Nonlinear Modulation of Periodic Waves in the Cylindrical Gardner Equation

Abstract

The propagation of the dispersive shock waves (DSWs) is investigated in the cylindrical Gardner (cG) equation, which is obtained by employing a similarity reduction to the two space one time (2+1) dimensional Gardner-Kadomtsev-Petviashvili (Gardner-KP) equation. We consider the step-like initial condition along a parabolic front. Then, the cG-Whitham modulation system, which is a description of DSW evolution in the cG equation, in terms of appropriate Riemann type variables is derived. Our study is supported by numerical simulations. The comparison is given between the direct numerical solution of the cG equation and the DSW solution obtained from the numerical solution of the Whitham system. According to this comparison, a good agreement is found between the solutions.

References in corpus (8)

• Resolution of a shock in hyperbolic systems modified by weak dispersion
• Undular bore theory for the Gardner equation
• Dispersive shock waves theory for non-integrable equations
• Nonlinear waves of polarization in two-component Bose-Einstein condensates
• Semiclassical solitons in strongly correlated systems of ultracold bosonic atoms in optical lattices
• Dispersive shock waves in the Kadomtsev-Petviashvili and Two Dimensional Benjamin-Ono equations
• Soliton Generation and Multiple Phases in Dispersive Shock and Rarefaction Wave Interaction
• Whitham modulation theory for (2+1)-dimensional equations of Kadomtsev-Petviashvili type