arXiv subjects

Xuezhu Lu

Publications and source records attributed to Xuezhu Lu.

3 recordsLinked to original sources

arXiv2023

Partial Data Inverse Problems for the Nonlinear Schrödinger Equation

In this paper we prove the uniqueness and stability in determining a time-dependent nonlinear coefficient $β(t, x)$ in the Schrödinger equation $(i\partial_t + Δ+ q(t, x))u + βu^2 = 0$, from the boundary Dirichlet-to-Neumann (DN) map. In particular, we are interested in the partial data problem, in which the DN-map is measured on a proper subset of the boundary. We show two results: a local uniqueness of the coefficient at the points where certain type of geometric optics (GO) solutions can reach; and a stability estimate based on the unique continuation property for the linear equation.

math.AP↗

arXiv2022

Inverse problems for nonlinear Helmholtz Schrödinger equations and time-harmonic Maxwell's equations with partial data

We consider Calderón's inverse boundary value problems for a class of nonlinear Helmholtz Schrödinger equations and Maxwell's equations in a bounded domain in $\R^n$. The main method is the higher-order linearization of the Dirichlet-to-Neumann map of the corresponding equations. The local uniqueness of the linearized partial data Calderón's inverse problem is obtained following \cite{DKSU}. The Runge approximation properties and unique continuation principle allow us to extend to global situations. Simultaneous recovery of some unknown cavity$/$boundary and coefficients are given as some applications.

math.AP↗

arXiv2015

General KAM theorems and their applications to invariant tori with prescribed frequencies

In this paper we develop some new KAM-technique to prove two general KAM theorems for nearly integrable hamiltonian systems without assuming any non-degeneracy condition. Many of KAM-type results (including the classical KAM theorem) are special cases of our theorems under some non-degeneracy condition and some smoothness condition. Moreover, we can obtain some interesting results about KAM tori with prescribed frequencies.

math.DS↗