Accumulation of embedded solitons in systems with quadratic nonlinearity

Previous numerical studies have revealed the existence of embedded solitons (ESs) in a class of multi-wave systems with quadratic nonlinearity, families of which seem to emerge from a critical point in the parameter space, where the zero solution has a fourfold zero eigenvalue. In this paper, the existence of such solutions is studied in a three-wave model. An appropriate rescaling casts the system in a normal form, which is universal for models supporting ESs through quadratic nonlinearities. The normal-form system contains a single irreducible parameter $\epsilon$, and is tantamount to the basic model of type-I second-harmonic generation. An analytical approximation of WKB type yields an asymptotic formula for the distribution of discrete values of $\epsilon$ at which the ESs exist. Comparison with numerical results shows that the asymptotic formula yields an exact value of the scaling index, -6/5, and a fairly good approximation for the numerical factor in front of the scaling term.