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Eigenfunctions are shown to constitute privileged coordinates of self-dual Einstein spaces with the underlying governing equation being revealed as the general heavenly equation. The formalism developed here may be used to link algorithmically a variety of known heavenly equations. In particular, the classical connection between Plebański’s first and second heavenly equations is retrieved and interpreted in terms of eigenfunctions. In addition, connections with travelling wave reductions of the recently introduced TED equation which constitutes a 4 + 4-dimensional integrable generalisation of the general heavenly equation are found. These are obtained by means of (partial) Legendre transformations. As a particular application, we prove that a large class of self-dual Einstein spaces governed by a compatible system of dispersionless Hirota equations is genuinely four-dimensional in that the (generic) metrics do not admit any (proper or non-proper) conformal Killing vectors. This generalises the known link between a particular class of self-dual Einstein spaces and the dispersionless Hirota equation encoding three-dimensional Einstein–Weyl geometries.

Singularities 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 are discussed. Hierarchy of singularities is analysed by double-scaling expansion method for the simplest $2$-component Jordan system. It is shown that flex is the lowest singularity while higher singularities are given by $(k+1,k+2)$ curves which are of cusp type for $k=2n+1$, $n=1,2,3,\dots$. Regularization of these singularities by deformation of plane into plane mappings into surface $S^{2+k} (\subset \mathbb{R}^{2+k} ) $ to plane is discussed. Applicability of the proposed approach to other parabolic type mappings is noted., Comment: 19 pages, 4 figures

Deformations of geometric characteristics of statistical hypersurfaces governed by the law of growth of entropy are studied. Both general and special cases of deformations are considered. The basic structure of the statistical hypersurface is explored through a differential relation for the variables, and connections with the replicator dynamics for Gibbs' weights are highlighted. Ideal and super-ideal cases are analysed, also considering their integral characteristics., 21 pages, 3 figures

Non-standard parabolic regularization of gradient catastrophes for the Burgers-Hopf equation is proposed. It is based on the analysis of all (generic and higher order) gradient catastrophes and their step by step regularization by embedding the Burgers-Hopf equation into multi-component parabolic systems of quasilinear PDEs with the most degenerate Jordan block. Probabilistic realization of such procedure is presented. The complete regularization of the Burgers-Hopf equation is achieved by embedding it into the infinite parabolic Jordan chain. It is shown that the Burgers equation is a particular reduction of the Jordan chain. Gradient catastrophes for the parabolic Jordan systems are also studied., 20 pages- typos corrected

Based on the commutativity of scalar vector fields, an algebraic scheme is developed which leads to a privileged multi-dimensionally consistent 2 n + 2 n -dimensional integrable partial differential equation with the associated eigenfunction constituting an infinitesimal symmetry. The ‘universal’ character of this novel equation of vanishing Pfaffian type is demonstrated by retrieving and generalizing to higher dimensions a great variety of well-known integrable equations such as the dispersionless Kadomtsev–Petviashvili and Hirota equations and various avatars of the heavenly equation governing self-dual Einstein spaces.

We 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 two-form defining the potential and equation for it. We develop the dressing scheme, calculate a class of special solutions and demonstrate that reduction from $4+4$-dimensional equation to four-dimensional general heavenly equation can be effectively performed on the level of the dressing data. We consider also the extension of the proposed scheme to $2N+2N$-dimensional case., 14 pages

Changes of type transitions for the two-component hydrodynamic type systems are discussed. It is shown that these systems generically assume the Jordan form (with 2 X 2 Jordan block) on the transition line with hodograph equations becoming parabolic. Conditions which allow or forbid the transition from hyperbolic domain to elliptic one are discussed. Hamiltonian systems and their special subclasses and equations, like dispersionless nonlinear Schroedinger, dispersionless Boussinesq, one-dimensional isentropic gas dynamics equations and nonlinear wave equations are studied. Numerical results concerning the crossing of transition line for the dispersionless Boussinesq equation are presented too., 18 pages, 7 figures

Quasi-classical approximation in the intrinsic description of the vortex filament dynamics is discussed. Within this approximation, the governing equations are given by elliptic system of quasi-linear partial differential equations of the first order. Dispersionless Da Rios system and dispersionless Hirota equation are among them. They describe motion of vortex filament with slow-varying curvature and torsion without or with axial flow. Gradient catastrophe for governing equations is studied. It is shown that geometrically this catastrophe manifests as a fast oscillation of a filament curve around the rectifying plane which resembles the flutter of airfoils. Analytically, it is the elliptic umbilic singularity in the terminology of the catastrophe theory. It is demonstrated that its double scaling regularization is governed by the Painleve I equation.

The singular sector of zero genus case for the Hermitian random matrix model in the large N limit is analyzed. It is proved that the singular sector of the hodograph solutions for the underlying dispersionless Toda hierarchy and the singular sector of the 1-layer Benney (classical long wave equation) hierarchy are deeply connected. This property is due to the fact that the hodograph equations for both hierarchies describe the critical points of solutions of Euler-Poisson-Darboux equations E(a,a), with a=-1/2 for the dToda hierarchy and a=1/2 for the 1-layer Benney hierarchy., Comment: 12 pages

Singular sectors $\mathcal{Z}_{\mathrm{sing}}$ (loci of zeros) for real-valued non-positively defined partition functions $\mathcal{Z}$ of $n$ variables are studied. It is shown that $\mathcal{Z}_{\mathrm{sing}}$ have a stratified structure and each stratum is a set of certain hypersurfaces in $\mathbb{R}^n$. The concept of statistical amoebas is introduced and their properties are studied. Relation with algebraic amoebas is discussed. Tropical limit of statistical amoebas is considered too., Comment: 41 pages, 25 figures. New results added

The algebraic and geometric properties of a novel generalization of Clifford's classical4point–circle configuration are analysed. A connection with the integrable quaternionic discrete Schwarzian Kadomtsev–Petviashvili equation is revealed.

Based on the Lenard relations, we completely classify integrable deformations of general algebraic curves. We construct the general solution of the Lenard relation from the invariance condition with respect to an element of the Galois group of the curve. We give some examples and also some associated conservation laws.

It is known that a large class of integrable hydrodynamic type systems can be constructed through the Lauricella function, a generalization of the classical Gauss hypergeometric function. In this paper, we construct novel class of integrable hydrodynamic type systems which govern the dynamics of critical points of confluent Lauricella type functions defined on finite dimensional Grassmannian Gr(2,n), the set of 2xn matrices of rank two. Those confluent functions satisfy certain degenerate Euler-Poisson-Darboux equations. It is also shown that in general, hydrodynamic type system associated to the confluent Lauricella function is given by an integrable and non-diagonalizable quasi-linear system of a Jordan matrix form. The cases of Grassmannian Gr(2,5) for two component systems and Gr(2,6) for three component systems are considered in details., 22 pages, PMNP 2015, added some comments and references

Symmetry constraints for dispersionless integrable equations are discussed. It is shown that under symmetry constraints the dispersionless Veselov-Novikov equation is reduced to the 1+1-dimensional hydrodynamic type systems., 14 pages, no figures, to appear on Fund.Prikl.Mat. (russian version), Jour.Math.Sciences (english version)

The $\dbar$-dressing scheme based on local nonlinear vector $\dbar$-problem is developed. It is applicable to multidimensional nonlinear equations for vector fields, and, after Hamiltonian reduction, to heavenly equation. Hamiltonian reduction is described explicitely in terms of the $\dbar$-data. An analogue of Hirota bilinear identity for heavenly equation hierarchy is introduced, $\tau$-function for the hierarchy is defined. Addition formulae (generating equations) for the $\tau$-function are found. It is demonstrated that $\tau$-function for heavenly equation hierarchy is given by the action for $\dbar$-problem evaluated on the solution of this problem., Comment: 11 pages

The conformal geometry of the Schwarzian Davey-Stewartson II hierarchy and its discrete analogue is investigated. Connections with discrete and continuous isothermic surfaces and generalised Clifford configurations are recorded. An interpretation of the Schwarzian Davey-Stewartson II flows as integrable deformations of conformally immersed surfaces is given.

Geometrical optics limit of the Maxwell equations for nonlinear media with the Cole-Cole dependence of dielectric function and magnetic permeability on the frequency is considered. It is shown that for media with slow variation along one axis such a limit gives rise to the dispersionless Veselov-Novikov equation for the refractive index. It is demonstrated that the Veselov-Novikov hierarchy is amenable to the quasiclassical DBAR-dressing method. Under more specific requirements for the media, one gets the dispersionless Kadomtsev-Petviashvili equation. Geometrical optics interpretation of some solutions of the above equations is discussed., Comment: 33 pages, 7 figures

It is shown that the geometrical optics limit of the Maxwell equations for certain nonlinear media with slow variation along one axis and particular dependence of dielectric constant on the frequency and fields gives rise to the dispersionless Veselov-Novikov equation for refractive index. It is demonstrated that the last one is amenable to the quasiclassical DBAR-dressing method. A connection is noted between geometrical optics phenomena under consideration and quasiconformal mappings on the plane., Comment: 10 pages, misprints corrected

Recently proposed quasi-classical DBAR-dressing method provides a systematic approach to study the weakly dispersive limit of integrable systems. We apply the quasi-classical DBAR-dressing method to describe dispersive corrections of any order. We show how calculate the DBAR problems at any order for a rather general class of integrable systems, presenting explicit results for the KP hierarchy case. We demonstrate the stability of the method at each order. We construct an infinite set of commuting flows at first order which allow a description analogous to the zero order (purely dispersionless) case, highlighting a Whitham type structure. Obstacles for the construction of the higher order dispersive corrections are also discussed., Comment: 24 pages

We use the semiclassical \(\bar \partial \)-dressing method to derive compact generating equations for dispersionless hierarchies. The considered illustrative examples are the dispersionless Kadomtsev–Petviashvili and two-dimensional Toda lattice hierarchies.

Tropical limit for macroscopic systems in equilibrium defined as the formal limit of Boltzmann constant k going to 0 is discussed. It is shown that such tropical limit is well-adapted to analyse properties of systems with highly degenerated energy levels, particularly of frustrated systems like spin ice and spin glasses. Tropical free energy is a piecewise linear function of temperature, tropical entropy is a piecewise constant function and the system has energy for which tropical Gibbs' probability has maximum. Properties of systems in the points of jump of entropy are studied. Systems with finite and infinitely many energy levels and phenomena of limiting temperatures are discussed., 16 pages, 6 figures

A large class of semi-Hamiltonian systems of hydrodynamic type is interpreted as the equations governing families of critical points of functions obeying the classical linear Darboux equations for conjugate nets.The distinguished role of the Euler-Poisson-Darboux equations and associated Lauricella-type functions is emphasised. In particular, it is shown that the classical g-phase Whitham equations for the KdV and NLS equations are obtained via a g-fold iterated Darboux-type transformation generated by appropriate Lauricella functions., Comment: 14 pages

A remarkable connection between soliton theory and an important and beautiful branch of the theory of graphical statics developed by Maxwell and his contemporaries is revealed. Thus, it is demonstrated that reciprocal triangles that constitute the simplest pair of reciprocal figures representing both a framework and a self-stress encapsulate the integrable discrete BKP equation and its Schwarzian version. The inherent Mobius invariant nature of the Schwarzian BKP equation is then exploited to define reciprocity in an inversive geometric setting. Integrable pairs of lattices of nontrivial combinatorics consisting of reciprocal triangles and their natural generalizations are discussed. Particular reductions of these BKP lattices are related to the integrable discrete versions of Darboux's (2+1)-dimensional sine-Gordon equation and the classical Tzitzeica equation of affine geometry. Furthermore, it is shown that octahedral figures and their hexahedral reciprocals as considered by Maxwell likewise give rise to discrete integrable systems and associated integrable lattices.

It is shown that the integrable discrete Schwarzian KP (dSKP) equation which constitutes an algebraic superposition formula associated with, for instance, the Schwarzian KP hierarchy, the classical Darboux transformation and quasi-conformal mappings encapsulates nothing but a fundamental theorem of ancient Greek geometry. Thus, it is demonstrated that the connection with Menelaus' theorem and, more generally, Clifford configurations renders the dSKP equation a natural object of inversive geometry on the plane. The geometric and algebraic integrability of dSKP lattices and their reductions to lattices of Menelaus-Darboux, Schwarzian KdV, Schwarzian Boussinesq and Schramm type is discussed. The dSKP and discrete Schwarzian Boussinesq equations are shown to represent discretizations of families of quasi-conformal mappings., 26 pages, 9 figures

A \(\overline \partial \) formalism for studying dispersionless integrable hierarchies is applied to the dispersionless KP (dKP) hierarchy. We relate this formalism to the theory of quasiconformal mappings on the plane and present some classes of explicit solutions of the dKP hierarchy.

The dispersionless limit of the scalar nonlocal ∂-problem is derived. It is given by a special class of nonlinear first-order equations. A quasiclassical version of the ∂-dressing method is presented. It is shown that the algebraic formulation of dispersionless hierarchies can be expressed in terms of properties of Beltrami-type equations. The universal Whitham hierarchy and, in particular, the dispersionless KP hierarchy turn out to be rings of symmetries for the quasiclassical ∂-problem. PACS number: 02.30.Uk

Analytic-bilinear approach is used to study continuous and discrete non-isospectral symmetries of the generalized KP hierarchy. It is shown that Mobius symmetry transformation for the singular manifold equation leads to continuous or discrete non-isospectral symmetry of the basic (scalar or multicomponent KP) hierarchy connected with binary Backlund transformation. A more general class of multicomponent Mobius-type symmetries is studied. It is demonstrated that symmetry constraints of KP hierarchy defined using multicomponent Mobius-type symmetries give rise to Calogero–Moser system.

Generalized Weierstrass representations for generic surfaces conformally immersed into four-dimensional Euclidean and pseudo-Euclidean spaces of different signatures are presented. Integrable deformations of surfaces in these spaces generated by the Davey–Stewartson hierarchy of integrable equations are proposed. The Willmore functional of a surface is invariant under such deformations.

Generalizations of the classical affine Lelieuvre's formula to surfaces in projective three-dimensional space and to hypersurfaces in multi-dimensional projective space are given. A discrete version of the projective Lelieuvre's formula is presented too.

Surfaces on which the lines of curvature form geodesics and parallels are discretized in a purely geometric manner. Discrete principal curvatures are defined and it is shown that the natural discrete Gaus equation is given by the standard discrete Schrodinger equation with the discrete Gausian curvature as its potential. The subclass of discrete surfaces of revolution is considered and used to establish algebraic and geometric properties which are reminiscent of those known in the continuous case. Important connections with integrable discrete equations are also recorded.

Surfaces of revolution in three-dimensional Euclidean space are considered. Several new examples of surfaces of revolution associated with well-known solvable cases of the Schrodinger equation (infinite well, harmonic oscillator, Coulomb potential, Bargmann potential, etc.) are analyzed and visualized. The properties of such surfaces are discussed. Two types of deformations (evolutions) of the surfaces of revolution, namely (1) preserving the Gaussian curvature and (2) via the dynamics of the Korteweg-de Vries (KdV) equation are discussed.

It is shown that the discrete Darboux system, descriptive of conjugate lattices in Euclidean spaces, admits constraints on the (adjoint) eigenfunctions which may be interpreted as discrete orthogonality conditions on the lattices. Thus, it turns out that the elementary quadrilaterals of orthogonal lattices are cyclic. Orthogonal lattices on lines, planes and spheres are discussed and the underlying integrable systems in one, two and three dimensions are derived explicitly. A discrete analogue of Bianchi's Ribaucour transformation is set down and particular orthogonal lattices are given. As a by–product, discrete Dini surfaces are obtained.

Analytic-bilinear approach for construction and study of integrable hierarchies is discussed. Generalized multicomponent KP and 2D Toda lattice hierarchies are considered. This approach allows to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations. Generalized hierarchy incorporates basic hierarchy, modified hierarchy, singularity manifold equation hierarchy and corresponding linear problems. Different levels of generalized hierarchy are connected via invariants of Combescure symmetry transformation. Resolution of functional equations also leads to the $\tau $-function and addition formulae to it., Comment: 43 pages, Latex

An analytic-bilinear approach for construction and study of integrable hierarchies, in particular, the KP hierarchy is proposed. It starts with the generalized Hirota identity for the Cauchy–Baker–Akhiezer (CBA) function and leads to a generalized KP hierarchy in the form of compact functional equations containing a special shift operator. A generalized KP hierarchy incorporates the basic KP hierarchy, modified KP hierarchy, KP singularity manifold equation hierarchy, and corresponding hierarchies of linear problems. Different “vertical” levels of the generalized KP hierarchy are connected via invariants of the Combescure symmetry group. The resolution of functional equations also gives rise to the τ function and the addition formulas for it.

It is shown that deformations of affine conormal according to the Nizhnik-Veselov-Novikov equation generate, via the Lelieuvre's formula, integrable deformations of affine surfaces. The total affine mean curvature is preserved by these deformations. It is shown that a stationary point of these deformations for centro-affine surfaces is given by the proper affine sphere.

We introduce a general setting for multidimensional dispersionless integrable hierarchy in terms of differential $m$-form $\Omega_m$ with the coefficients satisfying the Pl\"ucker relations, which is gauge-invariantly closed and its gauge-invariant coordinates (ratios of coefficients) are (locally) holomorphic with respect to one of the variables (the spectral variable). We demonstrate that this form defines a hierarchy of dispersionless integrable equations in terms of commuting vector fields locally holomorphic in the spectral variable. The equations of the hierarchy are given by the gauge-invariant closedness equations., Comment: 10 pages, the text of the talk at PMNP2013 (Gallipoli, Italy, 22-29 June 2013)

∂ operators with non-zero index are found to be associated with integrable hierarchies arising by restricting the KP hierarchy to submanifolds of finite codimension in the space of independent variables.

It is shown that the standard Gauss - Codazzi system of equations which describe generic surfaces immersed into the three-dimensional Euclidean space is equivalent to the two-dimensional Dirac equation (Davey - Stewartson linear problem) accompanied by a specific constraint. It is demonstrated that this system is linearizable via the parametrization of the moving trihedral by the Euler angles. Some special classes of surfaces are also considered.

Unifying hierarchies of integrable equations are discussed. They are constructed via generalized Hirota identity. It is shown that the Combescure transformations, known for a long time for the Darboux system and having a simple geometrical meaning, are in fact the symmetry transformations of generalized integrable hierarchies. Generalized equation written in terms of invariants of Combescure transformations are the usual integrable equations and their modified partners. The KP-mKP, DS-mDS hierarchies and Darboux system are considered., 17 pages, LaTeX

An intrinsic geometry of surfaces and three-dimensional Riemann spaces is discussed. In the geodesic coordinates the Gauss equation for two-dimensional Riemann spaces (surfaces) is reduced to the one-dimensional Schrodinger equation, where the Gaussian curvature plays a role of potential. The use of this fact provides an infinite set of explicit expressions for curvature and metric of surface. A special case is governed by the KdV equation for the Gaussian curvature. Integrable dynamics of curvature via the KdV equation, higher KdV equations, and 2+1-dimensional integrable equations with breaking solitons is considered. For a special class of three-dimensional Riemann spaces the relation between metric and scalar curvature is given by the two-dimensional stationary Schrodinger or perturbed string equations. This provides us an infinite family of Riemann spaces with explicit scalar curvature and metric. Particular class of spaces and their integrable evolutions are described by the Nizhnik–Veselov–Novikov equation and its higher analogs. Surfaces and three-dimensional Riemann spaces with large curvature and slow dependence on the variable are considered. They are associated with the Burgers and Kadomtsev–Petviashvili equations, respectively.