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1d Bose gas interacting through δ, δ ′ and double-δ function potentials is shown to be equivalent to a δ anyon gas allowing exact Bethe ansatz solution. In the noninteracting limit it describes an ideal gas with generalized exclusion statistics and solves some recent controversies.PACS numbers: 05.30.Jp, 03.70.+k 11.55.Ds, 71.10.Pm, The concept of particles with generalized exclusion statistics (GES) introduced by Haldane [1] has important consequences [2] in describing 1d non Fermi-liquids [3], which in turn is believed to be related [4] to the edge excitations in fractional quantum Hall effect [5]. On the other hand, inspired by the success of the Chern-Simon theory, an attempt was made recently [6] to describe a 1d ideal gas with GES in the framework of a gauge field model. However, in a subsequent paper [7] the previous result was shown to be wrong and some other conclusion was offered. Our aim here is to deal primarily with a 1d Bose gas interacting through double-δ function potentials together with the well known δ and derivative δ-function interactions. We show that this interacting model with several singular potentials is equivalent to a 1d gas with GES (which we call anyon for brevity) interacting via δ-function potential only. This δ-anyon gas is found to be exactly solvable by the coordinate Bethe ansatz (CBA) just like its bosonic counterpart, contradicting the common belief [8] that the CBA is applicable only to models with symmetric or antisymmetric wave functions. Remarkably, at the limit of vanishing interaction the anyon gas becomes free and gauge equivalent to a related model proposed in [6]. This shows that, though the explicit wave function and the N -body Hamiltonian of [6] are not exact, the conclusion it arrived at is basically correct. Therefore, while the error in the treatment of [6] was detected in [7], the source of this error and the possible way to rectify it becomes evident from our result.We start with a 1d Bose gas interacting through generalized δ function potentials as(1)This model was briefly considered and readily discarded in [9] as too difficult a problem to solve. Notice however that for γ a = 0, a = 1, 2, i.e. without the double-δ potentials, it has various exactly solvable limits. For example, for κ = 0, c = 0 the model becomes the well known δ-Bose gas [10], while for κ = 0, c = 0 it corresponds to Bose gas with δ ′ interaction [11]. Both these cases are not only exactly solvable by CBA, but also represent quantum integrable systems allowing R-matrix solution. This can be proved through their connection with the quantum integrable nonlinear Schrödinger equation (NLSE) [12] and derivative NLSE [13], respectively. Even the mixed case with κ = 0, c = 0 is solvable through CBA [8,11], though as a quantum model it does not allow a R-matrix solution. Nevertheless for γ a = 0, i.e. with the inclusion of highly singular double-δ function interactions, the solvability of the model is completely lost and the application of the CBA becomes problematic due to the presence of tree-b...

New Landau–Lifshitz (LL) and higher-order nonlinear systems gauge generated from nonlinear Schrödinger (NS) type equations are presented. The consequences of gauge equivalence between different dynamical systems are discussed. The gauge connections among various LL and NS equations are found and depicted through a schematic representation.

Abstract. We present an approach for joint inference of 3D scene structure and semantic labeling for monocular video. Starting with monocular image stream, our framework produces a 3D volumetric semantic + occupancy map, which is much more useful than a series of 2D semantic label images or a sparse point cloud produced by traditional semantic segmentation and Structure from Motion(SfM) pipelines respectively. We derive a Conditional Random Field (CRF) model defined in the 3D space, that jointly infers the semantic category and occupancy for each voxel. Such a joint inference in the 3D CRF paves the way for more informed priors and constraints, which is otherwise not possible if solved separately in their traditional frameworks. We make use of class specific semantic cues that constrain the 3D structure in areas, where multiview constraints are weak. Our model comprises of higher order factors, which helps when the depth is unobservable. We also make use of class specific semantic cues to reduce either the degree of such higher order factors, or to approximately model them with unaries if possible. We demonstrate improved 3D structure and temporally consistent semantic segmentation for difficult, large scale, forward moving monocular image sequences. Fig. 1. Overview of our system. From monocular image sequence, we first obtain 2D semantic segmentation, sparse 3D reconstruction and camera poses. We then build a volumetric 3D map which depicts both 3D structure and semantic labels.

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We define and illustrate the novel notion of dual integrable hierarchies, on the example of the nonlinear Schrödinger (NLS) hierarchy. For each integrable nonlinear evolution equation (NLEE) in the hierarchy, dual integrable structures are characterized by the fact that the zero-curvature representation of the NLEE can be realized by two Hamiltonian formulations stemming from two distinct choices of the configuration space, yielding two inequivalent Poisson structures on the corresponding phase space and two distinct Hamiltonians. This is fundamentally different from the standard bi-Hamiltonian or generally multitime structure. The first formulation chooses purely space-dependent fields as configuration space; it yields the standard Poisson structure for NLS. The other one is new: it chooses purely time-dependent fields as configuration space and yields a different Poisson structure at each level of the hierarchy. The corresponding NLEE becomes a space evolution equation. We emphasize the role of the Lagrangian formulation as a unifying framework for deriving both Poisson structures, using ideas from covariant field theory. One of our main results is to show that the two matrices of the Lax pair satisfy the same form of ultralocal Poisson algebra (up to a sign) characterized by an r-matrix structure, whereas traditionally only one of them is involved in the classical r-matrix method. We construct explicit dual hierarchies of Hamiltonians, and Lax representations of the triggered dynamics, from the monodromy matrices of either Lax matrix. An appeal-* Corresponding author. E-mail addresses: Jean.Avan@u-cergy.fr (J. Avan), v.caudrelier@city.ac.uk (V. Caudrelier), A.Doikou@hw.ac.uk (A. Doikou), Anjan.Kundu@saha.ac.in (A. Kundu). Funded by SCOAP 3 . 416J. Avan et al. / Nuclear Physics B 902 (2016) ing procedure to build a multi-dimensional lattice of Lax pair, through successive uses of the dual Poisson structures, is briefly introduced.

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Recently proposed nonholonomic deformation of the KdV equation is solved through inverse scattering method by constructing AKNS type Lax pair. Exact N-soliton solutions are found for the basic field and the deforming function showing unusual ac(de)celerated motion. Two-fold integrable hierarchy is revealed, one with usual higher order dispersion and the other with novel higher nonholonomic deformations.