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.