Dispersive shock waves in the Kadomtsev-Petviashvili and Two Dimensional Benjamin-Ono equations
arXiv:1507.08207 · doi:10.1016/j.physd.2016.01.013
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.
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/aXUNYKFlke0
References in corpus (4)
- On Dispersive and Classical Shock Waves in Bose-Einstein Condensates and Gas Dynamics
- Observation of a gradient catastrophe generating solitons
- Evolution of solitary waves and undular bores in shallow-water flows over a gradual slope with bottom friction
- Whitham method for Benjamin-Ono-Burgers equation and dispersive shocks in internal waves in deep fluid
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- Whitham modulation theory for the two-dimensional Benjamin-Ono equation
- Whitham modulation theory for (2+1)-dimensional equations of Kadomtsev-Petviashvili type
- Nonlinear ring waves in a two-layer fluid
- On critical behaviour in generalized Kadomtsev--Petviashvili equations
- Oblique spatial dispersive shock waves in nonlinear Schrödinger flows
- Evolution of truncated and bent gravity wave solitons: the Mach expansion problem
- Nonlinear Modulation of Periodic Waves in the Cylindrical Gardner Equation
- The KP equation of plane elastodynamics