On Asymptotics of the Solution to the Cauchy Problem for a Singularly Perturbed Operator-Differential Transport Equation with Weak Diffusion in the Case of Several Space Variables

A formal asymptotic expansion is constructed for the solution of the Cauchy problem for a singularly perturbed operator differential transport equation with small nonlinearities and weak diffusion in the case of several space variables. Under the conditions imposed on the data of the problem, the leading asymptotic term is described by the multidimensional generalized Burgers–Korteweg–de Vries equation. Under certain conditions, the remainder is estimated with respect to the residual.