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nlin.SI

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preprint2015arXiv

Quasi-exact-solvability of the $A_{2}/G_2$ Elliptic model: algebraic forms, $sl(3)/g^{(2)}$ hidden algebra, polynomial eigenfunctions

The potential of the $A_2$ quantum elliptic model (3-body Calogero-Moser elliptic model) is defined by the pairwise three-body interaction through Weierstrass $\wp$-function and has a single coupling constant. A change of variables has been found, which are $A_2$ elliptic invariants, such that the potential becomes a rational function, while the flat space metric as well as its associated vector are polynomials in two variables. It is shown that the model possesses the hidden $sl(3)$ algebra - the Hamiltonian is an element of the universal enveloping algebra $U_{sl(3)}$ for arbitrary coupling constant - thus, it is equivalent to $sl(3)$-quantum Euler-Arnold top. The integral, in a form of the third order differential operator with polynomial, is constructed explicitly, being also an element of $U_{sl(3)}$. It is shown that there exists a discrete sequence of the coupling constants for which a finite number of polynomial eigenfunctions, up to a (non-singular) gauge factor occur. The potential of the $G_2$ quantum elliptic model (3-body Wolfes elliptic model) is defined by the pairwise and three-body interactions through Weierstrass $\wp$-function and has two coupling constants. A ch

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preprint2013arXiv

Two charges on a plane in a magnetic field: hidden algebra, (particular) integrability, polynomial eigenfunctions

The quantum mechanics of two Coulomb charges on a plane $(e_1, m_1)$ and $(e_2, m_2)$ subject to a constant magnetic field $B$ perpendicular to the plane is considered. Four integrals of motion are explicitly indicated. It is shown that for two physically-important particular cases, namely that of two particles of equal Larmor frequencies, ${e_c} \propto \frac{e_1}{m_1}-\frac{e_2}{m_2}=0$ (e.g. two electrons) and one of a neutral system (e.g. the electron - positron pair, Hydrogen atom) at rest (the center-of-mass momentum is zero) some outstanding properties occur. They are the most visible in double polar coordinates in CMS $(R, ϕ)$ and relative $(ρ, φ)$ coordinate systems: (i) eigenfunctions are factorizable, all factors except one with the explicit $ρ$-dependence are found analytically, they have definite relative angular momentum, (ii) dynamics in $ρ$-direction is the same for both systems, it corresponds to a funnel-type potential and it has hidden $sl(2)$ algebra; at some discrete values of dimensionless magnetic fields $b \leq 1$, (iii) particular integral(s) occur, (iv) the hidden $sl(2)$ algebra emerges in finite-dimensional representation, thus, the system becomes {\it q

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preprint2017arXiv

The dilute Temperley-Lieb O($n=1$) loop model on a semi infinite strip: the sum rule

This is the second part of our study of the ground state eigenvector of the transfer matrix of the dilute Temperley-Lieb loop model with the loop weight $n=1$ on a semi infinite strip of width $L$. We focus here on the computation of the normalization (otherwise called the sum rule) $Z_L$ of the ground state eigenvector, which is also the partition function of the critical site percolation model. The normalization $Z_L$ is a symmetric polynomial in the inhomogeneities of the lattice $z_1,..,z_L$. This polynomial satisfies several recurrence relations which we solve independently in terms of Jacobi-Trudi like determinants. Thus we provide a few determinantal expressions for the normalization $Z_L$.

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preprint2017arXiv

Joint eigenfunctions for the relativistic Calogero-Moser Hamiltonians of hyperbolic type. II. The two- and three-variable cases

In a previous paper we introduced and developed a recursive construction of joint eigenfunctions $J_N(a_+,a_-,b;x,y)$ for the Hamiltonians of the hyperbolic relativistic Calogero-Moser system with arbitrary particle number $N$. In this paper we focus on the cases $N=2$ and $N=3$, and establish a number of conjectured features of the corresponding joint eigenfunctions. More specifically, choosing $a_+,a_-$ positive, we prove that $J_2(b;x,y)$ and $J_3(b;x,y)$ extend to globally meromorphic functions that satisfy various invariance properties as well as a duality relation. We also obtain detailed information on the asymptotic behavior of similarity transformed functions E$_2(b;x,y)$ and E$_3(b;x,y)$. In particular, we determine the dominant asymptotics for $y_1-y_2\to\infty$ and $y_1-y_2,y_2-y_3\to\infty$, resp., from which the conjectured factorized scattering can be read off.

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preprint2016arXiv

Matrix product formula for $U_q(A^{(1)}_2)$-zero range process

The $U_q(A^{(1)}_n)$-zero range processes introduced recently by Mangazeev, Maruyama and the authors are integrable discrete and continuous time Markov processes associated with the stochastic $R$ matrix derived from the well-known $U_q(A_n^{(1)})$ quantum $R$ matrix. By constructing a representation of the relevant Zamolodchikov-Faddeev algebra, we present, for $n=2$, a matrix product formula for the steady state probabilities in terms of $q$-boson operators.

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preprint1999arXiv

Whitham Deformations of Seiberg-Witten Curves for Classical Gauge Groups

Gorsky et al. presented an explicit construction of Whitham deformations of the Seiberg-Witten curve for the $SU(N+1)$ $\calN = 2$ SUSY Yang-Mills theory. We extend their result to all classical gauge groups and some other cases such as the spectral curve of the $A^{(2)}_{2N}$ affine Toda Toda system. Our construction, too, uses fractional powers of the superpotential $W(x)$ that characterizes the curve. We also consider the $u$-plane integral of topologically twisted theories on four-dimensional manifolds $X$ with $b_2^{+}(X) = 1$ in the language of these explicitly constructed Whitham deformations and an integrable hierarchy of the KdV type hidden behind.

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preprint2016arXiv

Similarity solutions of Fokker-Planck equation with time-dependent coefficients and fixed/moving boundaries

We consider the solvability of the Fokker-Planck equation with both time-dependent drift and diffusion coefficients by means of the similarity method. By the introduction of the similarity variable, the Fokker-Planck equation is reduced to an ordinary differential equation. Adopting the natural requirement that the probability current density vanishes at the boundary, the resulting ordinary differential equation turns out to be integrable, and the probability density function can be given in closed form. New examples of exactly solvable Fokker-Planck equations are presented.

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preprint2016arXiv

Gauge Equivalence, Supersymmetry and Classical Solutions of the ospu($1, 1/1$) Heisenberg Model and the Nonlinear Schrödinger Equation

An integrable generalization of the continuous classical O$(2,1)$ pseudospin Heisenberg model to the case of the ospu$(1,1/1)$ superalgebra is constructed. The gauge equivalence of the constructed model and the related NLSE is established. We indicate a method of generating classical solutions using the global ospu$(1,1/1)$ supersymmetry. The relationship between solutions of O$(2,1)$ HM and 'superpartners' of NLSE is obtained.

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preprint2000arXiv

Bethe Ansatz solutions for Temperley-Lieb Quantum Spin Chains

We solve the spectrum of quantum spin chains based on representations of the Temperley-Lieb algebra associated with the quantum groups ${\cal U}% _{q}(X_{n})$ for $X_{n}=A_{1},$ $B_{n},$ $C_{n}$ and $D_{n}$. The tool is a modified version of the coordinate Bethe Ansatz through a suitable choice of the Bethe states which give to all models the same status relative to their diagonalization. All these models have equivalent spectra up to degeneracies and the spectra of the lower dimensional representations are contained in the higher-dimensional ones. Periodic boundary conditions, free boundary conditions and closed non-local boundary conditions are considered. Periodic boundary conditions, unlike free boundary conditions, break quantum group invariance. For closed non-local cases the models are quantum group invariant as well as periodic in a certain sense.

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preprint2016arXiv

Stochastic Laplacian growth

A point source on a plane constantly emits particles which rapidly diffuse and then stick to a growing cluster. The growth probability of a cluster is presented as a sum over all possible scenarios leading to the same final shape. The classical point for the action, defined as a minus logarithm of the growth probability, describes the most probable scenario and reproduces the Laplacian growth equation, which embraces numerous fundamental free boundary dynamics in non-equilibrium physics. For non-classical scenarios we introduce virtual point sources, in which presence the action becomes the Kullback-Leibler entropy. Strikingly, this entropy is shown to be the sum of electrostatic energies of layers grown per elementary time unit. Hence the growth probability of the presented non-equilibrium process obeys the Gibbs-Boltzmann statistics, which, as a rule, is not applied out from equilibrium. Each layer's probability is expressed as a product of simple factors in an auxiliary complex plane after a properly chosen conformal map. The action at this plane is a sum of Robin functions, which solve the Liouville equation. At the end we establish connections of our theory with the tau-fu

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preprint2016arXiv

On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight

We study the asymptotics of recurrence coefficients for monic orthogonal polynomials $π_n(z)$ with the quartic exponential weight $\exp[-N(\frac 12 z^2+\frac 14 tz^4)]$, where $t\in {\mathbb C}$ and $N\in{\mathbb N}$, $N\to\infty$. Our goal is to describe these asymptotic behaviors globally for $t\in {\mathbb C}$ in different regions. We also describe the "breaking" curves separating these regions, and discuss their special (critical) points. All these pieces of information combined provide the global asymptotic "phase portrait" of the recurrence coefficients of $π_n(z)$, which was studied numerically in [Constr. Approx. 41 (2015), 529-587, arXiv:1108.0321]. The main goal of the present paper is to provide a rigorous framework for the global asymptotic portrait through the nonlinear steepest descent analysis (with the $g$-function mechanism) of the corresponding Riemann-Hilbert problem (RHP) and the continuation in the parameter space principle. The latter allows to extend the nonlinear steepest descent analysis from some parts of the complex $t$-plane to all noncritical values of $t$. We also provide explicit solutions for recurrence coefficients in terms of the Ri

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preprint2016arXiv

Geometry of Riccati equations over normed division algebras

This work presents and studies Riccati equations over finite-dimensional normed division algebras. We prove that a Riccati equation over a finite-dimensional normed division algebra $A$ is a particular case of conformal Riccati equation on a Euclidean space and it can be considered as a curve in a Lie algebra of vector fields $V\simeq\mathfrak{so}(\dim A+1,1)$. Previous results on known types of Riccati equations are recovered from a new viewpoint. A new type of Riccati equations, the octonionic Riccati equations, are extended to the octonionic projective line $\mathbb{O}{\rm P}^1$. As a new physical application, quaternionic Riccati equations are applied to study quaternionic Schrödinger equations on 1+1 dimensions.

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preprint2016arXiv

Generalized Sobolev orthogonal polynomials, matrix moment problems and integrable systems

We introduce a large class of Sobolev bi-orthogonal polynomial sequences arising from a $LU$-factorizable moment matrix and associated with a suitable measure matrix that characterizes the Sobolev bilinear form. A theory of deformations of Sobolev bilinear forms is also proposed. We consider both polynomial deformations and a class of transformations related to the action of linear operators on the entries of a given bilinear form. Transformation formulae among new and old polynomial sequences are determined. Finally, integrable hierarchies of evolution equations arising from the factorization of a time deformation of the moment matrix are presented.

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preprint2016arXiv

Note on integrability of certain homogeneous Hamiltonian systems in 2D constant curvature spaces

We formulate the necessary conditions for the integrability of a certain family of Hamiltonian systems defined in the constant curvature two-dimensional spaces. Proposed form of potential can be considered as a counterpart of a homogeneous potential in flat spaces. Thanks to this property Hamilton equations admit, in a general case, a particular solution. Using this solution we derive necessary integrability conditions investigating differential Galois group of variational equations.

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preprint2016arXiv

Nonlocal symmetries of Lax integrable equations: a comparative study

We continue here the study of Lax integrable equations. We consider four three-dimensional equations: (1) the rdDym equation $u_{ty} = u_x u_{xy} - u_y u_{xx}$, (2) the 3D Pavlov equation $u_{yy} = u_{tx} + u_y u_{xx} - u_x u_{xy}$; (3) the universal hierarchy equation $u_{yy} = u_t u_{xy} - u_y u_{tx}$, and (4) the modified Veronese web equation $u_{ty} = u_t u_{xy} - u_y u_{tx}$. For each equation, using the know Lax pairs and expanding the latter in formal series in spectral parameter, we construct two infinite-dimensional differential coverings and give a full description of nonlocal symmetry algebras associated to these coverings. For all the for pairs of coverings, the obtained Lie algebras of symmetries manifest similar (but not the same) structures: the are (semi) direct sums of the Witt algebra, the algebra of vector fields on the line, and loop algebras; all of them contain a component of finite grading. We also discuss actions of recursion operators on shadows (in the sense of [I.S.Krasil'shchik, A.M. Vinogradov, Acta Appl. Math., 15 (1989) 1-2, 161--209.]) of nonlocal symmetries.

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preprint2016arXiv

A symmetric difference-differential Lax pair for Painlevé VI

We present a Lax pair for the sixth Painlevé equation arising as a continuous isomonodromic deformation of a system of linear difference equations with an additional symmetry structure. We call this a symmetric difference-differential Lax pair. We show how the discrete isomonodromic deformations of the associated linear problem gives us a discrete version of the fifth Painlevé equation. By considering degenerations we obtain symmetric difference-differential Lax pairs for the fifth Painlevé equation and the various degenerate versions of the third Painlevé equation.

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preprint2016arXiv

Symmetry properties of conservation laws

Symmetry properties of conservation laws of partial differential equations are developed by using the general method of conservation law multipliers. As main results, simple conditions are given for characterizing when a conservation law and its associated conserved quantity are invariant (and, more generally, homogeneous) under the action of a symmetry. These results are used to show that a recent conservation law formula (due to Ibragimov) is equivalent to a standard formula for the action of an infinitesimal symmetry on a conservation law multiplier.

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preprint2004arXiv

Quantum bound states for a derivative nonlinear Schrodinger model and number theory

A derivative nonlinear Schrodinger model is shown to support localized N-body bound states for several ranges (called bands) of the coupling constant eta. The ranges of eta within each band can be completely determined using number theoretic concepts such as Farey sequences and continued fractions. For N > 2, the N-body bound states can have both positive and negative momentum. For eta > 0, bound states with positive momentum have positive binding energy, while states with negative momentum have negative binding energy.

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preprint2016arXiv

Nonexistence of small, odd breathers for a class of nonlinear wave equations

In this note, we show that for a large class of nonlinear wave equations with odd nonlinearities, any globally defined odd solution which is small in the energy space decays to $0$ in the local energy norm. In particular, this result shows nonexistence of small, odd breathers for some classical nonlinear Klein Gordon equations such as the sine Gordon equation and $ϕ^4$ and $ϕ^6$ models. It also partially answers a question of Soffer and Weinstein in \cite[p. 19]{MR1681113} about nonexistence of breathers for the cubic NLKG in dimension one.

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preprint2016arXiv

Liouville correspondence between the Short-Pulse Hierarchy and the Sine-Gordon Hierarchy

This paper considers the whole hierarchy of bi-Hamiltonian integrable equations associated to each of the Short-Pulse (SP) equation and the Sine-Gordon (SG) equation. We prove that the transformation that relates the SP equation with the SG equation also serves to establish the correspondence between their flows and Hamiltonian conservation laws in respective hierarchy.

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