In the present paper, we study the real and complex coupled dispersionless (CD) equations, the real and complex short pulse (SP) equations geometrically and algebraically. From the geometric point of view, we first establish the link of the motions of space curves to the real and complex CD equations, then to the real and complex SP equations via hodograph transformations. The integrability of these equations are confirmed by constructing their Lax pairs geometrically. In the second part of the paper, it is made clear for the connection between the real and complex CD and SP equations and the two-component extended Kadomtsew-Petviashvili (KP) hierarchy. As a by-product, the N -soliton solutions in the form of determinants for these equations are provided.

In this paper, we prove that an integrable nonlocal complex modified Korteweg-de Vries (mKdV) equation introduced by Ablowitz and Musslimani [Nonlinearity 29, 915–946 (2016)] is gauge equivalent to a spin-like model. From the gauge equivalence, one can see that there exists significant difference between the nonlocal complex mKdV equation and the classical complex mKdV equation. Through constructing the Darboux transformation for nonlocal complex mKdV equation, a variety of exact solutions including dark soliton, W-type soliton, M-type soliton, and periodic solutions are derived.

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