Hamiltonian systems with an infinite number of localized travelling waves

In many Hamiltonian systems, propagation of steadily travelling solitons or kinks is prohibited because of resonances with linear excitations. We show that Hamiltonian systems with resonances may admit an infinite number of travelling solitons or kinks if the closest to the real axis singularities in the complex upper half-plane of limiting asymptotic solution are of the form $z_\pm=\pm\alpha+i\beta$, $\alpha\ne 0$. This quite a general statement is illustrated by examples of the fifth-order Korteweg--de Vries-type equation, the discrete cubic-quintic Klein--Gordon equation, and the nonlocal double sine--Gordon equations.