Determination of the exact solutions to the inhomogeneous burgers equation with the use of the darboux transformation

A method of determining the exact solutions to the Burgers equation on the basis of the Darboux transformation is described. It is shown that a single application of the Darboux transformation to the homogeneous Burgers equation transforms the latter into the inhomogeneous equation describing acoustic wave propagation against transonic flow in the de Laval nozzle. In this case, the contraction ratio of the nozzle is fixed and determined by the viscosity coefficient of the medium. Based on the exact solution of the homogeneous Burgers equation, for the aforementioned problem of the flow in the nozzle, all the possible regular steady-state solutions are presented and the evolution of nonstationary solutions is investigated. The algorithm of a multiple Darboux transformation, which allows an increase in the strength of inhomogeneity, i.e., in the contraction ratio of the nozzle, is determined. This approach leads to a discrete set of possible contraction ratios at which exact solutions can be obtained. The Crum's theorem is used to derive a formula that allows determination of the exact solutions to the inhomogeneous Burgers equation from the solutions to the homogeneous heat transfer equation. It is noted that, in fact, the proposed algorithm of the multiple Darboux transformation makes it possible to decrease the viscosity coefficient of the medium in a discrete way.