Time-dependent nonlinearity

Recently there has been a great deal of interest in nonlinear phenomena, both from a fundamental point of view, and for the development of new nonlinear optical and optoelectronic devices. Even in the optical case, the nonlinearity is usually engendered by a solid or molecular medium whose properties are typically determined by nonlinear response of an interacting many-electron system. To be able to predict these response properties we need an efficient description of exchange and correlation phenomena in many-electron systems which are not necessarily near to equilibrium. The objective of this chapter is to develop the basic formalism of time-dependent nonlinear response within density functional theory, i.e., the calculation of the higher-order terms of the functional Taylor expansion Eq. (143). In the following this will be done explicitly for the second- and third-order terms... 

Finite-Difference Methods. The numerical analysis literature abounds with finite difference methods for the numerical solution of partial differential equations. While these methods have been successfully applied in the solution of two-dimensional problems in fluid mechanics and diffusion (24, 25), there is little reported experience in the solution of three-dimensional, time-dependent, nonlinear problems. Application of these techniques, then, must proceed by extending methods successfully applied in two-dimensional formulations to the more complex problem of solving (7). The various types of finite-difference methods applicable in the solution of partial differential equations and their advantages and disadvantages are discussed by von Rosenberg (26), Forsythe and Wasow (27), and Ames (2S). 

The main advantage in using a finite-difference method to solve (7), as compared with other approaches, is that there has been extensive experience in applying these techniques to various partial differential equations. Even though reported experience with three-dimensional, time-dependent, nonlinear problems is sparse, experience with simpler systems gives a sound basis to develop feasible approaches. The disadvantages of finite difference methods are well-known ... 

Figure 3-31 Damping function hiy) obtained by vertically shifting the time-dependent nonlinear moduli in Fig. 3-30a into superposition at long times. The data are from Fukuda et al. (1975). The solid and dashed lines are the prediction of the Doi-Edwards model, respectively, with and without the independent alignment approximation. (From Doi and Edwards 1978a, reproduced by permission of The Royal Society of Chemistry.)...

Macdonald, I. F. Time dependent nonlinear behavior of viscoelastic fluids. Ph. D. Thesis, Univ. of Wisconsin (1968). 

In physics such oscillatory objects are denoted as self-sustained oscillators. Mathematically, such an oscillator is described by an autonomous (i.e., without an explicit time dependence) nonlinear dynamical system. It differs both from linear oscillators (which, if a damping is present, can oscillate only due to external forcing) and from nonlinear energy conserving systems, whose dynamics essentially depends on initial state. Dynamics of oscillators is typically described in the phase (state) space. Periodic oscillations, like those of the clock, correspond to a closed attractive curve in the phase space, called the limit cycle. The limit cycle is a simple attractor, in contrast to a strange (chaotic) attractor. The latter is a geometrical image of chaotic self-sustained oscillations. 

Equation (25.85), the basis of every atmospheric model, is a set of time-dependent, nonlinear, coupled partial differential equations. Several methods have been proposed for their solution including global finite differences, operator splitting, finite element methods, spectral methods, and the method of lines (Oran and Boris 1987). Operator splitting, also called the fractional step method or timestep splitting, allows significant flexibility and is used in most atmospheric chemical transport models. 

XII. Appendix F Frequency and Time Dependent Nonlinear Polarizabilities... 

An alternative method for the realization of KLM uses the birefringent properties of the Kerr medium, which turns the plane of polarization of the light wave passing through the Kerr medium. This is illustrated in Fig. 11.23. The incident wave passes through a linear polarizer and is then elliptically polarized by a A/4-plate. The Kerr medium causes a time dependent nonlinear polarization rotation. A A/2-plate and a linear polarizer behind the Ken-medium can be arranged in such a way that the pulse transmission reaches its maximum at the peak of the incident pulse, thus shortening the pulse width [11.55]. This device acts similarly to a passive saturable absorber and is particularly useful for fiber lasers with ultrashort pulses. 

The full quantum statistical mechanical approach to solvent effects on dynamic processes has not been analyzed in detail. It is important to note recent developments by Banacky and Zajac [102, 103] on the theory of particle dynamics in solvated molecular complexes. A time-dependent nonlinear equation of motion for the probability density of a proton in a solvated symmetric H-bond system was derived. Earlier work has been overviewed by the present author in a recent paper [10]. 

Velarde, M. G., and Rednikov, A. Ye. (1998) Time-dependent Benard-Marangoni instability and waves, in Time-Dependent Nonlinear Convection, edited by PA. Tyvand, Computational Mechanics Publications, Southampton, 177-218. 

Velarde M.G. and Rednikov A. Ye. (1998). Time-dependent B nard-Marangoni instability. In Time-Dependent Nonlinear Conveetion, Chapter 6, 177-218, Tyvand P.A. Editor. Adv. in Fluid Mech., 19, Computational Mech. Publ., Southampton, UK. 

RDX ignition was modeled l by following the time evolution of the molecular species formed from gas-phase decomposition of RDX and subsequent chemical reactions. For the ignition studies, we took the thermodynamic conditions to be an adiabatic system at constant pressure The resulting time-dependent, nonlinear coupled equations were solved using a total of 49 chemical species and 236... 

Abstract This chapter presents the general aspects of the response theory for molecular solutes in the presence of time-dependent perturbing fields (i) the nonequilibrium solvation, (ii) the variational formulation of the time-dependent nonlinear QM problem, and (iii) the connection of the molecular response functions with their macroscopic counterparts. The linear and quadratic molecular response functions are described at the coupled-cluster level. 

From the theoretical point of view, the mapping function S( ) is a no time-dependent nonlinear function. 

File listlS 19.1ua Time dependent nonlinear diffusion ... 

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