On critical behaviour in generalized Kadomtsev–Petviashvili equations

Abstract

An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev-Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painlevé I hierarchy. Several examples are discussed numerically to provide strong evidence for the validity of the conjecture. The numerical study of the long time behavior of these examples indicates persistence of dispersive shock waves in solutions to the (subcritical) KP equations, while in the supercritical KP equations a blow-up occurs after the formation of the dispersive shock waves.

Whitham modulation theory for (2 + 1)-dimensional equations of Kadomtsev–Petviashvili type

“…Subsequently Whitham theory has been considered for the (2+1) dimensional KP [1] and 2DBO [2] without using any one-dimensional reductions. We also mention that the underlying structure of solutions that will develop dispersive shocks in the small dispersion limit of the generalized KP equations has been studied in [11]. We refer the reader to these papers for additional background and references.In this work we present an approach that applies equally well to integrable and non-integrable PDEs, since at the heart of it lies Whitham's WKB type expansion and separation of fast and slow scales.…”

On the dispersionless Kadomtsev–Petviashvili equation with arbitrary nonlinearity and dimensionality: exact solutions, longtime asymptotics of the Cauchy problem, wave breaking and shocks

“…Therefore this dependence involves in a simple way the degree m + 1 of the nonlinearity and the dimensionality n + 1 of the problem through the coefficient c m,n , and it is consistent with the case of the Riemann equation (1.2), for which c m,1 = 1, and with the dKP equation (1.3), for which c 1,2 = 1/2 (confirming the apparently mysterious factor 2 in (1.7)); it is not consistent, instead, with the formal dependence x − u m t used in [20].…”

“…The other examples of dKP (m, n) equations are not integrable; therefore the possibility to investigate a generic wave breaking through equations like (1.7) and the precise form that these equations should take are, in our opinion, challenging open problems; in addition, blow up of the solutions is expected for sufficiently large m [8] to complicate the picture. In the recent paper [20], f.i., the formal dependence x − u m t, motivated by the y independent limit, was used to study the generic breaking features of dKP (m, 2) and its dispersive shock formation.…”

Oblique Spatial Dispersive Shock Waves in Nonlinear Schrödinger Flows

“…The limited theoretical investigations of DSWs in multiple dimensions [15,16,11,29,12,21,22,27,28,2,5] and the experimental realization of multi-dimensional DSWs in ultracold atoms as superfluid matter waves [6,34,19] and in nonlinear optical diffraction patterns [36,13] provides motivation for this study of two-dimensional DSWs. Furthermore, with the exception of recent studies on Kadomtsev-Petviashvili and related (2+1)D equations [29,28,2,5], the remaining previous theoretical works invoke asymptotic reductions to the Korteweg-de Vries or (1+1)D nonlinear Schrödinger (NLS) equations. These equations' complete integrability enables a detailed analytical description via the existence of Riemann invariants for the associated Whitham modulation equations.…”