Exact result vs. dynamic renormalization group analysis for the nonlocal molecular-beam-epitaxy equation

In this paper, I analyze a generalization of the molecular-beam-epitaxy equation (with spatially correlated noise) that takes into account long-range interactions of a volume conserving surface. This equation is, therefore, called the non-local molecular-beam-epitaxy (NMBE) equation, sometimes also known as the non-local conserved Kardar–Parisi–Zhang equation. I find an exact result for a subfamily of the NMBE models in one dimension. I then compare this exact result to a previous result obtained by Jung et al. (Phys. Rev. E 62 (2000) 2949) who applied a dynamic renormalization group (DRG) analysis to this problem. I show explicitly that this approach does not yield the exact result I obtain. This example suggests that one should be careful when applying DRG to nonlinear continuum equations.