The Riemann–Hilbert approach for the integrable fractional Fokas–Lenells equation

In this paper, we propose a new integrable fractional Fokas–Lenells equation by using the completeness of the squared eigenfunctions, dispersion relation, and inverse scattering transform. To solve this equation, we employ the Riemann–Hilbert approach. Specifically, we focus on the case of the reflectionless potential with a simple pole for the zero boundary condition. And we provide the fractional $N$-soliton solution in determinant form. In addition, we prove the fractional one-soliton solution rigorously. Notably, we demonstrate that as $\left|t\right|\to \infty$, the fractional $N$-soliton solution can be expressed as a linear combination of $N$ fractional single-soliton solutions.