Wave Systems with an Infinite Number of Localized Traveling Waves

In many wave systems, propagation of steadily traveling solitons or kinks is prohibited because of resonances with linear excitations. We show that wave systems with resonances may admit an infinite number of traveling solitons or kinks if the closest to the real axis singularities of a limiting asymptotic solution in the complex upper half plane are of the form z±=±α+iβ, α≠0. This quite general statement is illustrated by examples of the fifth-order Korteweg-de Vries equation, the discrete cubic-quintic Klein-Gordon equation, and the nonlocal double sine-Gordon equations.