N Martinov

New class of running-wave solutions of the (2+1)-dimensional sine-Gordon equation

A new class of running-wave solutions of the (2+1)-dimensional sine-Gordon equation is investigated. The obtained waves require two spatial dimensions for their propagation, i.e. they generalize solutions of the (2+0)-dimensional sine-Gordon equation. The parameters of the waves strongly depend on the wave amplitude and there exist forbidden areas for the wavenumber and frequency. The obtained solutions describe a new class of Josephson waves whose velocity is smaller than the Swihart velocity. If omega =0 the running waves are reduced to the self-consistent phase, current and magnetic field distributions in a large two-dimensional Josephson junction. The self-restriction coefficient for the Josephson current corresponding to one of the structures is calculated.

On some solutions of the two-dimensional sine-Gordon equation

There exists an approach for finding of exact solutions of the two-dimensional sine-Gordon equation. Using this approach three classes of solutions have been found. One of the classes consists of running wave solutions which are a generalization of the solutions of the one-dimensional sine-Gordon equation. This class of solutions is studied here.

Running wave solutions of the two-dimensional sine-Gordon equation

After the Lamb substitution the 2D sine-Gordon equation was solved. Three classes of solutions for this equation were found. Ad hoc the 2D sine-Gordon equation was reduced to an algebraic system consisting of three equations. The solutions given of the 2D sine-Gordon equation are a generalization of the solutions of the 1D sine-Gordon equation. They are also a generalization of the solutions of the equation phi xx+yy=sin phi (x,y).

On the correspondence between the self-consistent 2D Poisson-Boltzmann structures and the sine-Gordon waves

By means of the transformation connecting the sine-Gordon equation and the two-dimensional Poisson-Boltzmann equation a correspondence between the self-consistent two-dimensional Poisson-Boltzmann structures and the solutions of the sine-Gordon equation representing standing waves has been established. In this way two new solutions of the sine-Gordon equation were obtained and studied and their application for description of waves into ferromagnetics was examined.

New types of polarisation following from the non-linear spherical radial Poisson-Boltzmann equation

All possible types of solutions of the non-linear spherical radial Poisson-Boltzmann equation, describing the spatial distribution of the particles in a classical two-component Coulomb gas, are investigated analytically and numerically. The obtained solutions show that every particle of the system and its screening cloud forms an atom- or ion-like structure and reveals a tendency towards condensation in the Coulomb gas.

An approach for solving the two-dimensional sine-Gordon equation

A new approach for finding analytical solutions of the two-dimensional sine-Gordon equation is presented. The essence of this approach is the established relation between the solutions of the one-dimensional wave equation having the form of running waves and solutions of the two-dimensional sine-Gordon equation.

A tendency to a formation of two-dimensional self-consistent structures in Coulomb systems

A class of periodic solutions of the two-dimensional Poisson-Boltzmann system was found. These Jacobi elliptic functions reveal a tendency to self-organisation, e.g. periodic spatial distribution, for self-consistent Coulomb ensembles.

Stimulated waves of electric and magnetic fields in a two-dimensional semi-infinite Josephson junction

A two-dimensional semi-infinite Josephson junction without damping is considered. Its interaction with an external oscillating electromagnetic field in the form of a running wave with a phase velocity equal to the Swihart velocity is investigated. The results obtained are based on an exact solution of the (2+1)-dimensional sine-Gordon equation, depending on an arbitrary function. The boundary conditions on the interface are provided by an external time-varying electric field consistent with the exact solution. Under these conditions, an electromagnetic structure arises inside the junction. It is shown how the existence of this formation may be proved experimentally. A method for measuring the Swihart velocity is proposed.

New class solutions of the 3‐D Laplace equation

The problem of finding a solution of the three‐dimensional Laplace equation is reduced to the problem of solving a system of nonlinear algebraic equations by a new ansatz. The obtained solutions are expressed in real Jacobi elliptic functions and are periodic in three dimensions. They depend on four free parameters. The space behavior and the periodicity of the solutions with respect to the values of the free parameters are investigated.

About one superposition of solutions of the Laplace equation

New exact solutions of the three-dimensional Laplace equation are found. They are obtained by a superposition of previously found 3D-periodic ones. The new solutions depend on more free parameters than those already known.