Vector nonlinear Schrödinger equation with an integrable defect and new integrable boundary conditions

Abstract

The main purpose of the present paper is to study integrable defects and new integrable boundaries associated with the vector nonlinear Schrödinger (NLS) equation. For the first issue, we present a type I defect condition for the vector NLS equation by using a Bäcklund transformation (BT) frozen at defect location. Based on the canonical property of the BT, the integrability of our defect model is proved by virtue of the classical r-matrix method. The generating function of the infinite set of conserved quantities to the defect system is also constructed. For the second issue, we present several new integrable boundary conditions associated with the K(λ) matrices that are characterized by the presence of the vector NLS fields. These boundary conditions are constructed by imposing U(N) symmetry reduction on our defect condition and by dressing a boundary with our defect condition. The integrability of our new boundary conditions is proved by virtue of the Sklyanin’s approach. The Hamiltonian description of the vector NLS equation in the presence of our new boundary conditions is also discussed.