Nonlinear theory of modulated standing waves; domain walls, kinks and breathers

Abstract

A nonlinear theory of modulated standing waves is developed in one dimension. The existence of various localized solutions is elucidated. These include: domain walls , which characterize the transition between regions of different wave number, kinks , which describe a shift in the phase of the oscillation, and lower cut-off breathers . All of these states correspond to a spontaneous breaking of translational invariance, while the domain wall in addition represents a broken parity. Except for the breathers, the modulational equations that describe these states take a form that differs from the sine-Gordon, nonlinear Schrödinger, Toda lattice and Korteweg de Vries equations. In addition to the hamiltonian limit, the case of damped parametrically driven motion is also discussed.