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Equation de Schrödinger non-linéaire et impuretés dans les systèmes intégrables

Abstract : This thesis deals with the area of theoretical physics known as integrable systems, combining fruitfully physics and mathematics, and is characterized by the possibility of computing exact results (i.e. non perturbative) which serve as a basis for physical predictions.
In this context, the nonlinear Schrödinger equation (in 1+1 dimensions) is a system of special interest. It models various phenomena, being classical (nonlinear optics, fluid mechanics...) as well as quantum (ultra cold gases, Bose-Einstein condensation...). It also contributed to the development of certain techniques to solve integrable systems: inverse scattering method, Bethe ansatz, identifying and using symmetries (quantum groups, Yangians). By using this system both as a testing and a predictive model, my work is devoted essentially to the following two issues:
- Including bosonic and fermionic degrees of freedom.
- Including a boundary or an impurity.
At first, I studied a "supersymmetric" version of this equation for which I established the validity of all the results known for the original scalar version: integrability, symmetry, explicit solution at the classical and quantum levels. The issue of including a boundary has been treated from another point of view. The idea is to start from the characteristic symmetry algebra for integrable systems with boundaries, the reflection algebra, and to construct a general integrable Hamiltonian whose symmetry structure is precisely the reflection algebra. As a particular case of this Hamiltonian, I recovered the nonlinear Schrödinger Hamiltonian in the presence of a boundary. Another particular case is the Sutherland Hamiltonian with boundary for which the symmetry algebra was not known.
The problem of including an impurity in an integrable system represents an important part of my work. I could show that it is possible to preserve integrability for an interacting system while including a transmitting and reflecting defect (an impurity) thanks to a new algebraic structure, the Reflection-Transmission algebra, applied to the nonlinear Schrödinger equation. This allows to compute explicitly the form of the field, the scattering matrix elements, the N-point correlation functions and to identify the symmetry of the problem.
Following this work, the exact equations controlling the energy spectrum of a gas of particles with contact interaction and in the presence of a four parameter tunable impurity were established. These results pave the way to applications in condensed matter physics.
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Contributor : Vincent Caudrelier <>
Submitted on : Monday, June 27, 2005 - 4:47:50 PM
Last modification on : Friday, November 6, 2020 - 3:35:09 AM
Long-term archiving on: : Friday, April 2, 2010 - 10:01:01 PM


  • HAL Id : tel-00009612, version 1



Vincent Caudrelier. Equation de Schrödinger non-linéaire et impuretés dans les systèmes intégrables. Physique mathématique [math-ph]. Université de Savoie, 2005. Français. ⟨tel-00009612⟩



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