Search

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 × 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 × 4 matrix Lax pair on the half-line will be analyzed. Supposing that the solution {u1(x, t), u2(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 × 4 matrix Riemann–Hilbert problem formulated in the complex λ-plane. Moreover, we will prove that some spectral functions s(λ) and S(λ) are not independent of each other but meet the global relationship.

The Kundu equation as a special case of the complex Ginzburg–Landau equation can be used to describe a slice of phenomena in physics and mechanics. In this paper, we analyzed the Kundu equation on the half-line by the Fokas method and proved that the potential function u(z, t) of the Kundu equation can be uniquely expressed by the solution of Riemann-Hilbert (RH) problem. It also includes the RH problem of the derivative nonlinear Schrodinger equation (also known as Kaup-Newell equation) (if e = 0 ), Chen–Lee–Liu equation (if e = 1 4 ) and Gerjikov–Ivanov equation (if e = 1 2 ) on the half-line.