Related papers: New explicit exact solutions for the Li\'enard equ…
For the 2D and 3D Euler equations, their existing exact solutions are often in linear form with respect to variables x,y,z. In this paper, the Clarkson-Kruskal reduction method is applied to reduce the 2D incompressible Euler equations to a…
By a generalized bidirectional decomposition method, we obtain many new Superluminal localized solutions to the wave equation (for the electromagnetic case, in particular) which are suitable for arbitrary frequency bands; various of them…
The Dirac equation is generalized to $D+1$ space-time.The conserved angular momentum operators and their quantum numbers are discussed. The eigenfunctions of the total angular momenta are calculated for both odd $D$ and even $D$ cases. The…
Exact solutions are found to the Dirac equation for a combination of Lorentz scalar and vector Coulombic potentials with additional non-Coulombic parts. An appropriate linear combination of Lorentz scalar and vector non-Coulombic…
In this paper, we provide a systematic way of finding explicit solutions for a class of continuous fragmentation equations with growth or decay in the state space and derive explicit solutions in the cases of constant and linear…
A method is given to construct globally analytic (in space and time) exact solutions to the focusing cubic nonlinear Schrodinger equation on the line. An explicit formula and its equivalents are presented to express such exact solutions in…
We derive the modified L\"uscher equation in the presence of the long-range force caused by the exchange of a light particle. It is shown that the use of this equation enables one to circumvent the problems related to the strong…
We obtain exact solution of the Dirac equation with the Coulomb potential as an infinite series of square integrable functions. This solution is for all energies, the discrete as well as the continuous. The spinor basis elements are written…
A combination of some weighted energy estimates is applied for the Cauchy problem of quasilinear wave equations with the standard null conditions in three spatial dimensions. Alternative proofs for global solutions are shown including the…
We obtain exact, simple and very compact expressions for the linearization coefficients of the products of orthogonal polynomials; both the conventional Clebsch-Gordan-type and the modified version. The expressions are general depending…
Consider a bounded solution of the focusing, energy-critical wave equation that does not scatter to a linear solution. We prove that this solution converges in some weak sense, along a sequence of times and up to scaling and space…
A class of Siklos waves, representing exact vacuum solutions of general relativity with a cosmological constant, is extended to a new class of Siklos waves with torsion, defined in the framework of the Poincar\'e gauge theory. Three…
The invariance for the equation of fast diffusion in the 2D coordinate space has been proved, and its reduction to the 1D (with respect to the spatial variable) analog is demonstrated. On the basis of these results, new exact…
We give a more detailed description of the new system of Pl\"ucker-like equations from [4], discuss how it relates to the usual Pl\"ucker equations, and correct a mistake in that article.
The Teukolsky Master Equation is the basic tool for study of perturbations of the Kerr metric in linear approximation. It admits separation of variables, thus yielding the Teukolsky Radial Equation and the Teukolsky Angular Equation. We…
The exact quantum-mechanical equations for beams of free particles including photons and for beams of Dirac and spin-1 particles in a uniform magnetic field have been derived. These equations present the exact generalizations of the…
The Regge-Wheeler equation describes axial perturbations of Schwarzschild metric in linear approximation. Teukolsky Master Equation describes perturbations of Kerr metric in the same approximation. We present here unified description of all…
It is shown that the Confluent Heun Equation (CHEq) reduces for certain conditions of the parameters to a particular class of Quasi-Exactly Solvable models, associated with the Lie algebra $sl (2,{\mathbb R})$. As a consequence it is…
For the first time exact analytical solutions to the eikonal equations in (1+1) dimensions with a refractive index being a saturated function of intensity are constructed. It is demonstrated that the solutions exhibit collapse; an explicit…
We derive explicit expressions of the homogeneous solutions in two dimensional cones for Pucci's extremal equations. As examples of possible applications, we obtain monotonicity formulas for all nonnegative supersolutions and necessary and…