Linearizability of the Perturbed Burgers Equation

Abstract

We show in this letter that the perturbed Burgers equation $u_t = 2uu_x + u_{xx} + ε( 3 α_1 u^2 u_x + 3α_2 uu_{xx} + 3α_3 u_x^2 + α_4 u_{xxx} )$ is equivalent, through a near-identity transformation and up to order ε, to a linearizable equation if the condition $3α_1 - 3α_3 - 3/2 α_2 + 3/2 α_4 = 0$ is satisfied. In the case this condition is not fulfilled, a normal form for the equation under consideration is given. Then, to illustrate our results, we make a linearizability analysis of the equations governing the dynamics of a one-dimensional gas.