A new class of linear second order hyperbolic partial differential operators satisfying Huygens' principle in Minkowski spaces is presented. The construction reveals a direct connection between Huygens' principle and the theory of solitary wave solutions of the Korteweg-de Vries equation.
A family of systems related to a linear and bilinear evolution of roots of polynomials in the complex plane is introduced. Restricted to the line, the evolution induces dynamics of the Coulomb charges (or point vortices) in external potentials, while its fixed points correspond to equilibriums of charges in the plane. The construction reveals a direct… (More)
We examine connections between rationality of certain indefinite integrals and equilibrium of Coulomb charges in the complex plane .
Some direct relations between soliton solutions of integrable hierarchies and thermody-namical quantities of the Coulomb plasmas on the plane are revealed. We find that certain soliton solutions of the Kadomtsev-Petviashvili (KP) and B-type KP (BKP) hierarchies describe two-dimensional one or two component plasmas at special boundary conditions and fixed… (More)
The theory of quadrature domains for harmonic functions and the Hele-Shaw problem of the fluid dynamics are related subjects of the complex variables and mathematical physics. We present results generalizing the above subjects for elliptic PDEs with variable coefficients, emerging in a class of the free-boundary problems for viscous flows in non-homogeneous… (More)
Similarity symmetries of the factorization chains for one-dimensional differential and finite-difference Schrödinger equations are discussed. Properties of the potentials defined by self-similar reductions of these chains are reviewed. In particular, their algebraic structure, relations to q-special functions, infinite soliton systems, supersymmetry,… (More)
We discuss a phase transition of the second order taking place in non-local 1D Ising chains generated by specific infinite soliton solutions of the KdV and BKP equations. To the memory of Vadim B. Kuznetsov 1 The Korteweg–de Vries solitonic spin chain In a series of papers [9, 10], we described a direct relation between soliton solutions of integrable… (More)
The paper concerns the Hamiltonian structure of the finite-dimensional reductions 2D dis-persionless Toda hierarchy constrained by the string equation. We derive the Hamiltonian structure of the reduced dynamics and show connections of integrals of " multi-finger " solutions of the Laplacian growth problem with the " Toda–Krichever " flows of the 2dToda… (More)
Finite-dimensional reductions of the 2D dispersionless Toda hierarchy constrained by the " string equation " are studied. These include solutions determined by polynomial, rational or logarithmic functions, which are of interest in relation to the " Laplacian growth " or Hele-Shaw problem governing interface dynamics. The consistency of such reductions is… (More)
The paper concerns the study of rational and logarithmic reductions of the 2D dispersionless Toda hierarchy of integrable equations. The subject is motivated by important applications to problems in interface dynamics and statistical physics. We prove the consistency of such reductions with respect to the " Toda–Krichever " flows of the 2dToda hierarchy… (More)