Eigenvalues of kinematical conservation laws (KCL) based 3-D weakly nonlinear ray theory (WNLRT)

• Published in: Applied mathematics and computation 2010-11, Vol.217 (5), p.2285-2288
• Main Authors: Arun, K.R., Prasad, Phoolan
• Format: Article
• Language: English
• Subjects: Conservation laws; Eigenvalues; Evolution; Exact sciences and technology; Global analysis, analysis on manifolds; Kinematical conservation laws; Mathematical analysis; Mathematical models; Mathematics; Nonlinear algebraic and transcendental equations; Nonlinear wave; Nonlinearity; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Partial differential equations; Polytropic gas; Ray theory; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Wave fronts; Wave propagation; Weakly hyperbolic system
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2010.06.041

System of kinematical conservation laws (KCL) govern evolution of a curve in a plane or a surface in space, even if the curve or the surface has singularities on it. In our recent publication [K.R. Arun, P. Prasad, 3-D kinematical conservation laws (KCL): evolution of a surface in R 3 -in particular propagation of a nonlinear wavefront, Wave Motion 46 (2009) 293–311] we have developed a mathematical theory to study the successive positions and geometry of a 3-D weakly nonlinear wavefront by adding an energy transport equation to KCL. The 7 × 7 system of equations of this KCL based 3-D weakly nonlinear ray theory (WNLRT) is quite complex and explicit expressions for its two nonzero eigenvalues could not be obtained before. In this short note, we use two different methods: (i) the equivalence of KCL and ray equations and (ii) the transformation of surface coordinates, to derive the same exact expressions for these eigenvalues. The explicit expressions for nonzero eigenvalues are important also for checking stability of any numerical scheme to solve 3-D WNLRT.