Viscous Regularization of Delta Shock Wave Solution for a Simplified Chromatography System

Abstract

The delta shock wave for a simplified chromatography system is obtained in the Riemann solution when −1 < V − < 0 < V +. In fact, the result in this paper is the reasonable generalization of the result in Sun (2011) with V − = 0 < V + which is exactly the critical situation for −1 < V − < 0 < V +. The self-similar viscosity vanishing approach is also used to check the delta shock wave solution.

Soliton‐type asymptotics for non‐integrable equations: a survey

“…Originally, such idea had been suggested by V. Danilov and V. Shelkovich for shock wave type solutions (1997, [15]), and after that, it has been developed and adapted for many other problems (V. Danilov, G. Omel'yanov, V. Shelkovich, D. Mitrovic and others, [14,[16][17][18][19][20][21][22][23][24] and references therein). We called this approach the 'weak asymptotics method' .…”

“…At the same time (18), and thus the equality (21), is the integral form of (53) mod O D 0 ." 2 /, calculated for (12), where Y 1 has the form (23) and u 1 satisfies the Equation (24).…”

The Asymptotic Behaviors of Solutions to the Perturbed Riemann Problem near the Singular Curve for the Chromatography System*

“…Sun [37] discovered that a delta shock wave solution appears in the situation v l = 0 < v r for the Riemann problem (1.1) and (1.2) by employing the self-similiar viscosity vanishing approach. Furthermore, Li and Shen [20] has generalized the result in [37] to the general situation −1 < v l ≤ 0 < v r recently. If the system (1.1) comes from (1.5) to describe the interaction of a competitive-cooperative relation between two species, then it is natural to consider the system (1.1)…”

Wave Collision for the gKdV-4 equation. Asymptotic Approach

“…Originally, such idea had been suggested by V. Danilov and V. Shelkovich for shock wave type solutions (1997, [14]), and after that it has been developed and adapted for many other problems (V. Danilov, G. Omel'yanov, V. Shelkovich, D. Mitrovic and others, [15] - [27] and references therein). We called this approach the "weak asymptotics method".…”