Asymptotics of the first correction in the perturbation of the $N$-soliton solution to the KdV equation

We consider a triple Fourier-type integral that represents a solution to the KdV equation linearized on an $N$-soliton potential. Assuming that the parameters of the potential depend on the slow time $t$, we construct an asymptotics of this integral as $\varepsilon\to0$ uniform with respect to $x$, $t$ up to large time $0<t\le O(\varepsilon^{-1})$.