# Global well-posedness for the Schrödinger equation coupled to a nonlinear oscillator

@article{Komech2007GlobalWF, title={Global well-posedness for the Schr{\"o}dinger equation coupled to a nonlinear oscillator}, author={Alexander Komech and Andrew Komech}, journal={Russian Journal of Mathematical Physics}, year={2007}, volume={14}, pages={164-173} }

The Schrödinger equation with the nonlinearity concentrated at a single point proves to be an interesting and important model for the analysis of long-time behavior of solutions, including asymptotic stability of solitary waves and properties of weak global attractors. In this note, we prove global well-posedness of this system in the energy space H1.

## 12 Citations

### On Asymptotic Stability of Solitary Waves in Schrödinger Equation Coupled to Nonlinear Oscillator

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### On the derivation of the Schrödinger equation with point-like nonlinearity

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In this report we discuss the problem of approximating nonlinear delta-interactions in dimensions one and three with regular, local or non-local nonlinearities. Concerning the one dimensional case,…

### The 1-D Dirac equation with concentrated nonlinearity

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We define and study the Cauchy problem for a 1-D nonlinear Dirac equation with nonlinearities concentrated at one point. Global well-posedness is provided and conservation laws for mass and energy…

### The One-Dimensional Dirac Equation With Concentrated Nonlinearity

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Several examples, including nonlinear Gesztesy--Seba models and the concentrated versions of the Bragg resonance and 1-D Soler type models, all within the scope of the present paper, are given.

### The point-like limit for a NLS equation with concentrated nonlinearity in dimension three

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- 2014

The nonlinear Schrödinger equation with spatially dependent nonlinearities has been considered in many contexts, both for its mathematical interest and for its relevance in several physical models.…

### Spectral stability and instability of solitary waves of the Dirac equation with concentrated nonlinearity

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We consider the nonlinear Dirac equation with Soler-type nonlinearity concentrated at one point and present a detailed study of the spectrum of linearization at solitary waves. We then consider two…

### Interactions of propagating waves in a one-dimensional chain of linear oscillators with a strongly nonlinear local attachment

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We study the interaction of propagating wavetrains in a one-dimensional chain of coupled linear damped oscillators with a strongly nonlinear, lightweight, dissipative local attachment which acts, in…

### The NLS Equation in Dimension One with Spatially Concentrated Nonlinearities: the Pointlike Limit

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AbstractIn the present paper, we study the following scaled nonlinear Schrödinger equation (NLS) in one space dimension:
$$ i\frac{\rm d}{{\rm d}t}\psi^{\varepsilon}(t)=-\Delta\psi^{\varepsilon}(t)…

### Global Attraction to Stationary States

- MathematicsAttractors of Hamiltonian Nonlinear Partial Differential Equations
- 2021

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