The classical nonlinear Schrödinger model with a new integrable boundary

Abstract

A new integrable boundary for the classical nonlinear Schrödinger model is derived by dressing a boundary with a defect. A complete investigation of the integrability of the new boundary is carried out in the sense that the boundary K matrix is derived and the integrability is proved via the classical r-matrix. The issue of conserved charges is also discussed. The key point in proving the integrability of the new boundary is the use of suitable modified Poisson brackets. Finally, concerning the kind of defect used in the present context, this investigation offers the opportunity to prove -beyond any doubts -their integrability.

Fusing defect for theN= 2 super sinh-Gordon model

“…From the results obtained in this paper and those previously found in [9], it would be interesting to explore the possibility of finding new integrable boundary conditions for the N = 1 and N = 2 sshG models, by performing a consistent half-line limit, specially following the reasoning of [13]- [15].…”

“…1 , (20) Now taking the limit x 0 → 0 in the Lagrangian density (15), the bulk Langrangian term L 0 vanishes, and then the resulting Lagrangian density for fused defect becomes of the form of eq.…”

Type-II super-Bäcklund transformation and integrable defects for the N = 1 super sinh-Gordon model

“…Also, it is possible to find new integrable boundary conditions for the sshG model by using the type-II integrable defect for the sshG following the line of [30]- [32]. Some of these questions are expected to be developed in future investigations.…”

Time-dependent defects in integrable soliton equations

“…In recent years, there arose some interest in the study of defects, or impurities, in classical (1+1)dimensional integrable field theories; see for example [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and references therein. The presence of defects usually spoil the integrability of a system.…”

“…The study of defects, or impurities, in classical (1 + 1)dimensional integrable field theories has been undertaken intensively in recent years; see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and references therein. A defect is usually introduced as a discontinuity at a specific point together with suitable sewing conditions across the defect relating the fields and their derivatives on either side of the defect location.…”