We discuss a class of evolution equations equivalent to the simplest Universal Field Equation, the so–called Bateman equation, and show that all of them possess (at least) biHamiltonian structure. The first few conserved charges are calculated.

We apply the method of simplest equation for obtaining exact solitary travelingwave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives. We consider first the general case of presence of monomials of the both (odd and even) grades and then turn to the two particular cases of nonlinear equations that contain...

Simplest results presented here are the stability criteria of collocation methods for the second-order Volterra integrodifferential equation (VIDE) by polynomial spline functions. The polynomial spline collocation method is stable if all eigenvalues of a matrix are in the unit disk and all eigenvalues with |λ| = 1 belong to a 1× 1 Jordan block. Also many other conditions are derived depending u...

We describe the method for finding the non-Gaussian tails of the probability distribution function ~PDF! for solutions of a stochastic differential equation, such as the convection equation for a passive scalar, the random driven Navier-Stokes equation, etc. The existence of such tails is generally regarded as a manifestation of the intermittency phenomenon. Our formalism is based on the WKB ap...

In this paper we develop a dressing method for constructing and solving some classes of matrix quasi-linear Partial Differential Equations (PDEs) in arbitrary dimensions. This method is based on a homogeneous integral equation with a nontrivial kernel, which allows one to reduce the nonlinear PDEs to systems of non-differential (algebraic or transcendental) equations for the unknown fields. In ...

In this paper, we study the Kuralay equations, namely Kuralay-I equation (K-IE) and Kuralay-II (K-IIE). The integrable motion of space curves induced by these equations is investigated. gauge equivalence between two established. With help Hirota bilinear method, simplest soliton solutions are also presented. nonlocal dispersionless versions considered. Some generalizations other related nonline...