On a Riemann-Hilbert problem for the Fokas-Lenells equation

The solution of the initial value problem (IVP) for the Fokas-Lenells equation (FLE) was constructed in terms of the solution M(x,t,k) of a 2 × 2 matrix Riemann-Hilbert problem (RHP) as k→∞, and the one-soliton solution of the FLE was derived based on this Riemann-Hilbert problem, in Lenells and Fokas (2009). However, in fact, the derivative with respect to x of the solution of the FLE (ux(x,t)) was recovered from the RHP as k→∞. In this paper, we construct the solution of the FLE in terms of the RHP as k→0, because the Lax pair of the FLE contains the negative order of the spectral variable k. We show that the one-soliton solution of the FLE obtained in this paper is the same as Lenells and Fokas (2009), but avoiding a complex integral.