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Separation of variables bases for integrable $gl_{\mathcal{M}|\mathcal{N}}$ and Hubbard modelsJul 18 2019We construct quantum Separation of Variables (SoV) bases for both the fundamental inhomogeneous $gl_{\mathcal{M}|\mathcal{N}}$ supersymmetric integrable models and for the inhomogeneous Hubbard model both defined with quasi-periodic twisted boundary conditions ... More

A PDE Approach to the Combinatorics of the Full Map Enumeration Problem: Exact Solutions and their Universal CharacterJul 18 2019Maps are polygonal cellular networks on Riemann surfaces. This paper completes a program of constructing closed form general representations for the enumerative generating functions associated to maps of fixed but arbitrary genus. These closed form expressions ... More

Bound state soliton gas dynamics underlying the noise-induced modulational instabilityJul 18 2019We investigate theoretically the fundamental phenomenon of the spontaneous, noise-induced modulational instability (MI) of a plane wave. The long-term statistical properties of the noise-induced MI have been previously observed in experiments and in simulations ... More

Quantum spin chains from Onsager algebras and reflection $K$-matricesJul 18 2019We present a representation of the generalized $p$-Onsager algebras $O_p(A^{(1)}_{n-1})$, $O_p(D^{(2)}_{n+1})$, $O_p(B^{(1)}_n)$, $O_p(\tilde{B}^{(1)}_n)$ and $O_p(D^{(1)}_n)$ in which the generators are expressed as local Hamiltonians of XXZ type spin ... More

Collective Heavy Top DynamicsJul 18 2019We construct a Poisson map $\mathbf{M}\colon T^{*}\mathbb{C}^{2} \to \mathfrak{se}(3)^{*}$ with respect to the canonical Poisson bracket on $T^{*}\mathbb{C}^{2} \cong T^{*}\mathbb{R}^{4}$ and the $(-)$-Lie--Poisson bracket on the dual $\mathfrak{se}(3)^{*}$ ... More

Confining Strings, Infinite Statistics and IntegrabilityJul 17 2019We study confining strings in massive adjoint two-dimensional chromodynamics. Off-shell, as a consequence of zigzag formation, the resulting worldsheet theory provides a non-trivial dynamical realization of infinite quon statistics. Taking the high energy ... More

An Index for Quantum IntegrabilityJul 16 2019The existence of higher-spin quantum conserved currents in two dimensions guarantees quantum integrability. We revisit the question of whether classically-conserved local higher-spin currents in two-dimensional sigma models survive quantization. We define ... More

Benney-Lin and Kawahara equations: a detailed study through Lie symmetries and Painlevé analysisJul 16 2019We perform a detailed study on the integrability of the Benney-Lin and KdV-Kawahara equations by using the Lie symmetry analysis and the singularity analysis. We find that the equations under our consideration admit integrable travelling-wave solutions. ... More

Algebra of Dunkl Laplace-Runge-Lenz vectorJul 15 2019We introduce Dunkl version of Laplace-Runge-Lenz vector associated with a finite Coxeter group $W$ acting geometrically in $\mathbb R^N$ with multiplicity function $g$. This vector commutes with Dunkl Laplacian with additional Coulomb potential $\gamma/r$, ... More

Toda lattice hierarchy and trigonometric Ruijsenaars-Schneider hierarchyJul 15 2019We consider solutions of the 2D Toda lattice hierarchy which are trigonometric functions of the ``zeroth'' time $t_0=x$. It is known that their poles move as particles of the trigonometric Ruijsenaars-Schneider model. We extend this correspondence to ... More

Wave solutions of Gilson-Pickering equationJul 14 2019In this work, we apply the (1/G')-expansion method to produce the novel soliton solution of the Gilson-Pickering equation. This method is fundamental on homogeneous balance procedure that gives the order of the estimating polynomial-type solution. Also ... More

Inverse scattering transform for two-level systems with nonzero backgroundJul 14 2019We formulate the inverse scattering transform for the scalar Maxwell-Bloch system of equations describing the resonant interaction of light and active optical media in the case when the light intensity does not vanish at infinity. We show that pure background ... More

New integrable two-centre problem on sphere in Dirac magnetic fieldJul 14 2019We present a new integrable version of the two-centre problem on two-dimensional sphere in the presence of the Dirac magnetic monopole. The new system can be written on the dual space of Lie algebra $e(3)$ and is integrable both in classical and quantum ... More

QES solutions of a two dimensional system with quadratic non-linearitiesJul 12 2019We consider a one parameter family of a PT symmetric two dimensional system with quadratic non-linearities. Such systems are shown to perform periodic oscillations due to existing centers. We describe this systems by constructing a non-Hermitian Hamiltonian ... More

Continued fractions and Hankel determinants from hyperelliptic curvesJul 11 2019Following van der Poorten, we consider a family of nonlinear maps which are generated from the continued fraction expansion of a function on a hyperelliptic curve of genus $\mathrm{g}$. Using the connection with the classical theory of J-fractions and ... More

Generalized primitive potentialsJul 11 2019In our previous work, we introduced a new class of bounded potentials of the one-dimensional Schr\"odinger operator on the real axis, and a corresponding family of solutions of the KdV hierarchy. These potentials, which we call primitive, are obtained ... More

$π$-type Fermions and $π$-type KP hierarchyJul 09 2019In this paper, we firstly construct $\pi$-type Fermions. According to these, we define $\pi$-type Boson-Fermion correspondence which is a generalization of the classical Boson-Fermion correspondence. We can obtain $\pi$-type symmetric functions $S_\lambda^\pi$ ... More

Quantum torus symmetries of multicomponent modified KP hierarchy and reductionsJul 09 2019In this paper, we construct the multicomponent modified KP hierarchy and its additional symmetries. The additional symmetries constitute an interesting multi-folds quantum torus type Lie algebra. By a reduction, we also construct the constrained multicomponent ... More

Symmetries and reductions on the noncommutative Kadomtsev-Petviashvili and Gelfand-Dickey hierarchiesJul 09 2019In this paper, we construct the additional flows of the noncommutative Kadomtsev-Petviashvili(KP) hierarchy and the additional symmetry flows constitute an infinite dimensional Lie algebra $W_{1+\infty}$. In addition, the generating function of the additional ... More

A strongly coupled extended Toda hierarchy and its Virasoro symmetryJul 09 2019As a generalization of the integrable extended Toda hierarchy and a reduction of the extended multicomponent Toda hierarchy, from the point of a commutative subalgebra of $gl(2,\mathbb{C})$, we construct a strongly coupled extended Toda hierarchy(SCETH) ... More

Separation of variables and scalar products at any rankJul 08 2019Separation of variables (SoV) is a special property of integrable models which ensures that the wavefunction has a very simple factorised form in a specially designed basis. Even though the factorisation of the wavefunction was recently established for ... More

Generalised reversible transformations and the inhomogeneous nonlinear Schrödinger equation hierarchyJul 08 2019Under investigation is the nonlinear Schr\"odinger equation hierarchies and the reversible transformations. We propose a generalized reversible transformation between the the generalized NLSE hierarchy with focussing and defocussing nonlinearity and the ... More

Matrix extension of the Manakov-Santini system and integrable chiral model on Einstein-Weyl backgroundJul 03 2019It was demonstrated recently [Dunajski, Ferapontov and Kruglikov (2014)] that the Manakov-Santini system describes a local form of general Lorentzian Einstein-Weyl geometry. We introduce integrable matrix extension of the Manakov-Santini system and show ... More

Darboux transformation and soliton solutions of the semi-discrete massive Thirring modelJul 03 2019A one-fold Darboux transformation between solutions of the semi-discrete massive Thirring model is derived using the Lax pair and dressing methods. This transformation is used to find the exact expressions for soliton solutions on zero and nonzero backgrounds. ... More

On the rigorous justification of b-modulation method and inclusion of discrete eigenvaluesJul 02 2019Addressing the optical communication systems employing the nonlinear Fourier transform (NFT) for the data modulation/demodulation, we provide the explicit proof for the properties of the signals emerging in the so-called b-modulation method, the nonlinear ... More

A new Painleve-integrable equation possessing KdV-type solitonsJul 02 2019A new three-dimensional second-order nonlinear wave equation is introduced which passes the Painleve test for integrability and possesses KdV-type multisoliton solutions. Lax integrability of this equation remains unknown.

Entwining Yang-Baxter maps related to NLS type equationsJun 28 2019We 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 Schr\"odinger type equations and study ... More

Bi-rational maps in four dimensions with two invariantsJun 28 2019In this paper we present a class of four-dimensional bi-rational maps with two invariants satisfying certain constraints on degrees. We discuss the integrability properties of these maps from the point of view of degree growth and Liouville integrability. ... More

Revisiting generalized Hulthén potentialsJun 27 2019A relation between the deformed Hulth\'en potential and the Eckart one is used to write the bound-state wavefunctions of the former in terms of Jacobi polynomials and to calculate their normalization coefficients. The shape invariance property of the ... More

$\mathbb{Z}_\mathcal{N}$ graded discrete integrable systems and Darboux transformationsJun 27 2019We present the Darboux transformations for a novel class of two-dimensional discrete integrable systems named as $\mathbb{Z}_\mathcal{N}$ graded discrete integrable systems, which were firstly proposed by Fordy and Xenitidis within the framework of $\mathbb{Z}_\mathcal{N}$ ... More

Einstein Metrics, Projective Structures and the $SU(\infty)$ Toda EquationJun 27 2019We establish an explicit correspondence between two--dimensional projective structures admitting a projective vector field, and a class of solutions to the $SU(\infty)$ Toda equation. We give several examples of new, explicit solutions of the Toda equation, ... More

On the discretization of Darboux Integrable SystemsJun 27 2019We obtain semi-discrete analogues of some Darboux integrable systems and show their Darboux integrability.

The $τ$-function of the Ablowitz-Segur family of solutions to Painlevé II as a Widom constantJun 27 2019$\tau$-functions of certain Painlev\'e equations (PVI,PV,PIII) can be expressed as a Fredholm determinant. Further, the minor expansion of these determinants provide an interesting connection to Random partitions. This paper is a step towards understanding ... More

The $τ$-function of the Ablowitz-Segur family of solutions to Painlevé II as a Widom constantJun 27 2019Jul 09 2019$\tau$-functions of certain Painlev\'e equations (PVI,PV,PIII) can be expressed as a Fredholm determinant. Further, the minor expansion of these determinants provide an interesting connection to Random partitions. This paper is a step towards understanding ... More

Orbital Bifurcations and Shoaling of Cnoidal WavesJun 27 2019We study the parameter space of cnoidal waves -- the periodic solitons of the Korteweg-de Vries equation -- from the perspective of Virasoro coadjoint orbits. The monodromy method familiar from inverse scattering implies that many, but not all, of these ... More

General stationary solutions of the nonlocal nonlinear Schrödinger equation and their relevance to the PT-symmetric systemsJun 26 2019With the stationary solution assumption, we establish the connection between the nonlocal nonlinear Schr\"{o}dinger (NNLS) equation and an elliptic equation. Then, we obtain the general stationary solutions and discuss the relevance of their smoothness ... More

Discrete Symmetries and Nonlocal ReductionsJun 26 2019We show that nonlocal reductions of systems of integrable nonlinear partial differential equations are the special discrete symmetry transformations.

The Unified Soliton System as the ${\rm AdS_2}$ SystemJun 26 2019We study the Riemann geometric approach to be aimed at unifying soliton systems. The general two-dimensional Einstein equation with constant scalar curvature becomes an integrable differential equation. We show that such Einstein equation includes KdV/mKdV/sine-Gordon ... More

The pentagram map, Poncelet polygons, and commuting difference operatorsJun 25 2019The pentagram map takes a planar polygon $P$ to a polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $P$. This map is known to interact nicely with Poncelet polygons, i.e. polygons which are simultaneously inscribed ... More

Reciprocal transformations and their role in the integrability and classification of PDEsJun 25 2019Reciprocal transformations mix the role of the dependent and independent variables of (nonlinear partial) differential equations to achieve simpler versions or even linearized versions of them. These transformations help in the identification of a plethora ... More

On purely nonlinear oscillators generalizing an isotonic potentialJun 25 2019In this paper we consider a nonlinear generalization of the isotonic oscillator in the same spirit as one considers the generalization of the harmonic oscillator with a truly nonlinear restoring force. The corresponding potential being asymmetric we invoke ... More

On purely nonlinear oscillators generalizing an isotonic potentialJun 25 2019Jun 26 2019In this paper we consider a nonlinear generalization of the isotonic oscillator in the same spirit as one considers the generalization of the harmonic oscillator with a truly nonlinear restoring force. The corresponding potential being asymmetric we invoke ... More

KP hierarchy and trigonometric Calogero-Moser hierarchyJun 24 2019We consider trigonometric solutions of the KP hierarchy. It is known that their poles move as particles of the Calogero-Moser model with trigonometric potential. We show that this correspondence can be extended to the level of hierarchies: the evolution ... More

Models of Continuous-Time Networks with Tie Decay, Diffusion, and ConvectionJun 22 2019The study of temporal networks in discrete time has yielded numerous insights into time-dependent networked systems in a wide variety of applications. For many complex systems, however, it is useful to develop continuous-time models of networks and to ... More

Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation with step-like initial dataJun 20 2019We study the Cauchy problem for the integrable nonlocal nonlinear Schr\"odinger (NNLS) equation \[ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 \] with a step-like initial data: $q(x,0)=q_0(x)$, where $q_0(x)=o(1)$ as $x\to-\infty$ and $q_0(x)=A+o(1)$ ... More

Nonlocal Hydrodynamic Type of EquationsJun 20 2019We show that the integrable equations of hydrodynamic type admit nonlocal reductions. We first construct such reductions for a general Lax equation and then give several examples. The reduced nonlocal equations are of hydrodynamic type and integrable. ... More

Bi-Hamiltonian systems in $(2+1)$ and higher dimensions defined by Novikov algebrasJun 19 2019The results from the article [I.A.B. Strachan and B.M. Szablikowski, Novikov algebras and a classification of multicomponent Camassa-Holm equations, Stud. Appl. Math. 133 (2014), 84-117] are extended over consideration of central extensions allowing the ... More

Conformal geodesics on gravitational instantonsJun 19 2019We study the integrability of the conformal geodesic flow (also known as the conformal circle flow) on the $SO(3)$--invariant gravitational instantons. On a hyper--K\"ahler four--manifold the conformal geodesic equations reduce to geodesic equations of ... More

Flux Tube S-matrix BootstrapJun 19 2019We bootstrap the S-matrix of massless particles in unitary, relativistic two dimensional quantum field theories. We find that the low energy expansion of such S-matrices is strongly constrained by the existence of a UV completion. In the context of flux ... More

Exact Bohr-Sommerfeld Conditions for the Quantum Periodic Benjamin-Ono EquationJun 19 2019In this paper we describe the spectrum of the quantum periodic Benjamin-Ono equation in terms of the multi-phase solutions of the underlying classical system (the periodic multi-solitons). To do so, we show that the semi-classical quantization of this ... More

Source identities and kernel functions for the deformed Koornwinder-van Diejen modelsJun 18 2019We consider generalizations of the $BC$-type relativistic Calogero-Moser-Sutherland models, comprising of the rational, trigonometric, hyperbolic, and elliptic cases, due to Koornwinder and van Diejen, and construct an explicit eigenfunction for these ... More

Modified method of simplest equation for obtaining exact solutions of nonlinear partial differential equations: past and presentJun 18 2019We present a short review of the evolution of the methodology of the Method of simplest equation for obtaining exact particular solutions of nonlinear partial differential equations (NPDEs) and the recent extension of a version of this methodology called ... More

The Hodge-FVH CorrespondenceJun 17 2019The Hodge-FVH correspondence establishes a relationship between the special cubic Hodge integrals and an integrable hierarchy, which is called the fractional Volterra hierarchy. In this paper we prove this correspondence. As an application of this result, ... More

Consistency around a cuboctahedronJun 16 2019In this paper, we describe new results arising from a search for lattice equations that are consistently placed on cuboctahedra. These results extend the well-known ABS equations that are consistent on cubes. Our search was motivated by $\tau$-functions ... More

Integrable boundary conditions for the Hirota-Miwa equation and Lie algebrasJun 14 2019Systems of discrete equations on a quadrilateral graph related to the series $D^{(2)}_N$ of the affine Lie algebras are studied. The systems are derived from the Hirota-Miwa equation by imposing boundary conditions compatible with the integrability property. ... More

Discrete Painlevé equations from singularity patterns: the asymmetric trihomographic caseJun 14 2019We derive the discrete Painlev\'e equations associated to the affine Weyl group E$_8^{(1)}$ that can be represented by an (in the QRT sense) "asymmetric" trihomographic system. The method used in this paper is based on singularity confinement. We start ... More

Perturbative analysis of the colored Alexander polynomial and KP soliton $τ$-functionsJun 13 2019In this paper we elaborate on the statement given in arXiv:1805.02761. Mainly, we study the relation between the colored Alexander polynomial and the famous KP hierarchy. We explain and prove this relation by exploring the fact that the dispersion equations ... More

The KdV equation on the half-line: Time-periodicity and mass transportJun 12 2019The work presented here emanates from questions arising from experimental observations of the propagation of surface water waves. The experiments in question featured a periodically moving wavemaker located at one end of a flume that generated unidirectional ... More

Control contribution identifies top driver nodes in complex networksJun 11 2019We propose a new measure to quantify the impact of a node $i$ in controlling a directed network. This measure, called `control contribution' $\mathcal{C}_{i}$, combines the probability for node $i$ to appear in a set of driver nodes and the probability ... More

Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEsJun 11 2019We investigate the relation between pluri-Lagrangian hierarchies of $2$-dimensional partial differential equations and their variational symmetries. The aim is to generalize to the case of partial differential equations the recent findings in [Petrera, ... More

On a series of Darboux integrable discrete equations on the square latticeJun 11 2019We present a series of Darboux integrable discrete equations on the square lattice. Equations of the series are numbered with natural numbers $M$. All the equations have a first integral of the first order in one of directions of the two-dimensional lattice. ... More

Direct Characterization of Spectral Stability of Small Amplitude Periodic Waves in Scalar Hamiltonian Problems Via Dispersion RelationJun 11 2019Various approaches to studying the stability of solutions of nonlinear PDEs lead to explicit formulae determining the stability or instability of the wave for a wide range of classes of equations. However, these are typically specialized to a particular ... More

Periodic billiards within conics in the Minkowski plane and Akhiezer polynomialsJun 10 2019We derive necessary and sufficient conditions for periodic and for elliptic periodic trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlining elliptic curve. We provide several examples of periodic and elliptic periodic ... More

On consistent systems of difference equationsJun 10 2019We consider overdetermined systems of difference equations for a single function $u$ which are consistent, and propose a general framework for their analysis. The integrability of such systems is defined as the existence of higher order symmetries in ... More

Trigonal Toda lattice EquationJun 10 2019In this article, we give the trigonal Toda lattice equation, $$ -\frac{1}{2}\frac{d^3}{du^3} q_{\ell}(u) = e^{q_{\ell+1}(u)} +e^{q_{\ell+\zeta_3}(u)} +e^{q_{\ell-1-\zeta_3}(u)}-3e^{q_\ell(u)}, $$ for a lattice points $\ell \in \mathbb{Z}[\zeta_3]$ of ... More

The equilibrium dynamics of the XX chain revisitedJun 07 2019The equilibrium dynamics of the spin-1/2 XX chain is re-examined within a recently developed formalism based on the quantum transfer matrix and a thermal form factor expansion. The transversal correlation function is evaluated in real time and space. ... More

The equilibrium dynamics of the XX chain revisitedJun 07 2019Jun 20 2019The equilibrium dynamics of the spin-1/2 XX chain is re-examined within a recently developed formalism based on the quantum transfer matrix and a thermal form factor expansion. The transversal correlation function is evaluated in real time and space. ... More

Invariant tori, action-angle variables and phase space structure of the Rajeev-Ranken modelJun 07 2019We study the classical Rajeev-Ranken model, a Hamiltonian system with three degrees of freedom describing nonlinear continuous waves in a 1+1-dimensional nilpotent scalar field theory pseudodual to the SU(2) principal chiral model. While it loosely resembles ... More

Phase diagram of helicoids in Chiral Liquid CrystalsJun 06 2019Cholesteric Liquid Crystals (CLCs), in presence of an external uniform electric field and confined between two parallel planes with strong homeotropic anchoring conditions, are found to admit different types of helicoidal solutions with disclinations. ... More

Phase diagram of helicoids in Chiral Liquid CrystalsJun 06 2019Jun 10 2019Cholesteric Liquid Crystals (CLCs), in presence of an external uniform electric field and confined between two parallel planes with strong homeotropic anchoring conditions, are found to admit different types of helicoidal solutions with disclinations. ... More

Hyperbolic spin Ruijsenaars-Schneider model from Poisson reductionJun 06 2019We rederive the Hamiltonian structure of the $N$-particle hyperbolic spin Ruijsenaars-Schneider model by means of Poisson reduction of a suitable initial phase space. This phase space is realised as the direct product of the Heisenberg double of a factorisable ... More

Quasi-periodic solutions to the negative-order KdV hierarchyJun 04 2019A complete algorithm is developed to deduce quasi-periodic solutions for the negative-order KdV (nKdV) hierarchy by using the backward Neumann systems. From the nonlinearization of Lax pair, the nKdV hierarchy is reduced to a family of backward Neumann ... More

Exact results from the geometry of couplings and the effective actionJun 03 2019We invent a method that exploits the geometry in the space of couplings and the known all-loop effective action, in order to calculate the exact in the couplings anomalous dimensions of composite operators for a wide class of integrable $\sigma$-models. ... More

Singularity Analysis of a Variant of the Painlev{é}--Ince EquationJun 03 2019We examine by singularity analysis an equation derived by reduction using Lie point symmetries from the Euler--Bernoulli Beam equation which is the Painlev\'{e}--Ince Equation with additional terms. The equation possesses the same leading-order behaviour ... More

Re-factorising a QRT mapJun 02 2019A QRT map is the composition of two involutions on a biquadratic curve: one switching the $x$-coordinates of two intersection points with a given horizontal line, and the other switching the $y$-coordinates of two intersections with a vertical line. Given ... More

Exact travelling solution for a reaction-diffusion system with a piecewise constant production supported by a codimension-1 subspaceJun 02 2019A generalisation of reaction diffusion systems and their travelling solutions to cases when the productive part of the reaction happens only on a surface in space or on a line on plane but the degradation and the diffusion happen in bulk are important ... More

Integrability, existence of global solutions and wave breaking criteria for a generalization of the Camassa-Holm equationJun 01 2019Jun 06 2019Recent generalizations of the Camassa-Holm equation are studied from the point of view of existence of global solutions, criteria for wave breaking phenomena and integrability. We provide conditions, based on lower bounds for the first spatial derivative ... More

Integrability, existence of global solutions and wave breaking criteria for a generalization of the Camassa-Holm equationJun 01 2019Recent generalizations of the Camassa-Holm equation are studied from the point of view of existence of global solutions, criteria for wave breaking phenomena and integrability. We provide conditions, based on lower bounds for the first spatial derivative ... More

Integrability of the $n$-dimensional axially symmetric Chaplygin sphereJun 01 2019Jun 08 2019We consider the $n$-dimensional Chaplygin sphere under the assumption that the mass distribution of the sphere is axisymmetric. We prove that for initial conditions whose angular momentum about the contact point is vertical, the dynamics is quasi-periodic. ... More

The axisymmetric Chaplygin sphereJun 01 2019We consider the $n$-dimensional Chaplygin sphere under the assumption that the mass distribution of the sphere is axisymmetric. We prove that for initial conditions whose angular momentum about the contact point is vertical, the dynamics is quasi-periodic. ... More

Superposition of the Coupled NLS and MKdV SystemsJun 01 2019Superpositions of hierarchies of integrable equations are also integrable. The superposed equations, such as the Hirota equations in the AKNS hierarchy, cannot be considered as new integrable equations. Furthermore if one applies the Hirota bilinear method ... More

Conjectures about the ground state energy of the Lieb-Liniger model at weak repulsionMay 31 2019In this paper we develop an alternative description to solve the problem of ground state energy of the Lieb-Liniger model that describes one-dimensional bosons with contact repulsion. For this integrable model we express the Lieb integral equation in ... More

Wire billiards, the first stepsMay 31 2019Wire billiard is defined by a smooth embedded closed curve of non-vanishing curvature $k$ in $\mathbb{R}^n$ (a wire). For a class of curves, that we call nice wires, the wire billiard map is area preserving twist map of the cylinder. In this paper we ... More

Foliated Lie systems: Theory and applicationsMay 30 2019A $\mathcal{F}$- foliated Lie system is a first-order system of ordinary differential equations whose particular solutions are contained in the leaves of the foliation $\mathcal{F}$ and all particular solutions within any leaf can be written as a certain ... More

Joint eigenfunctions for the relativistic Calogero-Moser Hamiltonians of hyperbolic type. III. Factorized asymptoticsMay 30 2019In the two preceding parts of this series of papers, we introduced and studied a recursion scheme for constructing joint eigenfunctions $J_N(a_+, a_-,b;x,y)$ of the Hamiltonians arising in the integrable $N$-particle systems of hyperbolic relativistic ... More

Classification of Real Solutions of the Fourth Painleve EquationMay 28 2019Painleve transcendents are usually considered as complex functions of a complex variable, but in applications it is often the real cases that are of interest. Under a reasonable assumption (concerning the behavior of a dynamical system associated with ... More

A connection between the classical r-matrix formalism and covariant Hamiltonian field theoryMay 28 2019We bring together aspects of covariant Hamiltonian field theory and of classical integrable field theories in $1+1$ dimensions. Specifically, our main result is to obtain for the first time the classical $r$-matrix structure within a covariant Poisson ... More

Rogue waves on the periodic wave background in the focusing nonlinear Schrodinger equationMay 28 2019We present exact solutions for rogue waves arising on the background of periodic waves in the focusing nonlinear Schrodinger equation. The exact solutions are obtained by characterizing the Lax spectrum related to the periodic waves and by using the one-fold ... More

Two-dimensional rogue waves on zero background of the Davey-Stewartson II equationMay 27 2019A prototypical example of a rogue wave structure in a two-dimensional model is presented in the context of the Davey-Stewartson~II (DS~II) equation arising in water waves. The analytical methodology involves a Taylor expansion of an eigenfunctionof the ... More

Vibrations of an elastic bar, isospectral deformations, and modified Camassa-Holm equationsMay 27 2019Vibrations of an elastic rod are described by a Sturm-Liouville system. We present a general discussion of isospectral (spectrum preserving) deformations of such a system. We interpret one family of such deformations in terms of a two-component modified ... More

Power laws in code repositories: A skeptical approachMay 27 2019Software development as done using modern methodologies and source control management systems, has been often established as an example of self-organization, with code growing and evolving organically, through activities that do not stem from entralized ... More

Elliptic solutions to integrable nonlinear equations and many-body systemsMay 26 2019We review elliptic solutions to integrable nonlinear partial differential and difference equations (KP, mKP, BKP, Toda) and derive equations of motion for poles of the solutions. The pole dynamics is given by an integrable many-body system (Calogero-Moser, ... More

Grothendieck's Dessins d'Enfants in a Web of DualitiesMay 26 2019In this paper we show that counting Grothendieck's dessins d'enfants is universal in the sense that some other enumerative problems are either special cases or directly related to it. Such results provide concrete examples that support a proposal made ... More

Non-commutative double-sided continued fractionsMay 24 2019We study double-sided continued fractions whose coefficients are non-commuting symbols. We work within the formal approach of the Mal'cev-Neumann series and free division rings. We start with presenting the analogs of the standard results from the theory ... More

Effect of local Peregrine soliton emergence on statistics of random waves in the 1-D focusing Nonlinear Schrödinger equationMay 23 2019The Peregrine soliton is often considered as a prototype of the rogue waves. After recent advances in the semi-classical limit of the 1-D focusing Nonlinear Schr\"odinger (NLS) equation this conjecture can be seen from another perspective. In the present ... More

Exact perturbative results for the Lieb-Liniger and Gaudin-Yang modelsMay 23 2019We present a systematic procedure to extract the perturbative series for the ground state energy density in the Lieb-Liniger and Gaudin-Yang models, starting from the Bethe ansatz solution. This makes it possible to calculate explicitly the coefficients ... More

Exact perturbative results for the Lieb-Liniger and Gaudin-Yang modelsMay 23 2019May 28 2019We present a systematic procedure to extract the perturbative series for the ground state energy density in the Lieb-Liniger and Gaudin-Yang models, starting from the Bethe ansatz solution. This makes it possible to calculate explicitly the coefficients ... More

Frobenius manifolds and a new class of extended affine Weyl groups $\widetilde{W}^{(k,k+1)}(A_l)$May 22 2019We show the existence of Frobenius manifold structures on the orbit spaces of a new class of extended affine Weyl groups $\widetilde{W}^{(k,k+1)}(A_l)$ for $1\leq k <l$. We also construct Landau--Ginzburg superpotentials for these Frobenius manifold structures. ... More

Frobenius manifolds and a new class of extended affine Weyl groups $\widetilde{W}^{(k,k+1)}(A_l)$May 22 2019May 24 2019We present a new class of extended affine Weyl groups $\widetilde{W}^{(k,k+1)}(A_l)$ for $1\leq k <l$ and obtain an analogue of Chevalley-type theorem for their invariants. We further show the existence of Frobenius manifold structures on the orbit spaces ... More

Variational approximations of soliton dynamics in the Ablowitz-Musslimani nonlinear Schrödinger equationMay 22 2019We study the integrable nonlocal nonlinear Schr\"odinger equation proposed by Ablowitz and Musslimani, that is considered as a particular example of equations with parity-time ($\mathcal{PT}$) symmetric self-induced potential. We consider dynamics (including ... More