Set partitions and integrable hierarchies
We demonstrate that statistics for several types of set partitions are described by generating functions which appear in the theory of integrable equations.
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arXiv subjects
V. E. Adler
Publications and source records attributed to V. E. Adler.
On the combinatorics of several integrable hierarchies
We demonstrate that statistics of certain classes of set partitions is described by generating functions related to the Burgers, Ibragimov--Shabat and Korteweg--de Vries integrable hierarchies.
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Integrability test for evolutionary lattice equations of higher order
A generalized summation by parts algorithm is presented for solving of difference equations of the form $T^m(y)-a[u]y=b[u]$ where $T$ denotes the shift $u_j\to u_{j+1}$. Solvability of such type of equations with respect to coefficients of formal symmetry (or formal recursion operator) provides a convenient integrability test for evolutionary differential-difference equations $u_{,t}=f(u_{-m},\dots,u_m)$. The algorithm is implemented in {\em Mathematica}.
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Necessary integrability conditions for evolutionary lattice equations
The structure of solutions is studied for the Lax equation $D_t(G)=[F,G]$ for formal power series with respect to the shift operator. It is proved that if the equation with a given series $F$ of degree $m$ admits a solution $G$ of degree $k$ then it admits, as well, a solution $H$ of degree $m$ such that $H^k=G^m$. This property is used for derivation of necessary integrability conditions for scalar evolutionary lattices.
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On discrete integrable equations of higher order
We study 2D discrete integrable equations of order 1 with respect to one independent variable and $m$ with respect to another one. A generalization of the multidimensional consistency property is proposed for this type of equations. The examples are related to the Bäcklund--Darboux transformations for the lattice equations of Bogoyavlensky type.
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Cauchy problem for integrable discrete equations on quad-graphs
Initial value problems for the integrable discrete equations on quad-graphs are investigated. A geometric criterion of the well-posedness of such a problem is found. The effects of the interaction of the solutions with the localized defects in the regular square lattice are discussed for the discrete potential KdV and linear wave equations. The examples of kinks and solitons on various quad-graphs, including quasiperiodic tilings, are presented.
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Towards the theory of integrable hyperbolic equations of third order
The examples are considered of integrable hyperbolic equations of third order with two independent variables. In particular, an equation is found which admits as evolutionary symmetries the Krichever--Novikov equation and the modified Landau--Lifshitz system. The problem of choice of dynamical variables for the hyperbolic equations is discussed.
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Quantum tops as examples of commuting differential operators
We study the quantum analogs of tops on Lie algebras $so(4)$ and $e(3)$ represented by differential operators.
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Differential-difference equations associated with the fractional Lax operators
We study integrable hierarchies associated with spectral problems of the form $Pψ=λQψ$ where $P,Q$ are difference operators. The corresponding nonlinear differential-difference equations can be viewed as inhomogeneous generalizations of the Bogoyavlensky type lattices. While the latter turn into the Korteweg--de Vries equation under the continuous limit, the lattices under consideration provide discrete analogs of the Sawada--Kotera and Kaup--Kupershmidt equations. The $r$-matrix formulation and several simplest explicit solutions are presented.
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On a discrete analog of the Tzitzeica equation
A discrete analog of the Tzitzeica equation is found in the form of quad-equation. Its continuous symmetry is an inhomogeneous Narita--Bogoyavlensky type lattice equation which defines a discretization of the Sawada--Kotera equation. The integrability of these discretizations is proven by construction of the Lax representations.
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Linear problems and Bäcklund transformations for the Hirota-Ohta system
The auxiliary linear problems are presented for all discretization levels of the Hirota-Ohta system. The structure of these linear problems coincides essentially with the structure of Nonlinear Schrödinger hierarchy. The squared eigenfunction constraints are found which relate Hirota-Ohta and Kulish-Sklyanin vectorial NLS hierarchies.
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The tangential map and associated integrable equations
The tangential map is a map on the set of smooth planar curves. It satisfies the 3D-consistency property and is closely related to some well-known integrable equations.
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On vector analogs of the modified Volterra lattice
Modified Volterra lattice admits two vector generalizations. One of them is studied for the first time. The zero curvature representations, Bäcklund transformations, nonlinear superposition principle and the simplest explicit solutions of soliton and breather type are presented for both vector lattices. The relations with some other integrable equations are established.
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Classification of integrable Volterra type lattices on the sphere. Isotropic case
The symmetry approach is used for classification of integrable isotropic vector Volterra lattices on the sphere. The list of integrable lattices consists mainly of new equations. Their symplectic structure and associated PDE of vector NLS-type are discussed.
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The model equation of soliton theory
We consider an hierarchy of integrable 1+2-dimensional equations related to Lie algebra of the vector fields on the line. The solutions in quadratures are constructed depending on $n$ arbitrary functions of one argument. The most interesting result is the simple equation for the generating function of the hierarchy which defines the dynamics for the negative times and also has applications to the second order spectral problems. A rather general theory of integrable 1+1-dimensional equations can be developed by study of polynomial solutions of this equation under condition of regularity of the corresponding potentials.
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On a class of third order mappings with two rational invariants
A novel family of integrable third order maps is presented. Each map possesses, by construction, a pair of rational invariants and a commuting map from the same class. The 3-dimensional invariant curve is parametrized, in general, by an elliptic curve.
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Dressing chain for the acoustic spectral problem
The iterations are studied of the Darboux transformation for the generalized Schroedinger operator. The applications to the Dym and Camassa-Holm equations are considered.
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Some incidence theorems and integrable discrete equations
Several incidence theorems of planar projective geometry are considered. It is demonstrated that generalizations of Pascal theorem due to Möbius give rise to double cross-ratio equation and Hietarinta equation. The construction corresponding to the double cross-ratio equation is a reduction to a conic section of some planar configuration $(20_3 15_4)$. This configuration provides a correct definition of the multidimensional quadrilateral lattices on the plane.
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