Riemann-Hilbert approach for an initial-boundary value problem of the two-component modified Korteweg-de Vries equation on the half-line

In this work, we investigate the two-component modified Korteweg-de Vries (mKdV) equation, which is a complete integrable system, and accepts a generalization of 4 x 4 matrix Ablowitz-Kaup-Newell-Segur (AKNS)-type Lax pair. By using of the unified transform approach, the initial-boundary value (IBV) problem of the two-component mKdV equation associated with a 4 x 4 matrix Lax pair on the half-line will be analyzed. Supposing that the solution {u(1)(x, t), u(2)(x, t)) of the two-component mKdV equation exists, we will show that it can be expressed in terms of the unique solution of a 4 x 4 matrix Riemann-Hilbert problem formulated in the complex lambda-plane. Moreover, we will prove that some spectral functions s(lambda) and S(lambda) are not independent of each other but meet the global relationship. (C) 2018 Elsevier Inc. All rights reserved.