2014/10/19 by Michael Shearer, Shearer, Michael, Kimberly Spayd +3
Physics and Astronomy · #Nonlinear Waves and Solitons #Cosmology and Gravitation Theories #Black Holes and Theoretical Physics
paper · pdf · doi:10.48550/arxiv.1410.5121
Scalar conservation laws with non-convex fluxes have shock wave solutions\nthat violate the Lax entropy condition. In this paper, such solutions are\nselected by showing that some of them have corresponding traveling waves for\nthe equation supplemented with dissipative and dispersive higher-order terms.\nFor a cubic flux, traveling waves can be calculated explicitly for linear\ndissipative and dispersive terms. Information about their existence can be used\nto solve the Riemann problem, in which we find solutions for some data that are\ndifferent from the classical Lax-Oleinik construction. We consider dispersive\nterms of a BBM type and show that the calculation of traveling waves is\nsomewhat more intricate than for a KdV-type dispersion. The explicit\ncalculation is based upon the calculation of parabolic invariant manifolds for\nthe associated ODE describing traveling waves. The results extend to the\np-system of one-dimensional elasticity with a cubic stress-strain law.\n