# Existence of dynamics for a 1D NLS equation perturbed with a generalized point defect

@article{Adami2009ExistenceOD, title={Existence of dynamics for a 1D NLS equation perturbed with a generalized point defect}, author={Riccardo Adami and Diego Noja}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2009}, volume={42}, pages={495302} }

In the present paper we study the well-posedness for the one-dimensional cubic NLS perturbed by a generic point interaction. Point interactions are described as the 4-parameter family of self-adjoint extensions of the symmetric 1D Laplacian defined on the regular functions vanishing at a point, and in the present context can be interpreted as localized defects interacting with the NLS field. A previously treated special case is given by an NLS equation with a δ defect which we generalize and…

## 39 Citations

### Standing waves and global well-posedness for the 2d Hartree equation with a point interaction

- Mathematics
- 2022

. We study a class of two-dimensional non-linear Schr¨odinger equa- tions with point-like singular perturbation and Hartree non-linearity. The point-like singular perturbation of the free Laplacian…

### Well posedness of the nonlinear Schrödinger equation with isolated singularities

- MathematicsJournal of Differential Equations
- 2021

### Ground state and orbital stability for the NLS equation on a general starlike graph with potentials

- Mathematics
- 2016

We consider a nonlinear Schrödinger equation (NLS) posed on a graph (or network) composed of a generic compact part to which a finite number of half-lines are attached. We call this structure a…

### Singular Hartree equation in fractional perturbed Sobolev spaces

- MathematicsJournal of Nonlinear Mathematical Physics
- 2018

We establish the local and global theory for the Cauchy problem of the singular Hartree equation in three dimensions, that is, the modification of the non-linear Schrödinger equation with Hartree…

### Ground states for the planar NLSE with a point defect as minimizers of the constrained energy

- MathematicsCalculus of Variations and Partial Differential Equations
- 2022

We investigate the ground states for the focusing, subcritical nonlinear Schrödinger equation with a point defect in dimension two, defined as the minimizers of the energy functional at fixed mass.…

### FAST SOLITONS ON STAR GRAPHS

- Mathematics
- 2011

We define the Schrodinger equation with focusing, cubic nonlinearity on one-vertex graphs. We prove global well-posedness in the energy domain and conservation laws for some self-adjoint boundary…

### Stability and Symmetry-Breaking Bifurcation for the Ground States of a NLS with a δ′ Interaction

- Mathematics
- 2013

We determine and study the ground states of a focusing Schrödinger equation in dimension one with a power nonlinearity |ψ|2μψ and a strong inhomogeneity represented by a singular point perturbation,…

### Non-linear Schrodinger equations with singular perturbations and with rough magnetic potentials

- Mathematics
- 2018

In this thesis we discuss thoroughly a class of linear and non-linear Schrödinger equations that arise in various physical contexts of modern relevance. First we work in the scenario where the main…

### Bifurcation and Stability for Nonlinear Schrödinger Equations with Double Well Potential in the Semiclassical Limit

- Mathematics
- 2011

We consider the stationary solutions for a class of Schrödinger equations with a symmetric double-well potential and a nonlinear perturbation. Here, in the semiclassical limit we prove that the…

## References

SHOWING 1-10 OF 50 REFERENCES

### Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate

- Mathematics
- 2004

Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential N 2 V (N(xi − xj)), where x = (x1, . . ., xN) denotes the positions of the particles. Let HN…

### The Initial Value Problem, Scattering and Inverse Scattering, for Non-Linear Schr\"odinger Equations with a Potential and a Non-Local Non-Linearity

- Mathematics
- 2005

We consider non-linear Schr\"odinger equations with a potential, and non-local non-linearities, that are models in mesoscopic physics, for example of a quantum capacitor, and that also are models of…

### A Class of Nonlinear Schro dinger Equations with Concentrated Nonlinearity

- Mathematics
- 2001

Abstract We consider the nonlinear Schrodinger equation in dimension one with a nonlinearity concentrated in a finite number of points. Detailed results on the local existence of the solution in…

### The transition from diffusion to blow-up for a nonlinear Schrodinger equation in dimension 1

- Mathematics
- 2005

We consider the time-dependent one-dimensional nonlinear Schrodinger equation with a pointwise singular potential. We prove that if the strength of the nonlinear term is small enough, then the…

### Rigorous Derivation of the Cubic NLS in Dimension One

- Mathematics, Physics
- 2007

We derive rigorously the one-dimensional cubic nonlinear Schrödinger equation from a many-body quantum dynamics. The interaction potential is rescaled through a weak-coupling limit together with a…

### Singular Perturbations of Differential Operators

- Physics
- 2000

Singular perturbations of Schrodinger type operators are of interest in mathematics, e.g. to study spectral phenomena, and in applications of mathematics in various sciences, e.g. in physics,…