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KINEMATICAL CONSERVATION LAWS (KCL): EQUATIONS OF EVOLUTION OF CURVES AND SURFACES

    https://doi.org/10.1142/9789814338820_0002Cited by:0 (Source: Crossref)
    Abstract:

    The d-dimensional (d-D) kinematical conservation laws (KCL) are the equations of evolution of a moving surface Ωt in ℝd. The KCL are derived in a specially defined ray coordinates (ξ1, ξ2, …, ξd-1, t), where ξ1, ξ2,…, ξd-1 are surface coordinates on Ωt and t > 0 is time. We discuss various properties of 2-D and 3-D KCL systems. We first review the important properties of 2-D KCL and some of its applications. The KCL are the most general equations in conservation form, governing the evolution of Ωt with special type of singularities, which we call kinks. The kinks are points on Ωt when Ωt is a curve in ℝ2 and curves on Ωt when it is a surface in ℝ3. Across a kink the normal n to Ωt and amplitude w on Ωt are discontinuous. From 3-D KCL we derive a system of six differential equations and show that the KCL system is equivalent to the ray equations for Ωt. The six independent equations and an energy transport equation for small amplitude waves in a polytropic gas involving an amplitude w (related to the normal velocity m of Ωt) forms a completely determined system of seven equations. We have determined eigenvalues of the system by a novel method and find that the system has two distinct nonzero eigenvalues and five zero eigenvalues and the dimension of the eigenspace associated with the multiple eigenvalue zero is only four. For an appropriately defined m, the two nonzero eigenvalues are real when m > 1 and pure imaginary when m < 1. Finally, we have presented an an application of the theory to get evolution of a nonlinear wavefront by solving the conservation laws numerically.