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.Comment: 25 Pages, 16 Figures. The movies showing dispersive shock wave
propagation in Kadomtsev-Petviashvili II and Two Dimensional Benjamin-Ono
equations are available at https://youtu.be/AExAQHRS_vE and
https://youtu.be/aXUNYKFlke
