Geodesic
Existence of Traveling Waves of Conservation Laws with Singular Diffusion and Nonlinear Dispersion
Bulletin of the Malaysian Mathematical Society, Tome 35 (2012) no. 2 Cet article a éte moissonné depuis la source Bulletin of the Malaysian Mathematical Society website

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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.
Classification : 35L65, 74N20, 76N10, 76L05. 
@article{BMMS_2012_35_2_a13,
     author = {Mai Duc Thanh},
     title = {Existence of {Traveling} {Waves}  of {Conservation} {Laws} with {Singular} {Diffusion} and {Nonlinear} {Dispersion}},
     journal = {Bulletin of the Malaysian Mathematical Society},
     year = {2012},
     volume = {35},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/BMMS_2012_35_2_a13/}
}
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Mai Duc Thanh. Existence of Traveling Waves  of Conservation Laws with Singular Diffusion and Nonlinear Dispersion. Bulletin of the Malaysian Mathematical Society, Tome 35 (2012) no. 2. http://geodesic.mathdoc.fr/item/BMMS_2012_35_2_a13/