Singular perturbation to shock wave solutions of Burgers KdV-equation.
The paper considers traveling-wave solutions to the mixed Korteweg-de Vries (KdV)-Burgers equation, in the regime which is close to the dissipative Burgers equation. In the latter case, well-known exact solutions are available, which describe traveling shock waves; from the viewpoint of the corresponding ODE, obtained from the underlying PDE by the traveling-wave substitution, these solutions are heteroclinic trajectories. The authors give a rigorous proof of the fact that shock-wave solutions exist as well in the general case of the mixed KdV-Burgers equation (when they are not available in an exact form), and these solutions smoothly carry over into the exact shock waves of the Burgers equation in the limit when the equation itself tends to the Burgers' form. Additionally, it is proved that, for fixed boundary values, the shock-wave solution (in the case of the general mixed equation) is a single possible solution.
scientific article; zbMATH DE number 2022060