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We obtain the the existence of global solutions to the Cauchy problem of theFokas-Lenells (FL) equation on the line \begin{align}&u_{xt}+\alpha\beta^2u-2i\alpha\beta u_x-\alphau_{xx}-i\alpha\beta^2|u|^2u_x=0,\nonumber \\ &u(x,t=0)=u_0(x), \nonumber\end{align} where without the small-norm assumption on initial data $u_0(x)\inH^3(\mathbb{R})\cap H^{2,1}(\mathbb{R})$. Our main technical tool is theinverse scattering transform method based on the representation of aRiemann-Hilbert (RH) problem associated with the above Cauchy problem. Theexistence and the uniqueness of the RH problem is shown via a general vanishinglemma. The spectral problem associated with the FL equation is changed into anequivalent Zakharov-Shabat-type spectral problem to establish the RH problemson the real axis. By representing the solutions of the RH problem via theCauchy integral protection and the reflection coefficients, the reconstructionformula is used to obtain a unique local solution of the FL equation. Further,the eigenfunctions and the reflection coefficients are shown Lipschitzcontinuous with respect to initial data, which provides a priori estimate ofthe solution to the FL equation. Based on the local solution and the uniformlypriori estimate, we construct a unique global solution in $H^3(\mathbb{R})\capH^{2,1}(\mathbb{R})$ to the FL equation.

A Riemann-Hilbert approach to existence of global solutions to the Fokas-Lenells equation on the line