Framed Curve Families Induced by Real and Complex Coupled Dispersionless-Type Equations

Abstract

In this study, we investigate coupled real and complex dispersionless equations for curve families, even if they have singular points. Even though the connections with the differential equations and regular curves were considered in various ways in the past, since each curve does not need to be regular, we establish the connections for framed base curves, which generalize regular curves with linear independent conditions. Also, we give the Lax pairs of the real and complex coupled dispersionless equations from the motions of any framed curve. These give us significant conditions based on the framed curvatures and associated curvatures of the framed curves for integrability since it is well known that the Lax pair provides the integrability of differential equations.