Dispersive Dynamics in the Characteristic Moving Frame

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Abstract

A mechanism for dispersion to automatically arise from the dispersionless Whitham Modulation equations (WMEs) is presented, relying on the use of a moving frame. The speed of this is chosen to be one of the characteristics which emerge from the linearisation of the Whitham system, and assuming these are real (and thus the WMEs are hyperbolic) morphs the WMEs into the Korteweg - de Vries (KdV) equation in the boosted co-ordinate. Strikingly, the coefficients of the KdV equation are universal, in the sense that they are determined by abstract properties of the original Lagrangian density. Two illustrative examples of the theory are given. The first being a revisitation of the derivation of the KdV equation from shallow water flows, to highlight how the theory of this paper fits into the existing literature. The second is a complex Klein-Gordon system, providing a case where the KdV equation may only arise with a moving frame.

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Is Open Access

On Dispersive and Classical Shock Waves in Bose-Einstein Condensates and Gas Dynamics

I Coddington, M. A. Hoefer, M. Ablowitz … (2006)

A Bose-Einstein condensate (BEC) is a quantum fluid that gives rise to interesting shock wave nonlinear dynamics. Experiments depict a BEC that exhibits behavior similar to that of a shock wave in a compressible gas, eg. traveling fronts with steep gradients. However, the governing Gross-Pitaevskii (GP) equation that describes the mean field of a BEC admits no dissipation hence classical dissipative shock solutions do not explain the phenomena. Instead, wave dynamics with small dispersion is considered and it is shown that this provides a mechanism for the generation of a dispersive shock wave (DSW). Computations with the GP equation are compared to experiment with excellent agreement. A comparison between a canonical 1D dissipative and dispersive shock problem shows significant differences in shock structure and shock front speed. Numerical results associated with the three dimensional experiment show that three and two dimensional approximations are in excellent agreement and one dimensional approximations are in good qualitative agreement. Using one dimensional DSW theory it is argued that the experimentally observed blast waves may be viewed as dispersive shock waves.