Abstract
We consider a nonlinear Schrödinger equation with Dirac interaction defect. Moreover, non-standard boundary conditions are introduced in connection to the behavior of the solutions. First, we prove that this kind of Schrödinger equation can be characterized by an autonomous dynamical system. Then, based on this result, we show that such an equation possesses a maximal compact attractor in the weak topology of mathbf{H}^{mathbf{1}}.
Highlights
In this paper, we consider a nonlinear Schrödinger equation (NLS) with Dirac interaction
In Section, we prove the precompactness of the attractor in the weak topology of H ( )
We introduce the nonlinear Schrödinger equation (NLS), and we discuss the relevant issue of its boundary conditions as well as its extended domain = ] – ∞, [
Summary
We consider a nonlinear Schrödinger equation (NLS) with Dirac interaction. The case q = was studied in [ ] in which it was shown that equation ( ) is well posed in H (R) for a simple particular potential function g(s) = s In this simple case, it proved in [ ] that the long-time behavior can be described by the existence of a global attractor in H (R). It proved in [ ] that the long-time behavior can be described by the existence of a global attractor in H (R) With regard to these previous works, the aim of this paper is to investigate the nonlinear Schrödinger (NLS) equation ( ) in the case = ] – ∞, [ with Dirac interaction defect. Section introduces an autonomous dynamical system in order to study the asymptotic behavior of the solutions to the Schrödinger equation.
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