Related papers: Critical points, Lauricella functions and Whitham-…

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…

Structure and properties of families of critical points for classes of functions $W(z,\bar{z})$ obeying the elliptic Euler-Poisson-Darboux equation $E(1/2,1/2)$ are studied. General variational and differential equations governing the…

Interrelation between Thom's catastrophes and differential equations revisited. It is shown that versal deformations of critical points for singularities of A,D,E type are described by the systems of Hamilton-Jacobi type equations. For…

Based on the well-established theory of discrete conjugate nets in discrete differential geometry, we propose and examine discrete analogues of important objects and notions in the theory of semi-Hamiltonian systems of hydrodynamic type. In…

We consider a broad class of systems of nonlinear integro-differential equations posed on the real line that arise as Euler-Lagrange equations to energies involving nonlinear nonlocal interactions. Although these equations are not readily…

In this work, we define and calculate critical exponents associated with higher order thermodynamic phase transitions. Such phase transitions can be classified into two classes: with or without a local order parameter. For phase transitions…

We prove the existence of critical points of the $N$-vortex Hamiltonian $H_\Omega (x_1,\ldots, x_N) =\sum\limits^N_{i=1}\Gamma^2_i h(x_i) + \sum\limits_{i,j=1\atop j\not= k}^N…

We prove the existence of critical points of vortex type Hamiltonians \[ H(p_1,\ldots, p_N) = \sum_{{i,j=1},{i\ne j}}^N \Gamma_i\Gamma_jG(p_i,p_j)+\psi(p_1,\dots,p_N) \] on a closed Riemannian surface $(\Sigma,g)$ which is not homeomorphic…

In this paper we define a Donaldson type functional whose Euler-Lagrange equations are a system of differential equations which corresponds to Hitchin's self-duality equations for a suitable choice of Higgs bundle on closed Riemann…

Critical analyses of well-known methods of derivation of kinetic and hydrodynamic equations is presented. Another method of derivation of kinetic and hydrodynamic equations from classic mechanics is described. It is shown that equations of…

Relative equilibria of Lagrangian and Hamiltonian systems with symmetry are critical points of appropriate scalar functions parametrized by the Lie algebra (or its dual) of the symmetry group. Setting aside the structures - symplectic,…

We consider critical points of a class of functionals on compact four-dimensional manifolds arising from Regularized Determinants for conformally covariant operators, whose explicit form was derived in [10], extending Polyakov's formula.…

We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient…

We study the minimizer u of a convex functional in the plane which is not G\^ateaux-differentiable. Namely, we show that the set of critical points of any C^1-smooth minimizer can not have isolated points. Also, by means of some appropriate…

We consider several types of quantum critical phenomena from finite-density gauge-gravity duality which to different degrees lie outside the Landau-Ginsburg-Wilson paradigm. These include: (1) a "bifurcating" critical point, for which the…

This paper establishes the theoretical foundation for statistical applications of an intriguing new type of spatial point processes called critical point processes. These point processes, residing in Euclidean space, consist of the critical…

The large-N saddle-point equations for the principal chiral models defined on a d-1 dimensional simplex are derived from the external field problem for unitary integrals. The saddle point equation are studied analytically and numerically in…

In this paper, we study some qualitative properties of Hardy-Littlewood-Sobolev type systems. The HLS type systems are categorized into three cases: critical, supercritical and subcritical. The critical case, the well known original HLS…

In the chiral limit the complicated many-body dynamics around the second-order chiral phase transition of two-flavor QCD can be understood by appealing to universality. We present a novel formulation of the real-time functional…

We review some essential aspects of classically integrable systems. The detailed outline of the lectures consists of: 1. Introduction and motivation, with historical remarks; 2. Liouville theorem and action-angle variables, with examples…