Phase shift in the Whitham zone for the Gurevich–Pitaevskii special solution of the Korteweg–de Vries equation

Abstract

We get the leading term of the Gurevich-Pitaevskii special solution of the KdV equation in the oscillation zone without using averaging methods.

“Quantizations” of higher Hamiltonian analogues of the Painlevé I and Painlevé II equations with two degrees of freedom

“…This special function plays a similar role not only in the case of simple waves, but also for generic solutions of two-component hydrodynamic systems with small dispersion (see [17], [18], and [24]- [26]). A series of particular two-component systems, largely nonintegrable, whose solutions confirm the last assertion was described in [18]; see also [20], [25], and [26]. According to a conjecture stated by Dubrovin in [25] and [26], the GP solution for problems with small dispersion may play an even more universal role.…”

“…This means that, in addition to representations in the form of "quantizations" (20), pairs (18) and (19) admit symbolic representations in the form of the equations…”

“…In the above description of "quantizations" of two higher analogues of Painlevé equations, the key role is played by change (21), which was previously used in close situations for somewhat different purposes by Novikov in [7] (see also (2.3.36) in [39]). It is not for certain that a similar change alone is sufficient to as easily clarify the question of whether "quantizations" of the form (20) are valid for all higher Hamiltonian isomonodromic analogues of Painlevé equations with two degrees of freedom to which 2× 2 matrix IDM equations correspond (in particular, for all analogues considered in [37]). …”

“…(The leading term of the asymptotics of the special GP solution as t → ∞ was described in the well-known paper [16]. Results of [16] were improved in [14] and [17]- [22]; the smoothness of the GP solution was proved in [23], and its behavior was numerically modeled in [20].) This special function plays a similar role not only in the case of simple waves, but also for generic solutions of two-component hydrodynamic systems with small dispersion (see [17], [18], and [24]- [26]).…”

“…According to a conjecture stated by Dubrovin in [25] and [26], the GP solution for problems with small dispersion may play an even more universal role. This nonlinear special function was also studied in [27] and [28] in relation to problems of quantum gravity theory (see [14] and [20]). …”