The Burgers equation can be understood as a simplification of the Navier-Stokes equations when the pressure term is neglected:
$$ \frac{\partial u_i}{\partial t}+\ u_j\frac{\partial u_i}{\partial x_j}=-\frac{1}{\rho}\frac{\partial p}{\partial x_i}+\nu\frac{\partial^2u_i}{\partial x_j^2} \quad\Rightarrow \quad \frac{\partial u_i}{\partial t}+\ u_j\frac{\partial u_i}{\partial x_j}=\nu\frac{\partial^2u_i}{\partial x_j^2} $$
Using the Burgers equation instead of the Navier Stokes equation has the advantage that there is an analytical solution for the Burgers equation. This can be obtained using the Hopf-Cole transformation, which reduces the Burgers PDE, which is nonlinear, to the heat equation, which is linear.
However, when is it possible to neglect the pressure term $\boldsymbol{\nabla}p$ in the Navier-Stokes equations? What could be a physical example of such a situation?