Linearizable Abel equations and the Gurevich-Pitaevskii problem

Applying symmetry reduction to a class of SL(2,R)$\mathrm{SL}(2,\mathbb {R})$-invariant third-order ordinary differential equations (ODEs), we obtain Abel equations whose general solution can be parameterized by hypergeometric functions. Particular case of this construction provides a general parametric solution to the Kudashev equation, an ODE arising in the Gurevich-Pitaevskii problem, thus giving the first term of a large-time asymptotic expansion of its solution in the oscillatory (Whitham) zone.