Third order differential equations describing pseudospherical surfaces

Third order differential equations which describe pseudospherical surfaces are considered. The complete classification of a class of such equations is given. A linear problem with one or more parameters, also known as zero curvature representation, for which the equation is the integrability condition, is explicitly given. The classification provides five large families of differential equations. Third order nonlinear dispersive wave equations, such as the Camassa-Holm equation and Degasperis-Procesi equation are examples contained in the classification. Many other explicit examples are included.