Yang-Baxter and reflection maps from vector solitons with a boundary

Based on recent results obtained by the authors on the inverse scattering method of the vector nonlinear Schr\"odinger equation with integrable boundary conditions, we discuss the factorization of the interactions of N-soliton solutions on the half-line. Using dressing transformations combined with a mirror image technique, factorization of soliton-soliton and soliton-boundary interactions is proved. We discover a new object, which we call reflection map, that satisfies a set-theoretical reflection equation which we also introduce. Two classes of solutions for the reflection map are constructed. Finally, basic aspects of the theory of set-theoretical reflection equations are introduced.Comment: 29 pages. Featured article in Nonlinearit

Multisymplectic approach to integrable defects in the sine-Gordon model

Ideas from the theory of multisymplectic systems, introduced recently in integrable systems by the author and Kundu to discuss Liouville integrability in classical field theories with a defect, are applied to the sine-Gordon model. The key ingredient is the introduction of a second Poisson bracket in the theory that allows for a Hamiltonian description of the model that is completely equivalent to the standard one, in the absence of a defect. In the presence of a defect described by frozen Bäcklund transformations, our approach based on the new bracket unifies the various tools used so far to attack the problem. It also gets rid of the known issues related to the evaluation of the Poisson brackets of the defect matrix which involve fields at coinciding space point (the location of the defect). The original Lagrangian approach also finds a nice reinterpretation in terms of the canonical transformation representing the defect conditions

A multisymplectic approach to defects in integrable classical field theory

We introduce the concept of multisymplectic formalism, familiar in covariant field theory, for the study of integrable defects in 1+1 classical field theory. The main idea is the coexistence of two Poisson brackets, one for each spacetime coordinate. The Poisson bracket corresponding to the time coordinate is the usual one describing the time evolution of the system. Taking the nonlinear Schr\"odinger (NLS) equation as an example, we introduce the new bracket associated to the space coordinate. We show that, in the absence of any defect, the two brackets yield completely equivalent Hamiltonian descriptions of the model. However, in the presence of a defect described by a frozen B\"acklund transformation, the advantage of using the new bracket becomes evident. It allows us to reinterpret the defect conditions as canonical transformations. As a consequence, we are also able to implement the method of the classical r matrix and to prove Liouville integrability of the system with such a defect. The use of the new Poisson bracket completely bypasses all the known problems associated with the presence of a defect in the discussion of Liouville integrability. A by-product of the approach is the reinterpretation of the defect Lagrangian used in the Lagrangian description of integrable defects as the generating function of the canonical transformation representing the defect conditions

The sine-Gordon model with integrable defects revisited

Application of our algebraic approach to Liouville integrable defects is proposed for the sine-Gordon model. Integrability of the model is ensured by the underlying classical r-matrix algebra. The first local integrals of motion are identified together with the corresponding Lax pairs. Continuity conditions imposed on the time components of the entailed Lax pairs give rise to the sewing conditions on the defect point consistent with Liouville integrability.Comment: 24 pages Latex. Minor modifications, added comment

On the Inverse Scattering Method for Integrable PDEs on a Star Graph

© 2015, Springer-Verlag Berlin Heidelberg. We present a framework to solve the open problem of formulating the inverse scattering method (ISM) for an integrable PDE on a star-graph. The idea is to map the problem on the graph to a matrix initial-boundary value (IBV) problem and then to extend the unified method of Fokas to such a matrix IBV problem. The nonlinear Schrödinger equation is chosen to illustrate the method. The framework unifies all previously known examples which are recovered as particular cases. The case of general Robin conditions at the vertex is discussed: the notion of linearizable initial-boundary conditions is introduced. For such conditions, the method is shown to be as efficient as the ISM on the full-line

Bosonization and Scale Invariance on Quantum Wires

We develop a systematic approach to bosonization and vertex algebras on quantum wires of the form of star graphs. The related bosonic fields propagate freely in the bulk of the graph, but interact at its vertex. Our framework covers all possible interactions preserving unitarity. Special attention is devoted to the scale invariant interactions, which determine the critical properties of the system. Using the associated scattering matrices, we give a complete classification of the critical points on a star graph with any number of edges. Critical points where the system is not invariant under wire permutations are discovered. By means of an appropriate vertex algebra we perform the bosonization of fermions and solve the massless Thirring model. In this context we derive an explicit expression for the conductance and investigate its behavior at the critical points. A simple relation between the conductance and the Casimir energy density is pointed out.Comment: LaTex 31+1 pages, 2 figures. Section 3.6 and two references added. To appear in J. Phys. A: Mathematical and Theoretica

Liouville integrable defects: the non-linear Schrodinger paradigm

A systematic approach to Liouville integrable defects is proposed, based on an underlying Poisson algebraic structure. The non-linear Schrodinger 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 time components of the Lax pair to all orders give rise to sewing conditions, which turn out to be compatible with the hierarchy of charges in involution. Coincidence of our results with the continuum limit of the discrete expressions obtained in earlier works further confirms our approach.Comment: 22 pages, Latex. Minor misprints correcte

The Quench Map in an Integrable Classical Field Theory: Nonlinear Schrödinger Equation

We study the non-equilibrium dynamics obtained by an abrupt change (a {\em quench}) in the parameters of an integrable classical field theory, the nonlinear Schr\"odinger equation. We first consider explicit one-soliton examples, which we fully describe by solving the direct part of the inverse scattering problem. We then develop some aspects of the general theory using elements of the inverse scattering method. For this purpose, we introduce the {\em quench map} which acts on the space of scattering data and represents the change of parameter with fixed field configuration (initial condition). We describe some of its analytic properties by implementing a higher level version of the inverse scattering method, and we discuss the applications of Darboux-B\"acklund transformations, Gelfand-Levitan-Marchenko equations and the Rosales series solution to a related, dual quench problem. Finally, we comment on the interplay between quantum and classical tools around the theme of quenches and on the usefulness of the quantization of our classical approach to the quantum quench problem

Interaction distance in the extended XXZ model

We employ the interaction distance to characterize the physics of a one-dimensional extended XXZ spin model, whose phase diagram consists of both integrable and nonintegrable regimes, with various types of ordering, e.g., a gapless Luttinger liquid and gapped crystalline phases. We numerically demonstrate that the interaction distance successfully reveals the known behavior of the model in its integrable regime. As an additional diagnostic tool, we introduce the notion of “integrability distance” and particularize it to the XXZ model to quantity how far the ground state of the extended XXZ model is from being integrable. This distance provides insight into the properties of the gapless Luttinger liquid phase in the presence of next-nearest-neighbor spin interactions which break integrability

Hydrodynamics of local excitations after an interaction quench in 1D cold atomic gases

We discuss the hydrodynamic approach to the study of the time evolution -induced by a quench- of local excitations in one dimension. We focus on interaction quenches: the considered protocol consists in creating a stable localized excitation propagating through the system, and then operating a sudden change of the interaction between the particles. To highlight the effect of the quench, we take the initial excitation to be a soliton. The quench splits the excitation into two packets moving in opposite directions, whose characteristics can be expressed in a universal way. Our treatment allows to describe the internal dynamics of these two packets in terms of the different velocities of their components. We confirm our analytical predictions through numerical simulations performed with the Gross-Pitaevskii equation and with the Calogero model (as an example of long range interactions and solvable with a parabolic confinement). Through the Calogero model we also discuss the effect of an external trapping on the protocol. The hydrodynamic approach shows that there is a difference between the bulk velocities of the propagating packets and the velocities of their peaks: it is possible to discriminate the two quantities, as we show through the comparison between numerical simulations and analytical estimates. In the realizations of the discussed quench protocol in a cold atom experiment, these different velocities are accessible through different measurement procedures. ArXI