Parabolic equation with exponential nonlinearity as the condition of joint venture linear system

Abstract

An attempt is made to construct a 2 + 1 - dimensional nonlinear model that is a condition for the compatibility of a system of two first - order linear differential equations. The resulting equation is related to the operator L, A to the pair and the Lax equation, when the operator L contains differentiation with respect to only one variable and depends parametrically on two additional variables differentiation with respect to which enters into the operator A. We study the possibility of reducing the linear part to a parabolic form, and also transformation of coordinates leading to exponential nonlinearity. The resulting equation has an applied value, since it can be attributed to Toda-type diffusion chain models, the main feature of which is the presence of nonlinearities of exponential type in them. Such equations describe the transfer of a passive impurity, for example, heat in a turbulent medium with a nonlinear turbulent thermal conductivity coeficient.