scholarly journals Forced cubic Schrödinger equation with Robin boundary data: continuous dependency result

2000 ◽  
Vol 41 (3) ◽  
pp. 301-311 ◽  
Author(s):  
Charles Bu
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Boundary Data ◽  
Schrodinger Equation ◽  
Cubic Schrödinger Equation ◽  
Robin Boundary

AbstractFor the cubic Schrödinger equation iut = uxx + k|u|2u, 0 ≤ x, t < ∞, initial data u(x, 0) = u0(x) ∈ H2[0, ∞), and Robin boundary data ux(0, t) + αu(0, t) = R(t) ∈ C2[0, ∞) (where α is real), we show that the solution u depends continuously on u0 and R.

scholarly journals Almost Sure Global Well-posedness for the Fractional Cubic Schrödinger Equation on the Torus

2015 ◽  
Vol 58 (3) ◽  
pp. 471-485 ◽  
Author(s):  
Seckin Demirbas
Keyword(s):  
Probability Measure ◽  
Initial Data ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Invariant Probability Measure ◽  
Well Posedness ◽  
Cubic Nonlinearity ◽  
Cubic Schrödinger Equation ◽  
Well Posed ◽  
Fractional Schrodinger Equation

AbstractIn a previous paper, we proved that the 1-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed inHsfors> 1 −α/2 and globally well-posed fors> 10α− 1/12. In this paper we define an invariant probability measureμonHsfors<α− 1/2, so that for any ∊ > 0 there is a set Ω ⊂Hssuch thatμ(Ωc) <∊and the equation is globally well-posed for initial data in Ω. We see that this fills the gap between the local well-posedness and the global well-posedness range in an almost sure sense forin an almost sure sense.

scholarly journals Forced cubic Schrödinger equation with Robin boundary data: large-time asymptotics

2013 ◽  
Vol 469 (2159) ◽  
pp. 20130341 ◽  
Author(s):  
Elena I. Kaikina
Keyword(s):  
Schrödinger Equation ◽  
Boundary Value Problem ◽  
Boundary Data ◽  
Schrodinger Equation ◽  
Large Time ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value ◽  
Robin Boundary

We consider the initial-boundary-value problem for the cubic nonlinear Schrödinger equation, formulated on a half-line with inhomogeneous Robin boundary data. We study traditionally important problems of the theory of nonlinear partial differential equations, such as the global-in-time existence of solutions to the initial-boundary-value problem and the asymptotic behaviour of solutions for large time.

scholarly journals Finite speed of disturbance for the nonlinear Schrödinger equation

2018 ◽  
Vol 149 (6) ◽  
pp. 1405-1419
Author(s):  
Simão Correia
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Finite Speed Of Propagation ◽  
Finite Speed ◽  
Nonlinear Schrödinger ◽  
The Cauchy Problem ◽  
Whole Space

AbstractWe consider the Cauchy problem for the nonlinear Schrödinger equation on the whole space. After introducing a weaker concept of finite speed of propagation, we show that the concatenation of initial data gives rise to solutions whose time of existence increases as one translates one of the initial data. Moreover, we show that, given global decaying solutions with initial data u0, v0, if |y| is large, then the concatenated initial data u0 + v0(· − y) gives rise to globally decaying solutions.

Sign in / Sign up

Export Citation Format

Share Document