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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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