Geodesic
Asymptotics of the first correction in the perturbation of the $N$-soliton solution to the KdV equation
Matematičeskie zametki, Tome 58 (1995) no. 2, pp. 204-217.

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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 $0$. 
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L. A. Kalyakin. Asymptotics of the first correction in the perturbation of the $N$-soliton solution to the KdV equation. Matematičeskie zametki, Tome 58 (1995) no. 2, pp. 204-217. http://geodesic.mathdoc.fr/item/MZM_1995_58_2_a3/

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