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Two kinds of exact Riemann solutions for a hyperbolic system arising from the steady 2D Helmholtz equation under a paraxial assumption are constructively achieved by using either delta shock wave or contact-vacuum-contact composite wave, which depends on the ordering relation between the left and right initial velocities. Moreover, the interaction between delta shock wave and contact-vacuum-contact composite wave is carefully explored by considering the initial data in three pieces separated by two jump discontinuities. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Riemann problem for a special three-component chromatography model with the mixed competitive-cooperative generalized Langmuir isotherm reactor, which belongs to the well-known Temple-type hyperbolic systems of conservation laws. All the possible structures of Riemann solutions are constructed in fully explicit forms, in which a singular solution including the so-called delta shock wave is also discovered where the Dirac delta measure is involved in the three state variables simultaneously. Such singular delta shock wave solution is obtained explicitly by using the generalized Rankine–Hugoniot conditions and the over-compressive delta entropy condition. Moreover, it is also proved rigorously that this singular delta shock wave solution satisfies the three-component chromatography model in the sense of distributions. Additionally, some typical numerical simulations are carried out to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

The Riemann problem for a hyperbolic system of conservation laws belonging to Temple class is considered within frame of α -solutions based on the concept of solutions defined in the setting of a distributional product. We obtain all the possible discontinuous solutions within a convenient space of distributions that contains Dirac delta measures and discontinuous functions. In addition, an interesting discontinuous solution is found for certain Riemann initial conditions, in which the Dirac delta measure is involved in both of the two state variables simultaneously. As an application, a detailed example for the mentioned system but involving a linear damping term is provided, in which the trajectories of discontinuous solutions including contact discontinuity, shock wave and delta shock wave change in exponential forms. [ABSTRACT FROM AUTHOR]

Abstract The exact solutions of the Riemann problem for a symmetric system of Keyfitz–Kranzer type are obtained in fully explicit forms. Furthermore, the global solutions of the double Riemann problems are also constructed explicitly when the initial data are taken to be three piecewise constant states. During the process of constructing the global solutions, all occurring wave interactions have been dealt with in detail by using the method of characteristics. In addition, it is shown that the Riemann solutions are stable with respect to the specific small perturbations of the Riemann initial data.

Abstract We investigate the Riemann problem for one-dimensional Temple class, obtain the Riemann solutions containing delta-shock wave, and discover that each variable of this system contains the Dirac delta function in different region. By the interaction of the delta-shock wave with the elementary waves we present the generalized Riemann problem for this system and construct the global solutions. By studying the limits of the solutions as perturbed parameter ε approaches zero we observe that the Riemann solutions are stable for such perturbations of the initial data.

The solutions to the Riemann problem for a nonsymmetric system of Keyfitz‐Kranzer type are constructed explicitly when the initial data are located in the quarter phase plane. In particular, some singular hyperbolic waves are discovered when one of the Riemann initial data is located on the boundary of the quarter phase plane, such as the delta shock wave and some composite waves in which the contact discontinuity coincides with the shock wave or the wave back of rarefaction wave. The double Riemann problem for this system with three piecewise constant states is also considered when the delta shock wave is involved. Furthermore, the global solutions to the double Riemann problem are constructed through studying the interaction between the delta shock wave and the other elementary waves by using the method of characteristics. Some interesting nonlinear phenomena are discovered during the process of constructing solutions; for example, a delta shock wave is decomposed into a delta contact discontinuity and a shock wave. [ABSTRACT FROM AUTHOR]

The Riemann solutions for a symmetric Keyfitz–Kranzer system are constructed explicitly, in which some singular hyperbolic waves are discovered such as the delta shock wave and the composite wave J R . The global solutions to the double Riemann problem are achieved when the delta shock wave is involved. It is shown that a delta shock wave is separated into a delta contact discontinuity and a shock wave during the process of constructing solutions. [ABSTRACT FROM AUTHOR]

The solutions of the perturbed Riemann problem for the chromatography system of Langmuir isotherm with one inert component are constructed in completely explicit forms when the initial data are taken as three piecewise constant states. The wave interaction problem is investigated in detail by using the method of characteristics. In addition, the generalized Riemann problem with the delta-type initial data is considered and the delta contact discontinuity is discovered. Moreover, the strength of delta contact discontinuity decreases linearly at a constant rate and then the delta contact discontinuity degenerates to be the contact discontinuity when across the critical point. [ABSTRACT FROM AUTHOR]

We study the one-dimensional Riemann problem for a hyperbolic system of three conservation laws of Temple class. The system is a simplification of a recently proposed system of five conservations laws by Bouchut and Boyaval that models viscoelastic fluids. An important issue is that the considered 3×3 system is such that every characteristic field is linearly degenerate. We study theRiemann problemfor this system and under suitable generalized Rankine-Hugoniot relation and entropy condition, both existence and uniqueness of particular delta shock type solutions are established. [ABSTRACT FROM AUTHOR]

The Riemann problem for the changed form of the chromatography system is considered here. It can be shown that the delta shock wave appears in the Riemann solution for exactly specified initial states. The generalized Rankine-Hugoniot relation of the delta shock wave is derived in detail. The existence and uniqueness of solutions involving the delta shock wave for the Riemann problem is proven by employing the self-similar viscosity vanishing approach. [ABSTRACT FROM AUTHOR]

Abstract: We prove that the Riemann solutions are stable for the chromatography system under the local small perturbations of the Riemann initial data. The proof is based on the detailed analysis of the wave interactions by applying the method of characteristic analysis. It is noteworthy that both the propagation directions of the shock wave S and rarefaction wave R are unchanged when they interact with the contact discontinuity J. Moreover, the global structures and large time asymptotic behaviors of the perturbed Riemann solutions are constructed and analyzed case by case. [Copyright &y& Elsevier]

The global solutions of an extended Riemann problem for a thin film model with a perfectly soluble anti-surfactant solution belonging to the Temple class with delta-type initial data are constructed in fully explicit forms for all possible situations. The method used is to perturb the delta-type initial data into initial data consisting of three piecewise constant states, such that the wave interaction problems have been extensively studied, including the interaction between a delta shock wave with a rarefaction (or shock) wave. The global solutions of the extended Riemann problem are obtained by letting the perturbation parameter tend to zero. We find that the initial Dirac delta function propagates along either the line of delta shock wave or the line of contact discontinuity. In addition, we find that some interesting nonlinear phenomena occur during the process of constructing the global solutions, such as a delta shock wave that separates into a shock wave and a delta contact discontinuity. • The global solutions of the extended Riemann problem for a thin film model with a perfectly soluble anti-surfactant solution with delta-type initial data are constructed in fully explicit forms. • It is found that the initial Dirac delta function propagates along either the line of delta shock wave or the line of contact discontinuity. • Some interesting nonlinear phenomena are observed, such as a delta shock wave is separated into a shock wave and a delta contact discontinuity. [ABSTRACT FROM AUTHOR]