Numerical Study of Oscillatory Regimes in the Kadomtsev–Petviashvili Equation

Abstract

The aim of this paper is the accurate numerical study of the KP equation. In particular we are concerned with the small dispersion limit of this model, where no comprehensive analytical description exists so far. To this end we first study a similar highly oscillatory regime for asymptotically small solutions, which can be described via the Davey-Stewartson system. In a second step we investigate numerically the small dispersion limit of the KP model in the case of large amplitudes. Similarities and differences to the much better studied Korteweg-de Vries situation are discussed as well as the dependence of the limit on the additional transverse coordinate.

Whitham modulation theory for the two-dimensional Benjamin-Ono equation

“…Following some earlier works [11][12][13], recently there has been considerable attention devoted to the study of small dispersion problems for (2+1)-dimensional systems [14][15][16]. One of the goals of this work is to develop tools that can be used to describe the behavior of DSWs in multi-dimensional settings.…”

Whitham modulation theory for the Kadomtsev– Petviashvili equation

“…The behavior of solutions of the KPI and KPII equations with small dispersion was recently studied numerically by Klein et al [28]; see also [20] for a study of shock formation in the dispersionless KP. Even though there have been a few works about the derivation of a Whitham system for the KP equation, [9,25,31], to the best of our knowledge there are no studies in which such systems were written in Riemann-type variables, nor studies regarding the use of these systems to study DSWs.…”

“…The KP equation is also the prototypical (2+1)-dimensional integrable system. As such, it has been heavily studied analytically over the last forty years; see for example [1,[3][4][5][6][8][9][10][11]19,[23][24][25][28][29][30][31][32][33][34]43,44,46,49,54,56] and references therein.…”

Extended two dimensional equation for the description of nonlinear waves in gas–liquid mixture

“…It is obvious that equation (44) has to be regularized for k x = 0 in order to give numerical sense to 1 k x . Following [26], we add to k x in the denominator a small imaginary part of appropriate sign iλ 0 . For λ 0 we use the smallest floating point number that MATLAB can represent 2.2×10 −16 .…”