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We consider 1D completely resonant nonlinear wave equations of the type vtt - vxx = -v3 + O(v4) with spatial periodic boundary conditions. We prove the existence of a new type of quasi-periodic small amplitude solutions with two frequencies, for more general nonlinearities. These solutions turn out to be, at the first order, the superposition of a traveling wave and a modulation of long period, depending only on time

Quasi-periodic solutions of the equation $v_{t t} - v_{x x} +v^3 = f(v)$