Riemann problem for kinematical conservation laws and geometrical features of nonlinear wavefronts.

Baskar, S and Prasad, Phoolan (2004) Riemann problem for kinematical conservation laws and geometrical features of nonlinear wavefronts. In: IMA Journal of Applied Mathematics, 69 (4). pp. 391-419.

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Abstract

A pair of kinematical conservation laws (KCL) in a ray coordinate system (ξ, t) are the basic equations governing the evolution of a moving curve in two space dimensions. We first study elementary wave solutions and then the Riemann problem for KCL when the metric g, associated with the coordinate ξ designating different rays, is an arbitrary function of the velocity of propagation m of the moving curve. We assume that m > 1 (m is appropriately normalized), for which the system of KCL becomes hyperbolic. We interpret the images of the elementary wave solutions in the (ξ, t)-plane to the (x, y)-plane as elementary shapes of the moving curve (or a nonlinear wavefront when interpreted in a physical system) and then describe their geometrical properties. Solutions of the Riemann problem with different initial data give the shapes of the nonlinear wavefront with different combinations of elementary shapes. Finally, we study all possible interactions of elementary shapes.

Item Type:	Journal Article
Publication:	IMA Journal of Applied Mathematics
Publisher:	Oxford Univeristy Press
Additional Information:	The copyright belongs to Institute of Mathematics and its Applications.
Keywords:	curved wavefront;elementary waves;hyperbolic conservation laws; kink;nonlinear waves;ray theory;Reimann problem;shock; wave interaction
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	31 Jan 2008
Last Modified:	19 Sep 2010 04:42
URI:	http://eprints.iisc.ac.in/id/eprint/12916

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