The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation

Inverse Problem for an Inhomogeneous Schrödinger Equation * †

Let (− k 2)u = −u + q(x)u − k 2 u = δ(x), x ∈ R, ∂u ∂|x| − iku → 0, |x| → ∞. Assume that the potential q(x) is real-valued and compactly supported: q(x) = q(x), q(x) = 0 for |x| ≥ 1, 1 −1 |q|dx < ∞, and that q(x) produces no bound states. Let u(−1, k) and u(1, k) ∀k > 0 be the data. Theorem.Under the above assumptions these data determine q(x) uniquely.

The Cauchy problem for the inhomogeneous porous medium equation

We consider the initial value problem for the filtration equation in an inhomogeneous medium ρ(x)ut = ∆u, m > 1. The equation is posed in the whole space R, n ≥ 2, for 0 < t < ∞ ; ρ(x) is a positive and bounded function with a certain behaviour at infinity. We take initial data u(x, 0) = u0(x) ≥ 0, and prove that this problem is well-posed in the class of solutions with finite “energy”, that is...

the algorithm for solving the inverse numerical range problem

برد عددی ماتریس مربعی a را با w(a) نشان داده و به این صورت تعریف می کنیم w(a)={x8ax:x ?s1} ، که در آن s1 گوی واحد است. در سال 2009، راسل کاردن مساله برد عددی معکوس را به این صورت مطرح کرده است : برای نقطه z?w(a)، بردار x?s1 را به گونه ای می یابیم که z=x*ax، در این پایان نامه ، الگوریتمی برای حل مساله برد عددی معکوس ارانه می دهیم.

Inhomogeneous Two-Parameter Abstract Cauchy Problem

Energy-conserving methods for the nonlinear Schrödinger equation

In this paper, we further develop recent results in the numerical solution of Hamiltonian partial differential equations (PDEs) [14], by means of energyconserving methods in the class of Line Integral Methods, in particular, the Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). We shall use HBVMs for solving the nonlinear Schrödinger equation (NLSE), of interest in many appl...