Competing wave-breaking mechanisms in second harmonic generation

Summary form only given. Since the early studies of dispersive nonlinear wave propagation, different mechanisms of breaking have been predicted and observed. Modulational (or Bejamin-Feir-Lighthill) instability (MI), in its basic manifestation entails breaking of a periodic wave (carrier) due to the exponential amplification of low frequency (long wavelength) modulations, observed in water waves and optics. When the instability builds up from noise, the breaking occurs at the most unstable frequency, determined by the underlying mechanism of nonlinear phase-matching. On the other hand, a completely different and universal mechanism involves, in the weakly dispersive limit, a gradient catastrophe, where a smooth envelope steepens until it develops an infinite gradient in finite time. Such breaking is conjectured to be generic for Hamiltonian models which possess a hyperbolic dispersionless (hydrodynamic) limit. Beyond the first point of infinite gradient, the regularizing action of dispersion leads to form unsteady dispersive shock waves (DSW), characterized by an expanding fan progressively filled with fast oscillations (the smaller the dispersion the shorter the oscillation wavelength). In settings described by the scalar nonlinear Schrödinger (NLS) equation, these two mechanims are mutually exclusive. Indeed the gradient catastrophe occurs in the defocusing regime characterized by a hyperbolic dispersionless limit, which has been also the focus of experimental work on DSW recently [1,2]. Conversely, MI takes place in the focusing regime where, however, the dispersionless limit turns out to be elliptic and ill-posed.The main aim of this contribution is to show that, when considering two modes interacting via the nonlinearity, the two mechanisms can coexist inducing a new scenario where generalized MI [3,4,5] can compete with a DSW, thus dramatically affecting the dynamics of the latter. SpeciIcally we discuss in details the most basic of the mixing optica- interactions, namely second harmonic generation (SHG). We predict for the Irst time experimentally accessible DSW in the regime of genuine parametric mixing involving a free phase mismatch parameter (previous analysis addressed only the highly mismatched case where SHG mimics the NLS dynamics, see [6]), and fully characterize its competition with MI. However, in order to show the ubiquity of such competition mechanism, we briefly illustrate also the case of the vector NLS model, which Inds application in contexts as different as optics [3], Bose-Einstein condensation [4], and ocean freak waves [5,7].