The authors start from the Riemann-Hilbert problem written in an uncoupled way as \(q_{t}+c_{j}q_{x}=0\), \(j=1,2\), and from the Lax pair \(\mu _{x}-ik\mu =q\); \(\mu _{t}+ic_{j}\mu =-c_{j}q\), \(j=1,2\), with the compatibility conditions \(\mu _{xt}=\mu _{tx}\). The authors gather the Riemann-Hilbert problem, the jump conditions and the normalization in an equivalent \(4\times 4\) matrix Riemann-Hilbert problem involving the operator \(M(t,x,k)=\left( \begin{matrix} I & \mu \\ 0 & I \end{matrix} \right) \) with \(\mu =\mathrm{diag}(\mu _{1},\mu _{2})\). Quoting from [A. S. Fokas, A unified approach to boundary value problems. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (2008; Zbl 1181.35002)] the dressing method, the authors look for two linear differential equations \(L\) and \(N\) such that \(LM\) and \(NM\) satisfy the same jump condition and are or order \(O(1/k)\) as \(k\rightarrow \infty \). The main purpose of the paper is to build these operators \(L\) and \(N\).