The coupled Hirota equations with a 3 × 3 $3\times 3$ Lax pair: Painlevé-type asymptotics in transition zone

Abstract

We consider the Painlevé asymptotics for a solution of the integrable coupled Hirota equations with a $3\times 3$ Lax pair whose initial data decay rapidly at infinity. Using the Riemann–Hilbert (RH) techniques and Deift–Zhou nonlinear steepest descent arguments, in a transition zone defined by $|x/t-1/\left(12\alpha \right)|t^{2/3}\le C$, where $C>0$ is a constant, it turns out that the leading-order term to the solution can be expressed in terms of the solution of a coupled Painlevé II equations, which are associated with a $3\times 3$ matrix RH problem and appear in a variety of random matrix models.