Burgers equation change of variable

Question

In this paper on the Burgers equation (1.1),

$$ u_t = \left(\nu u_{xx} - \sigma u^2\right)_x, $$

the authors use a simple change of variables (2.1a,b)

$$\begin{align} u(x,t) &= q(y,t) \\ x &= y + s(t) \end{align}$$

to write the equation as (2.2)

$$ q_t - \left(q_{yy} + 2\sigma qq_y\right)=\dot{s}(t)q_y. $$

However, should not the $2\sigma qq_y$ term have the opposite sign?

Answer 1

It may be because $\sigma$ is defined as $\pm 1$, so either sign is appropriate.