equation. The solution to this o.d.e. is explicit, and the instability mechanism is ...
متن کاملSlope Stability Analysis Using A Non-linear Optimization Technique (RESEARCH NOTE)
In this study, a limit equilibrium method has been developed that satisfies all conditions of equilibrium and assumes circular slip surfaces. All force and moment equilibrium equations are employed without using simplification assumptions. A non-linear optimization technique is used to solve the system of equations with the corresponding constraints. The proposed method is capable to determine ...
متن کاملGeometric Optics and Instability for Semi-classical Schrödinger Equations
We prove some instability phenomena for semi-classical (linear or) nonlinear Schrödinger equations. For some perturbations of the data, we show that for very small times, we can neglect the Laplacian, and the mechanism is the same as for the corresponding ordinary differential equation. Our approach allows smaller perturbations of the data, where the instability occurs for times such that the p...
متن کاملExact WKB analysis of second-order non-homogeneous linear ordinary differential equations
In this paper we consider the exact WKB analysis of a second-order non-homogeneous linear ordinary differential equation with a large parameter. We give a geometric criterion which guarantees the Borel summability of formal solutions of a non-homogeneous equation; this criterion is described in terms of exact steepest descent paths introduced in [AKT2]. An example related to the BNR equation ([...
متن کاملA New Stability Approach using Probabilistic Profile along Direction of Excavation
Estimation of the possible instability that may be encountered in the excavation slope(s) during the planning and application steps of the rock excavation processes is an important issue in geoengineering. In this paper, a modelling method is presented for assessing the probability of wedge failure involving new permanent or temporary slope(s) along the planned excavation dir...
متن کامل