Asymptotic approximation nonlinear systems

Hence, both the IET and Markovian theory provide the lowest-order approximation for the fluorescence quantum yield with respect to acceptor concentration. This approximation is the only limitation of the validity of IET. Because of this limitation, IET is unable to describe properly the long-time asymptote of the system response to instantaneous excitation (Fig. 3.56) and the nonlinearity of the Stern-Volmer law at high concentrations (Fig. 3.61). On... 

It is necessary to note the essential physical difference between the system (9) and its asymptotic approximation at <5 0 that is the equation (14). The system (9) at finite values S describes the physical waves and is suitable to comparison with experiments. The asymptotic equation (14) simulates the mathematical waves of unbounded length and infinitesimal amplitude and is not included parameters connecting with experimental conditions. This circumstance defines the preference of (9) before (14). It is important to note that mathematical model for nonlinear waves is reduced to single equation in the limiting case (5 0 only. That model includes a system of two equations for finite S values. 

The mathematical structure of the models is their unifying background systems of nonlinear coupled differential equations with eventually nonlocal terms. Approximate analytic solutions have been calculated for linearized or reduced models, and their asymptotic behaviors have been determined, while various numerical simulations have been performed for the complete models. The structure of the fixed points and their values and stability have been analyzed, and some preliminary correspondence between fixed points and morphological classes of galaxies is evident—for example, the parallelism between low and high gas content with elliptical and spiral galaxies, respectively. 

Using scaling analysis and perturbation methods, we have been able to derive approximate expressions for the momentum and energy flux in dilute gases and liquids. These methods physically involve formal expansions about local equilibrium states, and the particular asymptotic restrictions have been formally obtained. The flux expressions now involve the dependent transport variables of mass or number density, velocity, and temperature, and they can be utilized to obtain a closed set of transport equations, which can be solved simultaneously for any particular physical system. The problem at this point becomes a purely mathematical problem of solving a set of coupled nonlinear partial differential equations subject to the particular boundary and initial conditions of the problem at hand. (Still not a simple matter see interlude 6.2.)... 

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