On a Riemann–Hilbert problem for the Fokas–Lenells equation. (January 2019)

• Record Type: Journal Article
• Authors: Ai, Liping; Xu, Jian
• Is Part Of: Applied mathematics letters. Volume 87(2019)
• Page Start: 57
• Page End: 63
• Publication Date: 2019-01
• Subjects: Riemann–Hilbert problem -- Fokas–Lenells equation -- Initial value problem -- Negative order Lax pair; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1016/j.aml.2018.07.027
• Languages: English
• ISSNs: 0893-9659

Abstract

Abstract: 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 ( u x ( 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.