[Paper Review] The coupled Fokas-Lenells equations by a Riemann-Hilbert approach

Beibei Hu, Tiecheng Xia|arXiv (Cornell University)|Nov 9, 2017

Nonlinear Waves and Solitons30 references8 citations

TL;DR

This paper applies the unified transform method via a Riemann-Hilbert problem to solve the initial-boundary value problem for the coupled Fokas-Lenells equations on the half-line. It shows that the spatial derivatives $\{q_x(x,t), r_x(x,t)\}$ are determined by the solution of a $3\times3$ matrix Riemann-Hilbert problem in the complex spectral plane, and the full solution $\{q(x,t), r(x,t)\}$ is recovered by integration, establishing a constructive framework for integrable systems with $3\times3$ Lax pairs.

ABSTRACT

In this paper, we use the unified transform method to consider the initial-boundary value problem for the coupled Fokas-Lenells equations on the half-line, assuming that the solution $\{q(x,t),r(x,t)\}$ of the coupled Fokas-Lenells equations exists, we show that $\{q_x(x,t),r_x(x,t)\}$ can be expressed in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter $λ$. Thus, the solution $\{q(x,t),r(x,t)\}$ can be obtained by integration with respect to $x$.

Motivation & Objective

• To develop a systematic approach for solving the initial-boundary value problem of the coupled Fokas-Lenells equations on the half-line.
• To extend the unified transform method—previously used for $2\times2$ Lax pairs—to systems with $3\times3$ matrix Lax pairs.
• To establish a Riemann-Hilbert problem formulation that encodes initial and boundary data into spectral functions for reconstructing the solution.
• To provide a rigorous framework for the existence and construction of solutions to the coupled Fokas-Lenells system under Schwartz-class initial and boundary conditions.

Proposed method

• Formulate the initial-boundary value problem for the coupled Fokas-Lenells equations using a $3\times3$ matrix Lax pair.
• Define spectral functions $s(\lambda)$ and $S(\lambda)$ from initial and boundary data in the Schwartz class.
• Construct a sectionally meromorphic matrix $M(x,t,\lambda)$ on the Riemann $\lambda$-sphere with jumps across contours $\bar{D}_n \cap \bar{D}_m$.
• Derive the jump condition $M_n(\lambda) = M_m(\lambda) J_{m,n}$ for $n \neq m$, with jump matrix $J_{m,n}$ depending on spectral data.
• Use large-$\lambda$ asymptotics of $M(x,t,\lambda)$ to recover $q_x(x,t)$ and $r_x(x,t)$ via $-2i\lim_{\lambda\to\infty} \lambda M_{12}$ and $\lambda M_{13}$.
• Integrate the spatial derivatives to reconstruct the full solution $\{q(x,t), r(x,t)\}$.

Experimental results

Research questions

• RQ1 Can the unified transform method be extended to integrable systems with $3\times3$ Lax pairs, such as the coupled Fokas-Lenells equations?
• RQ2 How can initial and boundary conditions be encoded into a Riemann-Hilbert problem for such systems?
• RQ3 What is the precise structure of the spectral functions and jump matrices required for solution reconstruction?
• RQ4 Can the solution be recovered from the Riemann-Hilbert problem under Schwartz-class decay assumptions on initial and boundary data?
• RQ5 What is the role of the global relation in linking initial and boundary data through spectral functions?

Key findings

• The spatial derivatives $q_x(x,t)$ and $r_x(x,t)$ are expressed as residues of the large-$\lambda$ asymptotics of the solution $M(x,t,\lambda)$ to a $3\times3$ matrix Riemann-Hilbert problem.
• The solution $M(x,t,\lambda)$ satisfies a jump condition across contours separating the Riemann sphere into four domains $D_1, D_2, D_3, D_4$, with jump matrix $J_{m,n}$ derived from spectral data.
• The solution $M(x,t,\lambda)$ is normalized as $M(x,t,\lambda) = \mathbb{I} + O(\lambda^{-1})$ as $\lambda \to \infty$, ensuring proper spectral behavior.
• The Riemann-Hilbert problem is well-posed under the assumption that the spectral functions have no zeros at the poles $\lambda_j$, ensuring the existence of the solution.
• The full solution $\{q(x,t), r(x,t)\}$ is obtained by integrating $q_x(x,t)$ and $r_x(x,t)$ with respect to $x$, completing the reconstruction.
• The method establishes a constructive framework for solving initial-boundary value problems of integrable systems with higher-order Lax pairs, generalizing the inverse scattering approach to boundary value problems.