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Generic Hopf bifurcation from lines of equilibria without parameters: II. Systems of viscous hyperbolic balance laws. (English) Zbl 0980.35098
The authors consider solutions of systems of strictly hyperbolic balance laws of the form \[ u_t+ f(u)_x= \varepsilon\delta u_{xx}+ \varepsilon^{-1} g(u). \] Traveling wave solutions satisfy a system of ordinary differential equations from which \(\varepsilon\) can be scaled out. The linearization at an equilibrium can give rise to imaginary eigenvalues due to the balance between the flux \(f(u)\) and the source terms \(g(u)\). The authors explore the existence of oscillating traveling waves (viscous profiles) when this occurs. Such traveling waves arise from a bifurcation resembling Hopf bifurcation in that eigenvalues cross the imaginary axis transversally.

MSC:
35L65 Hyperbolic conservation laws
34C23 Bifurcation theory for ordinary differential equations
35L67 Shocks and singularities for hyperbolic equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
35L40 First-order hyperbolic systems
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