Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 39, pp. 1-15.
Title: Existence of traveling waves for diffusive-dispersive conservation laws
Authors: Cezar I. Kondo (Federal Univ. of Sao Carlos, SP, Brazil)
Alex F. Rossini (Federal Univ. of Sao Carlos, SP, Brazil)
Abstract:
In this work we show the existence existence and uniqueness of
traveling waves for diffusive-dispersive conservation laws with
flux function in $C^{1}(\mathbb{R})$, by using phase plane analysis.
Also we estimate the domain of attraction of the equilibrium point
attractor corresponding to the right-hand state.
The equilibrium point corresponding to the left-hand state is a
saddle point. According to the phase portrait close to the saddle point,
there are exactly two semi-orbits of the system. We establish that
only one semi-orbit come in the domain of attraction
and converges to $(u_{-},0)$ as $y\to -\infty$.
This provides the desired saddle-attractor connection.
Submitted October 1, 2012. Published February 01, 2013.
Math Subject Classifications: 35L65, 76N10.
Key Words: Scalar conservation law; diffusive-dispersive; weak solution;
traveling wave; phase portrait.