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- Yuri Yu, Igor Loutsenko
- 1997

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. Mathematics Subject Classification: 35Q51, 35Q53, 35L05, 35L15, 35Q05.

- Igor Loutsenko
- 2002

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)

Some direct relations between soliton solutions of integrable hierarchies and thermodynamical 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)

- Igor Loutsenko
- 2009

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)

- Igor Loutsenko
- 2008

We examine connections between rationality of certain indefinite integrals and equilibrium of Coulomb charges in the complex plane .

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.

- Igor Loutsenko, Oksana Yermolayeva
- 2006

Dynamics of planar domains with multiply connected moving boundaries driven by the gradient of a scalar field that satisfies an elliptic PDE is studied. We consider the question: For which kind of PDEs the domains are algebraic, provided the field has singularities at a finite number of fixed points? The construction reveals a direct connection with the… (More)

- Igor Loutsenko
- 2004

A new type of Coulomb gas is defined, consisting of arbitrary numbers of point charges of two species executing Brownian motions under the influence of their mutual electrostatic repulsion. Being a generalization of a model of identical particles introduced by Dyson as a dynamical system describing non-equilibrium state of various random matrix ensembles,… (More)

We discuss the possibility for a moving droplet of excitons and phonons to form a coherent state inside the packet. We describe such an inhomogeneous state in terms of Bose-Einstein condensation and prescribe it a macroscopic wave function. Existence and, thus, coherency of such a Bosecore inside the droplet can be checked experimentally by letting two… (More)

- Igor Loutsenko
- 2008

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)