non - linear Schrödinger hierarchy with defect
نویسندگان
چکیده
We introduce and solve the one-dimensional quantum non-linear Schrö-dinger (NLS) equation for an N-component field defined on the real line with a defect sitting at the origin. The quantum solution is constructed using the quantum inverse scattering method based on the concept of Reflection-Transmission (RT) algebras recently introduced. The symmetry of the model is generated by the reflection and transmission defect generators defining a defect subalgebra. We classify all the corresponding reflection and transmission matrices. This provides the possible boundary conditions obeyed by the canonical field and we compute these boundary conditions explicitly. Finally, we exhibit a phenomenon of spontaneous symmetry breaking induced by the defect and identify the unbroken generators as well as the exact remaining symmetry.
منابع مشابه
Liouville integrable defects: the non-linear Schrödinger paradigm
A systematic approach to Liouville integrable defects is proposed, based on an underlying Poisson algebraic structure. The non-linear Schrödinger model in the presence of a single particle-like defect is investigated through this algebraic approach. Local integrals of motions are constructed as well as the time components of the corresponding Lax pairs. Continuity conditions imposed upon the ti...
متن کاملOn the quantum KP hierarchy and its relation to the non-linear Schrödinger equation
We establish a relation between the classical non-linear Schrödinger equation and the KP hierarchy, and we extend this relation to the quantum case by defining a quantum KP hierarchy. We also present evidence that an integrable hierarchy of equations is obtained by quantizing the first Hamiltonian structure of the KdV equation. The connection between infinite-dimensional algebras and integrable...
متن کاملThe quantum non - linear Schrödinger model with point - like defect
We establish a family of point–like impurities which preserve the quantum integrability of the non–linear Schrödinger model in 1+1 space–time dimensions. We briefly describe the construction of the exact second quantized solution of this model in terms of an appropriate reflection–transmission algebra. The basic physical properties of the solution, including the space–time symmetry of the bulk ...
متن کاملDefects in the discrete non-linear Schrödinger model
The discrete non-linear Schrödinger (NLS) model in the presence of an integrable defect is examined. The problem is viewed from a purely algebraic point of view, starting from the fundamental algebraic relations that rule the model. The first charges in involution are explicitly constructed, as well as the corresponding Lax pairs. These lead to sets of difference equations, which include partic...
متن کامل