INTRODUCTION
We apply the well-known self-similar Ansatz of Zeldowich and Barenblatt to two different problems. It is known from a long time that via the self-similar Ansatz the global properties of the solutions of any kind of non-linear partial differential equations(PDE) could be investigated. Solutions with compact supports or discontinuities (shock waves) can be easily found and analyzed.
First we consider a non-linear heat conduction problem in solids where the heat conduction coefficient and the thermal relaxation depend on the temperature in a non-trivial power law. This can be considered as a generalization of the Cattaneo-Vernotte heat conduction law.
As the second – an independent but analogous system - we consider the two coupled time-dependent Maxwell equations with power-law dependent dielectric permittivity and magnetic permeability.
RESULTS
In the first case from such power law dependent material equations we cannot form a second order parabolic or telegraph-type equation. We consider the original non-linear hyperbolic system itself with the self-similar Ansatz for the temperature distribution and for the heat flux. As results continuous and shock-wave solutions are presented for varoious exponental coefficients. For physical establishment numerous materials with various temperature dependent heat conduction coefficients are mentioned1.
For the second case the method and the results are pretty much analogous. With the power-law dependent electric and magnetic material equations we investigate directly the Maxwell equations and not - the strongly related but not equivalent – second order wave equations.
In this sense - such in fluid mechanics - shock-waves can be formed an investigated2. Our solutions might be interesting for extreme strong lasers like the planned European ELIs.
CONCLUSIONS
We give two new and plastic examples for the statement that in first order PDEs shock-waves can be easily created and detected.
REFERENCES
1. Barna I.F. and R. Kersner, Heat conduction: hyperbolic self-similar shock-waves in solids http://arxiv.org/abs/1204.4386.
2. Barna I.F., Self-similar shock wave solutions of the non-linear Maxwell equations
Laser Phys. 24, 086002, (2014).
