Small-$x$ QCD evolution of $2n$ Wilson line correlator: the weak field limit

We write down explicit expressions for the $x$-evolution (equivalent to energy or rapidity evolution) of $2n$ ($n=1,2,\mathrm{.}\mathrm{.}\mathrm{.}$) Wilson lines using the JIMWLK equation and the Color Glass Condensate formalism. We investigate the equation in the weak gluon field limit (linear regime) by expanding the Wilson lines in powers of the gluon field and show that it reduces to the BJKP equation describing the evolution of a state of $2n$ Reggeized gluons with energy. We also make available for download a {\it Mathematica} program which provides this expression for any value of $n$.