An exact description of nonlinear wave interaction processes ruled by 2 × 2 hyperbolic systems

Nonlinear wave interactions in multicomponents chromatography

“…For 2 × 2 homogeneous systems nonlinear wave interaction has been exhaustively investigated by means of exact solutions to initial value problems and a special class of models allowing for solitonlike simple wave interactions has been also considered in [7]. For multicomponent chromatography with three components in [3] suitable boundary value problems useful for describing phenomena of interest in the applications have been solved.…”

“…In view of describing in the (τ, η) plane the propagation and interaction of simple waves travelling along characteristic curves of different type and according to the analysis carried on in [7] let us consider the "initial data"…”

“…In a very recent paper [3] the approach, worked out in [4]- [7] for 2 × 2 strictly hyperbolic systems, has been extended to diagonalizable semi-Hamiltonian homogeneous hyperbolic systems in order to perform an accurate description of the associated hyperbolic wave interaction processes. In this paper the attention is focussed on the governing model of chromatography with an "ideal column" in the case of Langmuir sorption and involving four components.…”

Nonlinear wave interaction problems in the three-dimensional case

“…Within such a theoretical framework, the Riemann method was extended to the nonhomogeneous case in [4,5,6] and a large new classes of solutions to quasilinear systems of PDEs have been obtained in [7,8]. Furthermore a combined use of the hodograph method and of the differential constraints technique was considered in [9] and quite recently was used in order to study nonlinear wave interactions [10,11] as well as discontinuous initial value problems [12,13,14] for homogeneous and nonhomogeneous 2 × 2 systems.…”

“…Actually, via the generalized hodograph method a general solution of these systems can be obtained [19,20]. Recently, in [21], the approach worked out in [10,11] for 2 × 2 hyperbolic systems, has been enlarged to these class of diagonalizable semi-Hamiltonian homogeneous hyperbolic systems in order to perform an accurate description of the associated hyperbolic wave interaction processes. Within such a theoretical framework, here our main aim is to extend to the three dimensional case the procedure proposed in [21] for hydrodynamic models involving two independent variables.…”