Integrable Abel equation and asymptotics of symmetry solutions of Korteweg-de Vries equation

Abstract

We provide a general solution to a first order ordinary differential equation with a rational right-hand side, which arises in constructing asymptotics for large time of simultaneous solutions of the Korteweg-de Vries equation and the stationary part of its higher non-autonomous symmetry. This symmetry is determined by a linear combination of the first higher autonomous symmetry of the Korteweg-de Vries equation and of its classical Galileo symmetry. This general solution depends on an arbitrary parameter. By the implicit function theorem, locally it is determined by the first integral explicitly written in terms of hypergeometric functions. A particular case of the general solution defines self-similar solutions of the Whitham equations, found earlier by G.V. Potemin in 1988. In the well-known works by A.V. Gurevich and L.P. Pitaevsky in early 1970s, it was established that these solutions of the Whitham equations describe the origination in the leading term of non-damping oscillating waves in a wide range of problems with a small dispersion. The result of this work supports once again an empirical law saying that under various passages to the limits, integrable equations can produce only integrable, in certain sense, equations. We propose a general conjecture: integrable ordinary differential equations similar to that considered in the present paper should also arise in describing the asymptotics at large times for other symmetry solutions to evolution equations admitting the application of the inverse scattering transform method.

Linearisable Abel equations and the Gurevich-Pitaevskii problem

“…This ODE was first derived by Vadim Kudashev in the late 1990s, but was never published during his lifetime. It has first appeared in [11], see also [21] where a peculiar hypergeometric integral was provided, thus confirming its integrability. We also refer to [8,7] for the universality property of system (1.2) and its rigorous asymptotic theory.…”

“…In [4], most of the known integrable Abel equations were categorised into 11 equivalence classes, with canonical representatives and their first integrals being provided therein. ) is known to possess a similar first integral, see [21]. Note that for both of the second hypergeometric functions in the numerator and the denominator, the first three parameters are greater by one than their counterparts on the left, and recall that it is the feature of the derivative of the hypergeometric function.…”

“…which takes a second-kind Abel form when written in terms of z(R). Equation (1.1) arises in the following context (see [11,21] and references therein): consider the KdV equation constrained by the stationary part of its higher-order non-autonomous symmetry,…”

Meromorphy of solutions for a wide class of ordinary differential equations of Painlevé type

“…], [24]) in connection with the problems of quantum gravitation theory ( [11], [25], [26]) and the description of blowup regimes in the random matrix theory ( [27], [28]). Many publications in the last quarter century were devoted to studying various properties of simultaneous solutions of ODE and KdV (first of all, the Gurevich-Pitaevskii special solution); see [29]- [39] and references therein in addition to those given above. Other representatives of the hierarchy of simultaneous solutions of ( 16) and ( 4) are also related to the description of formation of dispersive shock waves in degenerate cases [13], [29].…”

“…(The right-hand sides of this symmetry are obtained by formally applying the right-hand sides of the recursion relations (39) to the right-hand sides of the equations of the classical scaling symmetry (46). The basic fact of compatibility of the system (51) with the system of evolution equations (6) means that the following identities hold for the functions ( 52) and ( 53):…”