Vector rogue wave patterns associated with generalized Hermite and Okamoto polynomials

Abstract

We establish rogue wave patterns associated with the fourth Painlevé equation $(\mathrm{P}_{\mathrm{IV}})$ in the multi-component nonlinear Schrödinger and Hirota equations. The generalized Hermite and generalized Okamoto polynomials arise in representations of rational solutions of $\mathrm{P}_{\mathrm{IV}}$, and we show that their roots determine two classes of rogue wave patterns when one of the internal parameters of rogue wave solutions is large. Specifically, the generalized Hermite polynomials arise from rogue wave solutions represented by Schur-polynomial determinants with consecutive indices, whereas the generalized Okamoto polynomials arise from analogous determinants with index jumps of three. Numerical examples for both equations agree with the predictions.