College of Science & Engineering
The Kardar-Parisi-Zhang equation is a stochastic, second-order, nonlinear, partial differential equation that has been used to describe widely varied physical phenomena. It was created for the study of random interface growth, but has been applied to such problems as ballistic deposition, the behavior of polymers in stochastic environments, the growth of bacterial colonies, and interacting particle systems. In previous work, this group examined the KPZ equation in the weak noise theory. It transformed the problem into a Hamiltonian formulation, then solved the resulting equations numerically. A phase transition from a symmetric to an asymmetric configuration was found in the minimal-action solutions. The group also found certain solitonic, analytic solutions to the same formulation of the KPZ problem with the use of Hirota's method.
This work generalizes the previously-found analytic solutions and attempts to find in them the same phase transition that was found in the numerical ones. The researchers have already found the analytic solutions that minimize the action in a useful approximation; the code to be run on the supercomputer probes the parameter space in the neighborhood of these approximately minimal-action solutions to find truly minimal-action ones. The phase transition does not seem to appear in the approximation, but this is not surprising; it is hoped that it will emerge in the numeric probe of the true minimum.