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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

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

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

GL(NM) quantum dynamical $R$-matrix based on solution of the associative Yang-Baxter equationMay 21 2019In this letter we construct ${\rm GL}_{NM}$-valued dynamical $R$-matrix by means of unitary skew-symmetric solution of the associative Yang-Baxter equation in the fundamental representation of ${\rm GL}_{N}$. In $N=1$ case the obtained answer reproduces ... More

Non-crossing Brownian paths and Dyson Brownian motion under a moving boundaryMay 20 2019We compute analytically the probability $S(t)$ that a set of $N$ Brownian paths do not cross each other and stay below a moving boundary $g(\tau)= W \sqrt{\tau}$ up to time $t$. We show that for large $t$ it decays as a power law $S(t) \sim t^{- \beta(N,W)}$. ... More

On tau-functions for the Toda lattice hierarchyMay 20 2019We extend a recent result of [13] for the KdV hierarchy to the Toda lattice hierarchy. Namely, for an arbitrary solution to the Toda lattice hierarchy, we define a pair of wave functions, and use them to give explicit formulae for the generating series ... More

Remarks on intersection numbers and integrable hierarchies. I. Quasi-trivialityMay 20 2019Explicit expression for quasi-triviality of scalar non-linear PDE is under consideration.

Breather induced quantized superfluid vortex filaments and their characterizationMay 20 2019We study and characterize the breather-induced quantized superfluid vortex filaments which correspond to the Kuznetsov-Ma breather and super-regular breather excitations developing from localised perturbations. Such vortex filaments, emerging from an ... More

Generalized model of interacting integrable topsMay 19 2019We introduce a family of classical integrable systems describing dynamics of $M$ interacting ${\rm gl}_N$ integrable tops. It extends the previously known model of interacting elliptic tops. Our construction is based on the ${\rm GL}_N$ $R$-matrix satisfying ... More

Conformal mechanics of space curvesMay 16 2019May 23 2019Any conformally invariant energy associated with a curve possesses tension-free equilibrium states which are self-similar. When this energy is the three dimensional conformal arc-length, these states are the natural spatial generalizations of planar logarithmic ... More

Conformal mechanics of space curvesMay 16 2019Any conformally invariant energy associated with a curve possess tension-free equilibrium states which are self-similar. When this energy is the three dimensional conformal arc-length, these states are the natural spatial generalizations of planar logarithmic ... More

Inverse Hasimoto Map and Deformed Nonlinear Schrödinger EquationMay 16 2019A mapping from the solutions of vortex filament equation in the Local Induction Approximation (LIA) to soliton solutions of the NonLinear Schr\"odinger equation was obtained by Hasimoto [R. Hasimoto, J. Fluid Mechanics 51, (1972) 477]. We utilize it to ... More

$\mathbb{C}P^{2S}$ sigma models described through hypergeometric orthogonal polynomialsMay 15 2019The main objective of this paper is to establish a new connection between the Hermitian rank-1 projector solutions of the Euclidean $\mathbb{C}P^{2S}$ sigma model in two dimensions and the particular hypergeometric orthogonal polynomials called Krawtchouk ... More

Geometric Algorithm of Schrödinger Flow on a SphereMay 15 2019We construct the solution to the periodic Cauchy problem of the Schr\"odinger flow on the sphere. Such construction of solutions is formulated explicitly and therefore a geometric algorithm of solving this periodic Cauchy problem follows. Theoretical ... More

Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-lineMay 14 2019We consider the Kardar-Parisi-Zhang (KPZ) for the stochastic growth of an interface of height $h(x,t)$ on the positive half line with boundary condition $\partial_x h(x,t)|_{x=0}=A$. It is equivalent to a continuum directed polymer (DP) in a random potential ... More

Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-lineMay 14 2019May 17 2019We consider the Kardar-Parisi-Zhang (KPZ) for the stochastic growth of an interface of height $h(x,t)$ on the positive half line with boundary condition $\partial_x h(x,t)|_{x=0}=A$. It is equivalent to a continuum directed polymer (DP) in a random potential ... More

The direct scattering problem for the perturbed $\textrm{Gr}(1, 2)_{\ge 0}$ Kadomtsev-Petviashvili solitonsMay 14 2019Regular Kadomtsev-Petviashvili (KP) solitons have been investigated and classified successfully by the Grassmannian. We provide rigorous analysis for the direct scattering problem of perturbed $\textrm{Gr}(1, 2)_{\ge 0}$ KP solitons.

Superposition of fields of two rotating charged masses in General Relativity and existence of equilibrium configurationsMay 13 2019It is known that two Reissner-Nordstrom black holes or two overextreme Reissner-Nordstrom sources cannot be in physical equilibrium. In the static case such equilibrium is possible only if one of the sources is a black hole and another one is a naked ... More

Applications of Grassmannian and graph flows to nonlinear systemsMay 13 2019We show how many classes of partial differential systems with local and nonlocal nonlinearities are linearisable. By this we mean that solutions can be generated by solving a corresponding linear partial differential system together with a linear Fredholm ... More

Riemann-Hilbert approach to a generalised sine kernelMay 13 2019We derive the large distance asymptotics of the Fredholm determinant of the so-called generalised sine kernel at the critical point. This kernel corresponds to a generalisation of the pure sine kernel arising in the theory of random matrices and has potential ... More

Emergence of an Onion-like Network in Surface Growth and Its Strong RobustnessMay 13 2019We numerically investigate that optimal robust onion-like networks can emerge even with the constraint of surface growth in supposing a spatially embedded transportation or communication system. To be onion-like, moderately long links are necessary in ... More

Swirling fluid flow in flexible, expandable elastic tubes: variational approach, reductions and integrabilityMay 11 2019Many engineering and physiological applications deal with situations when a fluid is moving in flexible tubes with elastic walls. In the real-life applications like blood flow, there is often an additional complexity of vorticity being present in the ... More

The GGE averaged currents of the classical Toda chainMay 11 2019The Toda chain with random initial data is studied. Of particular interest are generalized Gibbs ensembles, their averaged conserved fields, and the averages of the corresponding currents. While averaged fields are well-understood, the description of ... More

On the solution of the Zakharov-Shabat system, which arises in the analysis of the largest real eigenvalue in the real Ginibre ensembleMay 08 2019Let $\lambda_{max}$ be a shifted maximal real eigenvalue of a random $N\times N$ matrix with independent $N(0,1)$ entries (the `real Ginibre matrix') in the $N\to\infty$ limit. It was shown by Poplavskyi, Tribe, Zaboronski \cite{PZT} that the limiting ... More

Integral formulas of ASEP and $q$-TAZRP on a RingMay 08 2019In this paper, we obtain the transition probability formulas for the Asymmetric Simple Exclusion Process (ASEP) and $q$-deformed Totally Asymmetric Zero Range Process ($q$-TAZRP) on the ring by applying the coordinate Bethe ansatz. We also compute the ... More

Dispersive Shock Wave, Generalized Laguerre Polynomials and Asymptotic Solitons of the Focusing Nonlinear Schrödinger EquationMay 07 2019We consider dispersive shock wave to the focusing nonlinear Schr\"odinger equation generated by a discontinuous initial condition which is periodic or quasi-periodic on the left semi-axis and zero on the right semi-axis. As an initial function we use ... More

Quantum cosmology for non-minimally coupled scalar field in FLRW space-time: A symmetry analysisMay 07 2019The present work deals with quantum cosmology for non-minimally coupled scalar field in the background of FLRW space--time model. The Wheeler-DeWitt equation is constructed and symmetry analysis is carried out. The Lie point symmetries are related to ... More

A simple-looking relative of the Novikov, Hirota-Satsuma and Sawada-Kotera equationsMay 06 2019We study the simple-looking scalar integrable equation $f_{xxt} - 3(f_x f_t - 1) = 0$, which is related (in different ways) to the Novikov, Hirota-Satsuma and Sawada-Kotera equations. For this equation we present a Lax pair, a B\"acklund transformation, ... More

Rational solutions for three semi-discrete modified Korteweg-de Vries type equationsMay 04 2019In this paper, we consider three semi-discrete modified Korteweg-de Vries type equations which are the nonlinear lumped self-dual network equation,the semi-discrete lattice potential modified Korteweg-de Vries equation and a semi-discrete modified Korteweg-de ... More

Integrability properties of symmetric 4+4-dimensional heavenly type equationMay 02 2019We demonstrate that the dispersionless $\bar\partial$-dressing method developed before for general heavenly equation is applicable to the $4+4$ and $2N+2N$ - dimensional symmetric heavenly type equations. We introduce generating relation and derive the ... More

Nonlocal transformations of the Generalized Liénard type equations and dissipative Ermakov-Milne-Pinney systemsMay 02 2019We employ the method of nonlocal generalized Sundman transformations to formulate the linearization problem for equations of the generalized Li\'enard type and show that they may be mapped to equations of the dissipative Ermakov-Milne-Pinney type. We ... More

Multi-component generalizations of mKdV equation and non-associative algebraic structuresMay 01 2019Relations between triple Jordan systems and integrable multi-component models of the modified Korteveg--de Vries type are established. The most general model is related to a pair consisting of a triple Jordan system and a skew-symmetric bilinear operation. ... More

Conformal Mechanics of Planar CurvesMay 01 2019It is possible to associate curvature-dependent M\"obius invariant energies with planar curves. They are of some interest because their tension-free stationary states, lacking a length scale, form self-similar curves. As such one would expect these energies ... More

Lie subalgebras of Differential Operators in one VariableMay 01 2019Let $\operatorname{Witt}$ be the Lie algebra generated by the set $\{L_i\,\vert\, i \in {\mathbb Z}\}$ and $\operatorname{Vir}$ its universal central extension. Let $\operatorname{Diff}(V)$ be the Lie algebra of differential operators on $V=\mathbb{C}[[z]]$, ... More

Superintegrable systems from block separation of variables and unified derivation of their quadratic algebrasMay 01 2019We present a new method for constructing $D$-dimensional minimally superintegrable systems based on block coordinate separation of variables. We give two new families of superintegrable systems with $N$ ($N\leq D$) singular terms of the partitioned coordinates ... More

Exact solutions of nonlinear Schrödinger equation including higher order dispersionMay 01 2019Exact solitary wave solution and periodic solutions expressed in terms of elliptic Jacobi's functions are obtained for the nonlinear Schr\"{o}dinger equation governing the propagation of pulses in optical fibers including the effects of second, third ... More

Dynamic stability for a system of ions in a Paul trapApr 30 2019We suggest a theoretical model to characterize regular and chaotic orbits for a system of two ions in a 3D Paul trap, depending on the chosen control parameters. When the electric potential is time independent or in case of the pseudopotential approximation ... More

Symmetries and hamiltonians of Ince's XXXVIII and XLIX equationsApr 29 2019We discuss symmetries of Hamiltonians of I$_{38}$ and I$_{49}$ equations that appear on Ince's list of fifty second-order differential equations with Painlev\'e property. This study is informed by structure of Weyl symmetries of Painlev\'e P$_{III}$ and ... More

Dissipative Quantum Ising chain as a non-Hermitian Ashkin-Teller modelApr 29 2019We study a quantum Ising chain with tailored bulk dissipation, which can be mapped onto a non-Hermitian Ashkin-Teller model. By exploiting the Kohmoto-den Nijs--Kadanoff transformation, we further map it to a staggered XXZ spin chain with pure-imaginary ... More

Traveling Electromagnetic Waves in Annular Josephson Tunnel JunctionsApr 27 2019It is well known that long Josephson tunnel junctions (JTJs) act as active transmission lines for the slow-mode propagation of magnetic flux-quanta (in the form of solitary waves) that is at the base of many superconducting circuits. At the same time, ... More

Classifying two-dimensional integrable spin chainsApr 26 2019May 13 2019We classify all two-dimensional fundamental integrable spin chains which have regular R-matrices of difference form. This means that the R-matrix underlying the integrable structures is of the form R(u,v)=R(u-v) and reduces to the permutation operator ... More

Classifying two-dimensional integrable spin chainsApr 26 2019We classify all two-dimensional fundamental integrable spin chains which have regular R-matrices of difference form. This means that the R-matrix underlying the integrable structures is of the form R(u,v)=R(u-v) and reduces to the permutation operator ... More

Homotopy Analysis Technique for a Generalised (1+1)-Dimensional KdV Equation of Variable CoefficientsApr 26 2019May 01 2019In this work, an exact solution to a new generalized nonlinear KdV partial differential equations has been investigated using homotopy analysis techniques. The mentioned partial differential equation has been solved using homotopy perturbation method ... More

On special limits of the Mixed Painlevé P$_{\mathbf{III-V}}$ ModelApr 26 2019The paper discusses P$_{III-V}$ equation for special values of its parameters for which this equation reduces to P$_{III}$, I$_{12}$, as well as, to some special cases of I$_{38}$ and I$_{49}$ equations from the Ince's list of $50$ second order differential ... More

Experimental evidence of a hydrodynamic soliton gasApr 26 2019We report on an experimental realization of a bi-directional soliton gas in a 34~m-long wave flume in shallow water regime. We take advantage of the fission of a sinusoidal wave to inject continuously solitons that propagate along the tank, back and forth. ... More

The spin Drude weight of the spin-1/2 $XXZ$ chain: An analytic finite size studyApr 25 2019The Drude weight for the spin transport of the spin-1/2 $XXZ$ Heisenberg chain in the critical regime is evaluated exactly for finite temperatures. We combine the thermodynamic Bethe ansatz with the functional relations of type $Y$-system satisfied by ... More

Branched Hamiltonians and time translation symmetry breaking in equations of the Lienard typeApr 25 2019Shapere and Wilczek ( Phys. Rev. Lett. 109, 160402 and 200402 (2012)) have recently described certain singular Lagrangian systems which display spontaneous breaking of time translation symmetry. We begin by considering the standard Lienard equation for ... More

Symmetries, integrals and hierarchies of new (3+1)-dimensional bi-Hamiltonian systems of Monge--Ampère typeApr 25 2019May 23 2019We study point symmetries, the corresponding conserved densities and hierarchies of four new bi-Hamiltonian heavenly systems in 3+1 dimensions which we discovered recently. We exhibit an important role played by the inverse recursion operators in the ... More

Symmetries, integrals and hierarchies of new (3+1)-dimensional bi-Hamiltonian systems of Monge--Ampère typeApr 25 2019We study point symmetries, the corresponding conserved densities and hierarchies of four new bi-Hamiltonian heavenly systems in 3+1 dimensions which we discovered recently. We exhibit an important role played by the inverse recursion operators in the ... More

Study of Non-Holonomic Deformations of Non-local integrable systems belonging to the Nonlinear Schrodinger familyApr 21 2019The non-holonomic deformations of non-local integrable systems belonging to the Nonlinear Schrodinger family are studied using the Bi-Hamiltonian formalism as well as the Lax pair method. The non-local equations are first obtained by symmetry reductions ... More

Complex plane representations and stationary states in cubic and quintic resonant systemsApr 21 2019Weakly nonlinear energy transfer between normal modes of strongly resonant PDEs is captured by the corresponding effective resonant systems. In a previous article, we have constructed a large class of such resonant systems (with specific representatives ... More

Induced dynamicsApr 20 2019Induced dynamics is defined as dynamics of real zeros with respect to $x$ of equation $f(q_1-x,\ldots,q_N-x,p_1,\ldots,p_N)=0$, where $f$ is a function, and $q_i$ and $p_j$ are canonical variables obeying some (free) evolution. Identifying zero level ... More

Bifurcation Diagram of One Generalized Integrable Model of Vortex DynamicsApr 20 2019The article is devoted to the results of a phase topology research on a generalized mathematical model, which covers such two problems as dynamics of two point vortices enclosed in a harmonic trap in a Bose-Einstein condensate and dynamics of two point ... More

Classification of integrable vector equations of geometric typeApr 19 2019Apr 29 2019A complete classification of isotropic vector equations of the geometric type that possess higher symmetries is proposed. New examples of integrable multi-component systems of the geometric type and their auto-Backlund transformations are found.

Classification of integrable vector equations of geometric typeApr 19 2019A complete classification of isotropic vector equations of the geometric type that possess higher symmetries is proposed. New examples of integrable multi-component systems of the geometric type and their auto-Backlund transformations are found.

A Kähler Compatible Moyal Deformation of the First Heavenly EquationApr 19 2019We construct a noncommutative K\"ahler manifold based on a non-linear perturbations of Moyal integrable deformations of $D=4$ self-dual gravity. The deformed K\"ahler manifold preserves all the properties of the commutative one, and we obtain the associated ... More

Squared eigenfunction symmetry of the D$Δ$mKP hierarchy and its constraintApr 17 2019In this paper squared eigenfunction symmetry of the differential-difference modified Kadomtsev-Petviashvili (D$\Delta$mKP) hierarchy and its constraint are considered. Under the constraint, the Lax triplets of the D$\Delta$mKP hierarchy, together with ... More

Integrable semi-discretizations of the Davey-Stewartson system and a $(2+1)$-dimensional Yajima-Oikawa system. IApr 16 2019The integrable Davey-Stewartson system is a linear combination of the two elementary flows that commute: $\mathrm{i} q_{t_1} + q_{xx} + 2q\partial_y^{-1}\partial_x (|q|^2) =0$ and $\mathrm{i} q_{t_2} + q_{yy} + 2q\partial_x^{-1}\partial_y (|q|^2) =0$. ... More

Conformal symmetry breaking and self-similar spiralsApr 15 2019Apr 29 2019Self-similar curves are shown to arise naturally as tension-free stationary states of conformally invariant energies. Planar logarithmic spirals are associated with the simplest such energy, the conformal arc-length, and their remarkable properties follow ... More

Conformal symmetry breaking and self-similar spiralsApr 15 2019Self-similar curves are shown to arise naturally as tension-free stationary states of conformally invariant energies. Planar logarithmic spirals are associated with the simplest such energy, the conformal arc-length, and their remarkable properties follow ... More

Duality in a hyperbolic interaction model integrable even in a strong confinement: Multi-soliton solutions and field theoryApr 14 2019Models that remain integrable even in confining potentials are extremely rare and almost non-existent. Here, we consider a one-dimensional hyperbolic interaction model, which we call as the Hyperbolic Calogero (HC) model. This is classically integrable ... More

Boundary correlations for the six-vertex model with reflecting end boundary conditionApr 12 2019We consider the six-vertex model with reflecting end boundary condition. We compute analytically boundary correlation functions, such as the boundary polarization and the emptiness formation probability. In order to do that, we use the Sklyanin's reflection ... More

Effects of vorticity on the travelling waves of some shallow water two-component systemsApr 12 2019In the present study we consider three two-component (integrable and non-integrable) systems which describe the propagation of shallow water waves on a constant shear current. Namely, we consider the two-component Camassa-Holm equations, the Zakharov-Ito ... More

Reflection $K$ matrices associated with an Onsager coideal of $U_p(A^{(1)}_{n-1})$, $U_p(B^{(1)}_n)$, $U_p(D^{(1)}_n)$ and $U_p(D^{(2)}_{n+1})$Apr 11 2019We determine the intertwiners of a family of Onsager coideal subalgebras of the quantum affine algebra $U_p(A^{(1)}_{n-1})$ in the fundamental representations and $U_p(B^{(1)}_{n}), U_p(D^{(1)}_{n}), U_p(D^{(2)}_{n+1})$ in the spin representations. They ... More

Reflection $K$ matrices associated with an Onsager coideal of $U_p(A^{(1)}_{n-1})$, $U_p(B^{(1)}_n)$, $U_p(D^{(1)}_n)$ and $U_p(D^{(2)}_{n+1})$Apr 11 2019Apr 20 2019We determine the intertwiners of a family of Onsager coideal subalgebras of the quantum affine algebra $U_p(A^{(1)}_{n-1})$ in the fundamental representations and $U_p(B^{(1)}_{n}), U_p(D^{(1)}_{n}), U_p(D^{(2)}_{n+1})$ in the spin representations. They ... More

N=2 supersymmetric extensions of relativistic Toda latticeApr 08 2019N=2 supersymmetric extensions of both the periodic and non-periodic relativistic Toda lattices are built within the framework of the Hamiltonian formalism. A geodesic description in terms of a non-metric connection is discussed.

Rationally weighted Hurwitz numbers, Meijer $G$-functions and matrix integralsApr 07 2019The quantum spectral curve equation associated to KP $\tau$-functions of hypergeometric type serving as generating functions for rationally weighted Hurwitz numbers is solved by generalized hypergeometric series. The basis elements spanning the corresponding ... More

Rationally weighted Hurwitz numbers, Meijer $G$-functions and matrix integralsApr 07 2019Apr 17 2019The quantum spectral curve equation associated to KP $\tau$-functions of hypergeometric type serving as generating functions for rationally weighted Hurwitz numbers is solved by generalized hypergeometric series. The basis elements spanning the corresponding ... More

Generating mechanism and dynamic of the smooth positons for the derivative nonlinear Schrödinger equationApr 07 2019Based on the degenerate Darboux transformation, the $n$-order smooth positon solutions for the derivative nonlinear Schr\"{o}dinger equation are generated by means of the general determinant expression of the $N$-soliton solution, and interesting dynamic ... More

New developments of the methodology of the Modified method of simplest equation with applicationApr 06 2019We discuss an extension of the modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations. The extension includes the possibility for use of: (i) more than one simplest equation; (ii) relationship ... More

On the classification of rational K-matricesApr 05 2019This paper presents a derivation of the possible residual symmetries of rational K-matrices which are invertible in the "classical limit" (the spectral parameter goes to infinity). This derivation uses only the boundary Yang-Baxter equation and the asymptotic ... More

On the classification of rational K-matricesApr 05 2019May 22 2019This paper presents a derivation of the possible residual symmetries of rational K-matrices which are invertible in the "classical limit" (the spectral parameter goes to infinity). This derivation uses only the boundary Yang-Baxter equation and the asymptotic ... More

A family of integrable and non-integrable difference equations arising from cluster algebrasApr 05 2019The one-parameter family of second order nonlinear difference equations each of which is given by $$ x_{n-1}x_nx_{n+1}=x_{n-1}+(x_n)^{\beta-1}+x_{n+1} \qquad(\beta\in\mathbb{N}) $$ is explored. Since the equation above is arising from seed mutations of ... More

Symmetries and reductions of integrable nonlocal partial differential equationsApr 03 2019In this paper, symmetry analysis is extended to nonlocal differential equations, in particular for two integrable nonlocal equations, the nonlocal nonlinear Schr\"odinger equation and the nonlocal modified Korteweg--de Vries equation. Obtained symmetries ... More

Graph clustering in industrial networksApr 03 2019The present work investigates clustering of a graph-based representation of industrial connections derived from international trade data by Hidalgo et al (2007) and confirms existence of around ten industrial clusters that are reasonably consistent with ... More

The multi-component modified nonlinear Schrödinger system with nonzero boundary conditionsApr 03 2019In a previous paper (Matsuno 2011 {\it J. Phys. A: Math. Theore.} {\bf 44} 495202), we have presented a determinantal expression of the bright $N$-soliton solution for a multi-component modified nonlinear Schr\"odinger (NLS) system with zero boundary ... More

The symmetry approach to integrability: recent advancesApr 02 2019We provide a concise introduction to the symmetry approach to integrability. Some results on integrable evolution and systems of evolution equations are reviewed. Quasi-local recursion and Hamiltonian operators are discussed. We further describe non-abelian ... More

Compacton equations and integrability: the Rosenau-Hyman and Cooper-Shepard-Sodano equationsApr 02 2019We study integrability --in the sense of admitting recursion operators-- of two nonlinear equations which are known to possess compacton solutions: the $K(m,n)$ equation introduced by Rosenau and Hyman \[ D_t(u) + D_x(u^m) + D_x^3(u^n) = 0 \; , \] and ... More

On the plane into plane mappings of hydrodynamic type. Parabolic caseApr 01 2019Singularities of plane into plane mappings described by parabolic two-component systems of quasi-liner partial differential equations of the first order are studied. Impediments arising in the application of the original Whitney's approach to such case ... More

On Separation of Variables for Reflection AlgebrasApr 01 2019We implement our new Separation of Variables (SoV) approach for open quantum integrable models associated to higher rank representations of the reflection algebras. We construct the (SoV) basis for the fundamental representations of the $Y(gl_n)$ reflection ... More

Solution structure of discrete integrable systems associated with $\mathbb{Z}_\mathcal{N}$ graded Lax pairsApr 01 2019Fordy and Xenitidis [J. Phys. A: Math. Theor. 50 (2017) 165205] recently proposed a large class of discrete integrable systems which include a number of novel integrable difference equations, from the perspective of $\mathbb{Z}_\mathcal{N}$ graded Lax ... More

Integrable Hamiltonian Hierarchies and Lagrangian 1-FormsApr 01 2019We present a new description of one-dimensional integrable systems called "Lagrangian 1-form" in both discrete and continuous levels. A key feature of integrability called a closure relation together with a generalised Euler-Lagrange equation and constraint ... More

Lax representations with non-removable parameters and integrable hierarchies of PDEs via exotic cohomology of symmetry algebrasMar 30 2019This paper develops the technique of constructing Lax representations for PDEs via non-central extensions generated by non-triivial exotic 2-cocycles of their contact symmetry algebras. We show that the method is applicable to the Lax representations ... More

Nonlocal Reductions of a Generalized Heisenberg Ferromagnet EquationMar 29 2019We study nonlocal reductions of coupled equations in $1+1$ dimensions of the Heisenberg ferromagnet type. The equations under consideration are completely integrable and have a Lax pair related to a linear bundle in pole gauge. We describe the integrable ... More

Quadrangular sets in projective line and in Moebius space, and geometric interpretation of the non-commutative discrete Schwarzian Kadomtsev-Petviashvili equationMar 28 2019We present geometric interpretation of the discrete Schwarzian Kadomtsev-Petviashvili equation in terms of quadrangular set of points of a projective line. We give also the corresponding interpretation for the projective line considered as a Moebius chain ... More

Some exact solutions of the Volterra latticeMar 28 2019We study solutions of the Volterra lattice satisfying the stationary equation for its non-autonomous symmetry. It is shown that the dynamics in $t$ and $n$ are governed by the continuous and discrete Painlev\'e equations, respectively. The class of initial ... More

Integrable modifications of the Ito-Narita-Bogoyavlensky equationMar 28 2019We consider five-point differential-difference equations. Our aim is to find integrable modifications of the Ito-Narita-Bogoyavlensky equation related to it by non-invertible discrete transformations. We enumerate all modifications associated to transformations ... More

Matrix resolvent and the discrete KdV hierarchyMar 27 2019Mar 28 2019Based on the matrix-resolvent approach, for an arbitrary solution to the discrete KdV hierarchy, we define the tau-function of the solution, and compare it with another tau-function of the solution defined via reduction of the Toda lattice hierarchy. ... More

Two-species hardcore reversible cellular automaton: matrix ansatz for dynamics and nonequilibrium stationary stateMar 25 2019May 20 2019In this paper we study the statistical properties of a reversible cellular automaton in two out-of-equilibrium settings. In the first part we consider two instances of the initial value problem, corresponding to the inhomogeneous quench and the local ... More

Two-species hardcore reversible cellular automaton: matrix ansatz for dynamics and nonequilibrium stationary stateMar 25 2019In this paper we study the statistical properties of a reversible cellular automaton in two out-of-equilibrium settings. In the first part we consider two instances of the initial value problem, corresponding to the inhomogeneous quench and the local ... More

Laws of large numbers in the Raise and Peel modelMar 25 2019We establish the exact laws of large numbers for two time additive quantities in the Raise and Peel model, the number of tiles removed by avalanches and the number of global avalanches happened by given time. The validity of conjectures for the related ... More

On symmetries of generalized Calogero model and Polychronakos-Frahm chainMar 24 2019The symmetry of the generalized Polychronakos-Frahm chain is obtained from the Dunkl-operator deformation of the unitary algebra, which describes the symmetry of the generalized Calogero model.

Non-interacting gravity waves on the surface of a deep fluidMar 24 2019We study the interaction of gravity waves on the surface of an infinitely deep ideal fluid. Starting from Zakharov's variational formulation for water waves we derive an expansion of the Hamiltonian to an arbitrary order, in a manner that avoids a laborious ... More

On bifurcation of the four Liouville tori in one generalized integrable model of the vortex dynamicsMar 24 2019The article deals with a generalized mathematical model of the dynamics of two point vortices in the Bose-Einstein condensate enclosed in a harmonic trap, and of the dynamics of two point vortices in an ideal fluid bounded by a circular region. In the ... More

The elliptic Painlevé Lax equation vs. van Diejen's 8-coupling elliptic HamiltonianMar 23 2019The 8-parameter elliptic Sakai difference Painlev\'e equation admits a Lax formulation. We show that a suitable specialization of the Lax equation gives rise to the time-independent Schr\"odinger equation for the $BC_1$ 8-parameter `relativistic' Calogero-Moser ... More

Spectral instability of the peaked periodic wave in the reduced Ostrovsky equationMar 22 2019We show that the peaked periodic traveling wave of the reduced Ostrovsky equations with quadratic and cubic nonlinearity is spectrally unstable in the space of square integrable periodic functions with zero mean and the same period. The main novelty is ... More

Curvature as an integrable deformationMar 22 2019The generalization of (super)integrable Euclidean classical Hamiltonian systems to the two-dimensional sphere and the hyperbolic space by preserving their (super)integrability properties is reviewed. The constant Gaussian curvature of the underlying spaces ... More

Some connections between the Classical Calogero-Moser model and the Log GasMar 22 2019In this work we discuss connections between a one-dimensional system of $N$ particles interacting with a repulsive inverse square potential and confined in a harmonic potential (Calogero-Moser model) and the log-gas model which appears in random matrix ... More

Wronskian solutions of integrable systemsMar 22 2019Wronski determinant (Wronskian) provides a compact form for $\tau$-functions that play roles in a large range of mathematical physics. In 1979 Matveev and Satsuma, independently, obtained solutions in Wronskian form for the Kadomtsev-Petviashvili equation. ... More

Thermodynamic limit of the two-spinon form factors for the zero field XXX chainMar 21 2019In this paper we propose a method based on the algebraic Bethe ansatz leading to explicit results for the form factors of quantum spin chains in the thermodynamic limit. Starting from the determinant representations we retrieve in particular the formula ... More