Involute evolute curve family induced by the coupled dispersionless equations

Abstract

In this paper, we derive the coupled dispersionless (CD) equation from the motion of the involute evolute curve pairs, which provides a favourable interpretation for the integrability conditions. This nonlinear differential equation allows us to introduce the Lax pair providing the integrability of this equation. The gauge equivalence between the CD equation and the Zhaidary-III equation is established. Integrable generalized Heisenberg ferromagnet type equations for the tangent, principal normal, and binormal unit vectors {T, N, B} are presented. Each of these three integrable nonlinear equations for the orthogonal frame is, separately, geometrical equivalent counterparts of the CD equation. Finally, we study the link between the involute evolute curves and the CD equation.