We study the large time behavior of random fields which are solutions of a nonlinear partial differential equation, called Burgers' equation, under stochastic initial conditions. These are assumed to be of the shot noise type with the Gibbs-Cox process driving the spatial distribution of the "bumps." In certain cases, this work extends an earlier effort by Surgailis and Woyczynski, where only noninteracting "bumps" driven by the traditional doubly stochastic Poisson process were considered. In contrast to the previous work by Bulinski and Molchanov, a non-Gaussian scaling limit of the statistical solutions is discovered. Burgers' equation is known to describe various physical phenomena such as nonlinear and shock waves, distribution of self-gravitating matter in the universe and so forth.