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We consider KdV-type equations with $C^1$ nonhomogeneous nonlinearities andsmall dispersion $\varepsilon$. The first result consists in the conclusionthat, in the leading term with respect to $\varepsilon$, the solitary waves inthis model interact like KdV solitons. Next it turned out that there exists avery interesting scenario of instability in which the short-wave solitonremains stable whereas a small long-wave part, generated by perturbations oforiginal equation, turns to be unstable, growing and destroying the leadingterm. At the same time, such perturbation can eliminate the collision ofsolitons.

Collision of solitons for a non-homogenous version of the KdV equation