Symmetry-breaking instabilities of spatial parametric solitons

For three decades optical spatial solitons confined in the transverse plane were commonly believed to be a prerogative of media with cubiclike nonlinearities @1#. A remarkable exception is the work carried out in Ref. @2#, where the possibility to achieve diffraction-free propagation via threephoton interactions in quadratic media ~henceforth, parametric solitons! was first pointed out. The field of quadratic solitons has acquired importance only recently @3#, also stimulated by experiments in second-harmonic generation ~SHG! in bulk media ~211 dimensions! and planar waveguides ~111 dimensions! @4#. Since the parametric solitons are strictly speaking solitary waves ~the model equations are not integrable!, a crucial issue is their stability. Two main types of instabilities can be distinguished: ~i! longitudinal instability against perturbations that share the soliton symmetry @5#; ~ii! symmetrybreaking instabilities ~reminiscent of modulational instabilities of plane-waves @6#!, that take place whenever the solitons are embedded in a higher dimensional ‘‘space’’ with respect to the subspace in which they are localized @7,8#. For the former type of problem, stability criteria have been recently developed @5#, through asymptotic techniques @9#: both ~111!and ~211!-dimensional parametric solitons are stable in the largest portion of their existence domain in the parameter space. Moreover, the global stability ~no collapse! of ~211!and ~311!-dimensional parametric solitons and bullets is supported by the Liapunov-type stability analysis @10#. Conversely, the symmetry breaking of parametric solitons is still an open issue, even though the problem has been widely studied for cubic media @7#. Here we investigate the stability of the whole one-parameter families of ground-state planar SHG solitons. We anticipate that the development of the instability leads either to the formation of lattices of higher dimensional solitons, or to the complete disintegration ~radiative decay! of the soliton. The transverse instability of soliton stripes belongs to the former case, whereas the dynamics of temporal instabilities of both ~111!and ~211!dimensional solitons depends on the dispersive regime. Our results are of great importance for recent experiments in transverse pattern formation occurring via SHG @11,12#. In particular, the filamentation of beams with strongly elliptical cross sections ~i.e., pseudostripe! was already observed @11#, using nonsoliton input conditions ~i.e., SHG from the funda-