Linearizability of the perturbed Burgers equation

We show in this report that the perturbed Burgers equation u(t)=2uu(x) + u(xx)+ epsilon(3 alpha(1)u(2)u(x) + 3 alpha(2)uu(xx) + 3 alpha(3)u(x)(2) + alpha(4)u(xxx)) is equivalent, through a near-identity transformation and up to O(epsilon), to a linearizable equation if the condition 3 alpha(1)-3 alpha(3)-3/2 alpha(2)+3/2 alpha(4)=0 is satisfied. In the case this condition is not fulfilled, a normal form for the equation under consideration is given. We show, furthermore, that nonlinearizable cases lead to perturbative expansions with secular-type behavior. Then, to illustrate our results, we make a linearizability analysis of the equations governing the dynamics of a one-dimensional gas.