Linearizability of the Perturbed Burgers Equation
Abstract
We show in this letter that the perturbed Burgers equation
u
t
=2u
u
x
+
u
xx
+ϵ(3
α
1
u
2
u
x
+3
α
2
u
u
xx
+3
α
3
u
2
x
+
α
4
u
xxx
)
is equivalent, through a near-identity transformation and up to order \epsilon, to a linearizable equation if the condition
3
α
1
−3
α
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.