On a nonlocal modified Korteweg-de Vries equation: Integrability, Darboux transformation and soliton solutions

• An integrable nonlocal modified Korteweg-de Vries equation (mKdV) is proposed.
• Darboux transformation for the nonlocal mKdV equation is constructed.
• Exact solutions for the nonlocal mKdV equation including soliton, kink, antikink, complexiton, and rogue-wave are given.
• It is demonstrated that these solutions possess new properties which are different from the ones for mKdV equation.

Very recently, Ablowitz and Musslimani introduced a new integrable nonlocal nonlinear Schrödinger equation. In this paper, we investigate an integrable nonlocal modified Korteweg-de Vries equation (mKdV) which can be derived from the well-known AKNS system. We construct the Darboux transformation for the nonlocal mKdV equation. Using the Darboux transformation, we obtain its different kinds of exact solutions including soliton, kink, antikink, complexiton, rogue-wave solution, and nonlocalized solution with singularities. It is shown that these solutions possess new properties which are different from the ones for mKdV equation.