Modulational instability of solitary waves in nondegenerate three-wave mixing: the role of phase symmetries

We show how the analytical approach of Zakharov and Rubenchik [Sov. Phys. JETP 38, 494 (1974)] to modulational instability (MI) of solitary waves in the nonlinear Schrödinger equation can be generalized for models with two phase symmetries. MI of three-wave parametric spatial solitons due to group velocity dispersion (GVD) is investigated as a typical example of such models. We reveal a new branch of neck instability, which dominates the usual snake type MI found for normal GVD. The resultant nonlinear evolution is thereby qualitatively different from cases with only a single phase symmetry.