The Kardar-Parisi-Zhang equation and universality class

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Stochastic partial differential equations (SPDE) are used throughout the sciences to provide more precise models than partial differential equations when randomness or uncertainty are present. However, our understanding of these equation is in a primitive stage. The investigator is studying relevant examples which fall well out of the domain of present theory, in order to develop an applicable theory of SPDE. The most important example is the Kardar-Parisi-Zhang equation. Although the equation is widely used in physics to model randomly growing interfaces, very little was known about well-posedness, or what the solutions looked like. In the last few years, the investigator has discovered several exact solutions, and this research aims to build on these and uncover the integrable structure of the equation.