{{[u]}\over{[t]}} . \,\,$ On the other hand the expression $\,\, a \, {{\partial u}\over{\partial x}} \,\,$ is homogeneous to the dimension $\,\, [a]\,\,$ of scalar $\, a \,$ multiplied by the ratio $\,\, {{[u]}\over{[x]}} \,\,$ and we have $\,\, a \,  {{\partial
u}\over{\partial x}} \sim  [a]\, {{[u]}\over{[x]}} \,.\, $ From equation (1.1.1), the two previous terms $\,\,  {{\partial u}\over{\partial t}} \,\,$ and $\,\, a \,
{{\partial u }\over{\partial x}}\,\,$ have the same dimension and we deduce from the previous formulae the equality : $\,\, {{1}\over{[t]}} \sim  {{[a]}\over{[x]}} \,. \,$ Then we have established that the constant $\, a \,$ is homogeneous to a [**celerity**]{} :

(1.1.2) $\qquad \displaystyle 
[a]\,\, \sim \,\,  {{[x]}\over{[t]}} \,. \,$

$\bullet \qquad \,\,\, $ The Cauchy problem for the model equation (1.1.1) is composed by the equation (1.1.1) itself and the following initial condition :

(1.1.3) $\qquad \displaystyle 
u(x, \,0) \,\,= \,\, u_0(x) \,,\qquad x \in \R \,,\,$

where $\,\, \R \, \ni x \longmapsto u_0(x) \in \R \,\, $ is some given function. We observe that the solution of equation (1.1.1) is constant along the characteristic (straight) lines that satisfy the differential equation

(1.1.4) $\qquad \displaystyle 
{{{\rm d}x}\over{{\rm d}t}} \,\,= \,\, a \,.\, $

[**Proposition 