Existence of Traveling Waves of Conservation Laws with Singular Diffusion and Nonlinear Dispersion

We establish the existence of traveling waves for diffusive-dispersive conservation laws with locally Lipschitz flux function, singular diffusion and nonlinear dispersion. Because of the singular diffusion, the linearized traveling wave system at the equilibrium corresponding to the right-hand state of the shock has purely imaginary eigenvalues. We use a Lyapunov-type function and LaSalle’s invariance principle to show that this equilibrium is attracting. The level sets of the Lyapunov-type function enables us to estimate its domain of attraction. The equilibrium corresponding to the left-hand state of the shock is a saddle. We show that exactly one of the two trajectories leaving the saddle enters the domain of attraction of the attractor, thus giving a traveling wave.