Kadomtsev-Petviashvili equation

Here u = u(x, y, t) is a scalar function, x and y are respectively the longitudinal and transverse spatial coordinates, subscripts x, y, t denote partial derivatives, and σ2 = ±1. The case σ = 1 is known as the KPII equation, and models, for instance, water waves with small surface tension. The case σ = i is known as the KPI equation, and may be used to model waves in thin films with high surface tension. The equation is often written with different coefficients in front of the various terms, but the particular values are inessential, since they can be modified by appropriately rescaling the dependent and independent variables.