Asymptotic solutions of a generalized Burgers equation

Abstract

The travelling wave solutions of the generalized Burgers equation \[ ∂ u ∂ t = ∂ ∂ x [ D ( u ) ∂ u ∂ x ] − ∂ ∂ x [ K ( u ) ] \frac {{\partial u}}{{\partial t}} = \frac {\partial }{{\partial x}}\left [ {D\left ( u \right ) \frac {{\partial u}}{{\partial x}}} \right ] - \frac {\partial }{{\partial x}}\left [ {K\left ( u \right )} \right ] \] are related to the solution of the initial boundary value problems for the same equation, subject to initial boundary conditions relevant to the physical problem of infiltration of moisture into a homogeneous soil. The theoretical prediction of the emergence of the travelling wave solutions as intermediate asymptotics is confirmed by numerical solutions of the problem for some specific choices of the functions D ( u ) D\left ( u \right ) and K ( u ) K\left ( u \right ) .