Existence of global solutions to the nonlocal mKdV equation on the line

Abstract

In this paper, we address the existence of global solutions to the Cauchy problem for the integrable nonlocal modified Korteweg-de vries (nonlocal mKdV) equation with the initial data $u_0 \in H^{3}(\mathbb{R}) \cap H^{1,1}(\mathbb{R}) $ with the $L^1(\mathbb{R})$ small-norm assumption. A Lipschitz $L^2$-bijection map between potential and reflection coefficient is established by using inverse scattering method based on a Riemann-Hilbert problem associated with the Cauchy problem. The map from initial potential to reflection coefficient is obtained in direct scattering transform. The inverse scattering transform goes back to the map from scattering coefficient to potential by applying the reconstruction formula and Cauchy integral operator. The bijective relation naturally yields the existence of a global solutions in a Sobolev space $H^3(\mathbb{R})\cap H^{1,1}(\mathbb{R})$ to the Cauchy problem.