Linearizable Abel equations and the Gurevich–Pitaevskii problem

摘要

Abstract 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.