Riemann-Hilbert problem and N-soliton solutions for the n-component derivative nonlinear Schr?dinger equations

The n-component derivative nonlinear Schrodinger (DNLS) equations are discussed by the Riemann-Hilbert approach. By developing an improved Riemann-Hilbert problem, explicit soliton solutions of the n-component DNLS equations are obtained. Different from soliton solutions in many literatures, our expression does not include unfriendly integrals. An invariant and its general formula of the single-soliton solution for any ncomponent DNLS equations are derived. For the 2-component DNLS equation, we obtain one type of interesting solution - double hump soliton. The results play important roles in the vector solitons in many nonlinear physical systems, such as Bose-Einstein condensates, nonlinear optical fibers, etc.