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

Abstract

AbstractWe address the existence of global solutions to the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (nonlocal NLS) equation under the initial data with the small‐norm. The nonlocal NLS equation was first introduced by Ablowitz and Musslimani as a new nonlocal reduction of the well‐known Ablowitz–Kaup–Newell–Segur system. The main technical difficulty for proving its global well‐posedness on the line in is due to the fact that mass and energy conservation laws, being nonlocal, do not preserve any reasonable norm and may be negative. In this paper, we use the inverse scattering transform approach to prove the existence of global solutions in based on the representation of a Riemann–Hilbert (RH) problem associated with the Cauchy problem of the nonlocal NLS equation. A key of this approach is, by applying the Volttera integral operator and Cauchy integral operator, to establish a Lipschitz bijective map between the solution of the nonlocal NLS equation and reflection coefficients associated with the RH problem. By using the reconstruction formula and estimates on the solution of the time‐dependent RH problem, we further affirm the existence of a unique global solution to the Cauchy problem for the nonlocal NLS equation.