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

Type-II B\"acklund Transformations via Gauge Transformations

The construction of type II Backlund transformation for the sine-Gordon and the Tzitzeica-Bullough-Dodd models are obtained from gauge transformation. An infinite number of conserved quantities are constructed from the defect matrices. This guarantees that the introduction of type II defects for these models does not spoil their integrability. In particular, modified energy and momentum are derived and compared with those presented in recent literature.Comment: Latex 19 pages, 2 tables. v2: Comments and two references adde

From Hamiltonian to zero curvature formulation for classical integrable boundary conditions

We reconcile the Hamiltonian formalism and the zero curvature representation in the approach to integrable boundary conditions for a classical integrable system in 1  +  1 space-time dimensions. We start from an ultralocal Poisson algebra involving a Lax matrix and two (dynamical) boundary matrices. Sklyanin's formula for the double-row transfer matrix is used to derive Hamilton's equations of motion for both the Lax matrix and the boundary matrices in the form of zero curvature equations. A key ingredient of the method is a boundary version of the Semenov-Tian-Shansky formula for the generating function of the time-part of a Lax pair. The procedure is illustrated on the finite Toda chain for which we derive Lax pairs of size for previously known Hamiltonians of type BC N and D N corresponding to constant and dynamical boundary matrices respectively

Vector Nonlinear Schr\"odinger Equation on the half-line

We investigate the Manakov model or, more generally, the vector nonlinear Schr\"odinger equation on the half-line. Using a B\"acklund transformation method, two classes of integrable boundary conditions are derived: mixed Neumann/Dirichlet and Robin boundary conditions. Integrability is shown by constructing a generating function for the conserved quantities. We apply a nonlinear mirror image technique to construct the inverse scattering method with these boundary conditions. The important feature in the reconstruction formula for the fields is the symmetry property of the scattering data emerging from the presence of the boundary. Particular attention is paid to the discrete spectrum. An interesting phenomenon of transmission between the components of a vector soliton interacting with the boundary is demonstrated. This is specific to the vector nature of the model and is absent in the scalar case. For one-soliton solutions, we show that the boundary can be used to make certain components of the incoming soliton vanishingly small. This is reminiscent of the phenomenon of light polarization by reflection.Comment: 23 pages, 5 figures, some clarifications in propositions 3.1 and 3.2, added appendix with detailed comparison between linear and nonlinear cases. Accepted in J. Phys.

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

Entwining Yang–Baxter maps related to NLS type equations

We construct birational maps that satisfy the parametric set-theoretical Yang–Baxter equation and its entwining generalisation. For this purpose, we employ Darboux transformations related to integrable nonlinear Schrödinger type equations and study the refactorisation problems of the product of their associated Darboux matrices. Additionally, we study various algebraic properties of the derived maps, such as invariants and associated symplectic or Poisson structures, and we prove their complete integrability in the Liouville sense

Multidimensional Signals and Analytic Flexibility: Estimating Degrees of Freedom in Human-Speech Analyses

Recent empirical studies have highlighted the large degree of analytic flexibility in data analysis that can lead to substantially different conclusions based on the same data set. Thus, researchers have expressed their concerns that these researcher degrees of freedom might facilitate bias and can lead to claims that do not stand the test of time. Even greater flexibility is to be expected in fields in which the primary data lend themselves to a variety of possible operationalizations. The multidimensional, temporally extended nature of speech constitutes an ideal testing ground for assessing the variability in analytic approaches, which derives not only from aspects of statistical modeling but also from decisions regarding the quantification of the measured behavior. In this study, we gave the same speech-production data set to 46 teams of researchers and asked them to answer the same research question, resulting in substantial variability in reported effect sizes and their interpretation. Using Bayesian meta-analytic tools, we further found little to no evidence that the observed variability can be explained by analysts’ prior beliefs, expertise, or the perceived quality of their analyses. In light of this idiosyncratic variability, we recommend that researchers more transparently share details of their analysis, strengthen the link between theoretical construct and quantitative system, and calibrate their (un)certainty in their conclusions

Tangential beam IMRT versus tangential beam 3D-CRT of the chest wall in postmastectomy breast cancer patients: A dosimetric comparison

<p>Abstract</p> <p>Background</p> <p>This study evaluates the dose distribution of reversed planned tangential beam intensity modulated radiotherapy (IMRT) compared to standard wedged tangential beam three-dimensionally planned conformal radiotherapy (3D-CRT) of the chest wall in unselected postmastectomy breast cancer patients</p> <p>Methods</p> <p>For 20 unselected subsequent postmastectomy breast cancer patients tangential beam IMRT and tangential beam 3D-CRT plans were generated for the radiotherapy of the chest wall. The prescribed dose was 50 Gy in 25 fractions. Dose-volume histograms were evaluated for the PTV and organs at risk. Parameters of the dose distribution were compared using the Wilcoxon matched pairs test.</p> <p>Results</p> <p>Tangential beam IMRT statistically significantly reduced the ipsilateral mean lung dose by an average of 21% (1129 cGy versus 1437 cGy). In all patients treated on the left side, the heart volume encompassed by the 70% isodose line (V70%; 35 Gy) was reduced by an average of 43% (5.7% versus 10.6%), and the mean heart dose by an average of 20% (704 cGy versus 877 cGy). The PTV showed a significantly better conformity index with IMRT; the homogeneity index was not significantly different.</p> <p>Conclusions</p> <p>Tangential beam IMRT significantly reduced the dose-volume of the ipsilateral lung and heart in unselected postmastectomy breast cancer patients.</p

Data for: Transfer of sensorimotor learning reveals phoneme representations in preliterate children

This dataset contains adaptation, transfer and after-effect magnitude data by subject and by tria