A note on solutions for the intrinsic generalized wave and sine-Gordon equations (Q1191790)

In the classical theory of differential geometry, the sine-Gordon equation was associated to surfaces of constant negative curvature contained in the Euclidean space \(\mathbb{R}^ 3\). A generic solution of this equation represents the angle between the asymptotic curves of the surface. We consider the unit vector fields determined by two angle functions and rewrite the intrinsic generalized sine-Gordon and wave equations in terms of these functions. Moreover, we obtain explicit 1-soliton solutions and provide a graph of the angle functions.