Existence of traveling waves for diffusive-dispersive conservation laws

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 $y o -infty$. This provides the desired saddle-attractor connection.