The solution to the Hamilton-Jacobi equation which approximates the master equation of a nonlinear chemical system is, in general, impossible to obtain explicitly. In this work, we introduce a natural method for approximating the solution of the Hamilton-Jacobi equation, called "Poissonian gluing," which has a general range of application. We show on a specific two-dimensional example (autocatalytic reaction in two cells coupled by diffusion) that this new approximation leads to explicit analytic results which are in good agreement with numerical simulations.