Shadow wave solution for the generalized Langmuir isotherm in chromatography

The shadow wave solution of the Riemann problem for the chromatography system under the mixed competitive-cooperative generalized Langmuir isotherm is constructed. It is shown that this shadow wave solution is weakly unique in the sense that all the entropy shadow wave solutions have the same distributional limit which is exactly the corresponding delta shock wave solution of the Riemann problem. Furthermore, the generalized Riemann problem with the delta type initial data is also considered and the existence and weak uniqueness of a solution are obtained in the framework of shadow wave solution.