Stability of Three-Wave Parametric Solitons in Diffractive Quadratic Media.

Analysis of wave interaction in nonlinear media is a fundamental problem of nonlinear physics. Resonant interaction is especially interesting since it leads to an energy exchange between waves of different carrier frequencies. In a medium without the center of symmetry, quadratic nonlinearity results in parametric three-wave mixing when two waves create a third wave at a combined frequency v3 ­ v1 1 v2. This phenomenon occurs in a wide range of nonlinear wave systems, e.g., surface waves in fluids and solids, plasma waves, optics, etc. [1–4]. A particular case of the three-wave interaction is known in nonlinear optics as type II second-harmonic generation (SHG). In a degenerate case, when two identical waves of the fundamental frequency create a second-harmonic wave, such interaction is known as type I SHG. When dispersion or diffraction become important (e.g., for sufficiently short pulses or focused beams), parametric interaction of two waves can lead to beam self-trapping and solitary waves. Spatial optical solitons due to type I SHG in optical materials with quadratic [or x s2d] nonlinearity have been extensively investigated during the last two years for both (1 1 1) dimensional (planar waveguides) [5–7] and (2 1 1) dimensional (self-guided beams in a bulk medium) [8] geometries. However, the first experimental observation of optical beam self-trapping and stationary propagation of (2 1 1) dimensional spatial solitons in a bulk x s2d medium has been reported [9] for the case of type II SHG (see also Ref. [10]). In the theoretical analysis of three-wave mixing little progress has been achieved even for the case of (1 1 1) dimensional geometry [11]. The similar three-wave mixing equations appear in many different branches of physics (e.g., [2–4]) and also for describing Fermi resonances in multilayer superlattices [12], near-resonant interactions of elastic waves on corrugated surfaces [13], etc. This Letter aims to present the families of two-parameter solitary waves for both (1 1 1) and (2 1 1) dimensional cases and to analyze their stability. Moreover, taking the problem of three-wave mixing as an important example, we derive a novel type of analytical stability criterion for solitary waves with more than one internal parameter.