Dispersive shock waves in the Kadomtsev–Petviashvili and two dimensional Benjamin–Ono equations

Abstract

Dispersive shock waves (DSWs) in the Kadomtsev-Petviashvili (KP) equation and two dimensional Benjamin-Ono (2DBO) equation are considered using parabolic front initial data. Employing a front tracking type ansatz exactly reduces the study of DSWs in two space one time (2 + 1) dimensions to finding DSW solutions of (1 + 1) dimensional equations. With this ansatz, the KP and 2DBO equations can be exactly reduced to cylindrical Korteweg-de Vries (cKdV) and cylindrical Benjamin-Ono (cBO) equations, respectively. Whitham modulation equations which describe DSW evolution in the cKdV and cBO equations are derived in general and Riemann type variables are introduced. DSWs obtained from the numerical solutions of the corresponding Whitham systems and direct numerical simulations of the cKdV and cBO equations are compared with excellent agreement obtained. In turn, DSWs obtained from direct numerical simulations of the KP and 2DBO equations are compared with the cKdV and cBO equations, again with remarkable agreement. It is concluded that the (2 + 1) DSW behavior along parabolic fronts can be effectively described by the DSW solutions of the reduced (1 + 1) dimensional equations.

Whitham modulation theory for the Kadomtsev– Petviashvili equation

“…The cKdV reduction, obtained with n = 1, is the simplest type of such ICs, and does indeed generate (2+1)-dimensional DSWs [2].…”

“…In fact, for the Riemann problem for the KdV equation, the limits of the velocity V 2 as m → 0 and m → 1 yield the velocities of the leading and trailing edges of the DSW, respectively [22]. Similarly, for the cylindrical KdV equation, the Whitham system is inhomogeneous [2], but the velocities V 1 , V 2 and V 3 have the same form as those for the KdV equation, and the inhomogeneous terms also have finite limits when m → 0 and m → 1.…”

“…where V 1 , V 2 and V 3 are given by (2.32a) together with a PDE for the fourth variable q(x, y, t) 2) and the constraint (1.4c). (Note however that the system (3.1) is independent of q, whose value is only needed if one wants to recover the solution of the KP equation from that of the KP-Whitham system.)…”

“…Accordingly, we consider a step-like initial datum for u as Similarly to the case of the KdV equation and the cKdV equation [2], it is convenient to regularize the jump and choose the corresponding initial conditions for the Riemann variables r 1 , r 2 and r 3 to be…”

Circular dispersive shock waves in colloidal media

“…Models such as the Euler water wave equations, Kadomtsev-Petviashvili equation and the two dimensional Benjamin-Ono (BO) equation have been considered [7,8]. Simplified cylindrical KdV and BO equations were derived and circular DSW were studied via both numerical solutions and modulation theory.…”