A new generalization of Tzitzeica curves in arbitrary dimension

Abstract

We generalize the notion of a Tzitzeica curve r in the nth dimension by using the constant ratio of the αth power of the Wronskians W(r′) of the derivative curve r′ and the βth power of the Wronskian W(r) of the original curve. In this context, the powers α and β may be seen as control parameters whose interplay determine special classes of (α, β)-Tzitzeica curves. Our paper is intended to be example–oriented and, therefore, several intriguing, new families of generalized Tzitzeica curves in the nth dimension, such as the power and exponential curves, are introduced and discussed in detail. In particular, for n = 2, we show how the (α, β)-Tzitzeica curves are related to the second order homogeneous linear Schr¨odinger equation.