Related papers: Symmetry and Integrability of Classical Field Equa…
The problems of Classical Electrodynamics with the electron equation of motion and with non-integrable singularity of its self-field stress tensor are well known. They are consequences, we show, of neglecting terms that are null off the…
The discrete symmetry's dressing chains of the nonlinear Schrodinger equation (NLS) and Davey-Stewartson equations (DS) are consider. The modified NLS (mNLS) equation and the modified DS (mDS) equations are obtained. The explicitly…
The model of dark matter is presented where the dark matter is a \emph{classical} gauge field. A spherical symmetric solution of Yang-Mills equation is obtained. The asymptotic behavior of the gauge fields and matter density is…
We represent a version of multidimensional quasilinear partial differential equation (PDE) together with large manifold of particular solutions given in an integral form. The dimensionality of constructed PDE can be arbitrary. We call it…
A discrete nonlinear system is analysed in case of open chain boundary conditions at the ends. It is shown that the integrability of the system remains intact, by obtaining a modified set of Lax equations which automatically take care of…
Our recent results on {\em extended crystal PDE's} are generalized to PDE's in the category $\mathfrak{Q}_S$ of quantum supermanifolds. Then obstructions to the existence of global quantum smooth solutions for such equations are obtained,…
Symmetries of the field equations are used to construct infinitely many nontrivial linearly independent new solutions to different partial differential equations such as the Schroedinger, the diffusion, and the paraxial equations, among…
Functional relation for commuting quantum transfer matrices of quantum integrable models is identified with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice…
We discuss the solution of eigenvalue problems associated with partial differential equations that can be written in the generalized form $\m{A}x=\lambda\m{B}x$, where the matrices $\m{A}$ and/or $\m{B}$ may depend on a scalar parameter.…
We investigate real solutions of a C-integrable non-evolutionary partial differential equation in the form of a scalar conservation law where the flux density depends both on the density and on its first derivatives with respect to the…
For static reductions of isotropic and anisotropic Magnetohydrodynamics plasma equilibrium models, a complete classification of admitted point symmetries and conservation laws up to first order is presented. It is shown that the symmetry…
We describe a variant of the dressing method giving alternative representation of multidimensional nonlinear PDE as a system of Integro-Differential Equations (IDEs) for spectral and dressing functions. In particular, it becomes single…
In this paper we introduce an observer design framework for ordinary differential equation (ODE) systems based on various types of existing or even novel one-parameter symmetries (exact, asymptotic and variational) ending up with a certain…
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…
We present a pedagogical review of old inconsistencies of Classical Electrodynamics and of some new ideas that solve them. Problems with the electron equation of motion and with the non-integrable singularity of its self-field energy tensor…
It is shown that all strongly symmetric elements are solutions of constant classical Yang-Baxter equation in Lie algebra, or of quantum Yang-Baxter equation in algebra. Otherwise, all solutions of constant classical Yang-Baxter equation…
We represent an integration algorithm combining the characteristics method and Hopf-Cole transformation. This algorithm allows one to partially integrate a large class of multidimensional systems of nonlinear Partial Differential Equations…
The reduction by symmetry of the linear system of the self-dual Yang-Mills equations in four-dimensions under representatives of the conjugacy classes of subgroups of the connected part to the identity of the corresponding Euclidean group…
We prove the existence of classical solutions to parabolic linear stochastic integro-differential equations with adapted coefficients using Feynman-Kac transformations, conditioning, and the interlacing of space-inverses of stochastic flows…
The general stationary cylindrically symmetric solution of Einstein-massless scalar field system with a non-positive cosmological constant is presented. It is shown that the general solution is characterized by four integration constants.…