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