Energy of the soliton internal modes and broken symmetries in nonlinear optics

Applications of the concepts of the energy and, in particular, of the negative energy of the internal modes are described in the context of nonlinear optics. It is shown that the transformation of a soliton propagating in a bulk into solitons propagating in a coherently pumped cavity is followed by emergence of the negative-energy modes. These modes are directly associated with a locking of the soliton phase to the phase of the coherent pump, i.e., with a breaking of the phase symmetry. Energy of the Vakhitov–Kolokolov modes is also calculated. This energy is positive, providing that conditions of the Vakhitov–Kolokolov stability criterion are met. Mutual annihilation of the positive- and the negative-energy modes leads to oscillatory (Hopf) instability of the cavity solitons.