Fourth order evolution equations which describe pseudospherical surfaces

Differential equations that describe pseudospherical surfaces are considered. These equations are equivalent to the structure equations of a metric with Gaussian curvature K=−1K=−1. They can also be described as the compatibility condition of an associated linear problem also referred to as a zero curvature representation. A complete and explicit classification of a class of fourth order evolution equations is given. The classification provides four huge classes (referred to as Types I–IV) of fourth order evolution equations that describe pseudospherical surfaces, together with the associated one (or more) parameter linear problems. The differential equations of each type are determined by choosing certain arbitrary differentiable functions. Fourth-order member of the Burgers hierarchy and a modified Kuramoto–Sivashinsky equation are examples of equations described by Types I and IV, respectively. Many other explicit examples are presented.