In this article, we focus on the inverse scattering transformation for the Fokas-Lenells (FL) equation with nonzero boundary conditions via the Riemann-Hilbert (RH) approach. Based on the Lax pair of the FL equation, the analyticity and symmetry, asymptotic behavior of Jost solutions and scattering matrix are discussed in detail. With these results, we further present a generalized RH problem, from which a reconstruction formula between the solution of the FL equation and the Riemann-Hilbert problem is obtained. The N-soliton solutions of the FL equation is obtained via solving the RH problem.

中文翻译：

---

具有非零边界条件的 Fokas-Lenells 方程的逆散射变换

在本文中，我们将重点介绍通过 Riemann-Hilbert (RH) 方法对具有非零边界条件的 Fokas-Lenells (FL) 方程进行逆散射变换。基于FL方程的Lax对，详细讨论了Jost解的解析性和对称性、渐近行为和散射矩阵。有了这些结果，我们进一步提出了一个广义的 RH 问题，从中得到了 FL 方程的解与 Riemann-Hilbert 问题之间的重构公式。FL 方程的 N 孤子解是通过求解 RH 问题获得的。