Nonlinear theoretical models

Graduate-level introduction mainly to theoretical modelling of nonlinear reactions Scott S K 1993 Chemical Chaos (Oxford Oxford University Press)... 

The subject of liquid jet and sheet atomization has attracted considerable attention in theoretical studies and numerical modeling due to its practical importance.[527] The models and methods developed range from linear stability models to detailed nonlinear numerical models based on boundary-element methods 528 5291 and Volume-Of-Fluid (VOF) method. 530 ... 

The more rigorous stress/strain nonlinear material model, oflen referred to as the plastic zone method, is theoretically capable of handling any general cross section Both isotropic and kinematic hardening rules are usually available. This method is... 

Our theoretical model for SHG in bulk nonlinear crystals, which describes the process of SHG under conditions of large GVM, assumes an... 

In the case of SHG in waveguide nonlinear crystals, we describe a theoretical model which accounts for the temporal behavior of the interacting pulses and the possible z-dependence of the phasematching condition. The model also describes the observed saturation and subsequent decrease in SHG conversion efficiency in the waveguide samples, as a result of two-photon absorption (TPA) of the second harmonic (SH) wave. The results of this model are later compared with experimental data from SHG experiments using femtosecond pulses in the waveguide nonlinear crystals of periodically-poled potassium titanyl phosphate (ppKTP) and appKTP. This model is presented in section 2.3. 

Theoretical model to describe SHG of focused femtosecond pulses in periodically-poled and aperiodically-poled bulk nonlinear crystals... 

The results of experimental studies of the sorption and diffusion of light hydrocarbons and some other simple nonpolar molecules in type-A zeolites are summarized and compared with reported data for similar molecules in H-chabazite. Henry s law constants and equilibrium isotherms for both zeolites are interpreted in terms of a simple theoretical model. Zeolitic diffusivitiesy measured over small differential concentration steps, show a pronounced increase with sorbate concentration. This effect can be accounted for by the nonlinearity of the isotherms and the intrinsic mobilities are essentially independent of concentration. Activation energies for diffusion, calculated from the temperature dependence of the intrinsic mobilitieSy show a clear correlation with critical diameter. For the simpler moleculeSy transition state theory gives a quantitative prediction of the experimental diffusivity. 

Many challenging materials issues addressable by chemists currently exist. Chemists bring to the field a chemical understanding of materials that physicists and engineers do not have. Chemists form the bridge between theoretical models for nonlinear polarizability and real NLO materials. As a result, they have an excellent opportunity to make a large, positive, impact on the field of nonlinear optics. 

The theoretical models discussed above indicate that the sulfonyl group, although slightly weaker in electron acceptor strength, is indeed a viable alternative to the nitro group. In particular, sulfonyl derivatives of stilbene and azobenzene display large molecular hyperpolarizabilities and can be used as bifunctional chromophores for the construction of materials with nonlinear optical properties. 

Solving the nonlinear mathematic models by a numerical method depends on the trial values that are chosen by experience and is not based upon a particular theory. The results obtained are greatly influenced by the initial values selected that is a characteristic of strong nonlinear problems. By using numerical techniques to solve nonlinear models, iterations must be implemented. With the incorrect selection of a trial value, divergence in solution can appear. The accuracy of the numerical results cannot be estimated theoretically for nonlinear problem since there is no analytical solution available. This is why an approximate analytical solution is extremely useful for a theoretical analysis of nonlinear problem If an approximate analytical solution can be obtained, then this has a number of benefits ... 

Electro- and magnetooptical phenomena in colloids and suspensions are widely used for structure and kinetics analysis of those media as well as practical applications in optoelectronics [143,144]. The basic theoretical model used to study optical anisotropy of the disperse systems is the noninteracting Brownian particle ensemble. In the frame of this general approximation, several special cases according to the actual type of particle polarization response to the applied field may be distinguished (1) particles with permanent dipole moments, (2) linearly polarizable particles, (3) nonlinearly polarizable particles, and (4) particles with hysteretic dipole moment reorientation. 

In this contribution, we will try to give an overview of the possible mechanistic origin of chiral amplification in the Soai reaction. We will present a reaction network derived from simple theoretical models of chiral amplification that can give rise to a plausible description of the combined experimental observations of this reaction. Following our interest in describing the dynamic features of nonlinear reaction systems [20-23], we will emphasize the possible kinetic rea-... 

Stefan Balint, Analysis and Numerical Computation of Solutions of Nonlinear Systems Modeling Physical Phenomena, Especially Nonlinear Optics, Inverse Problems, Mathematical Materials Science and Theoretical Fluid Mechanics, Proceedings of a conference held 19-21 May 1997, in Timisoara, Romania, University of the West Timisoara, Timisoara, Romania, 1997. 

It is easily seen that for such trial functions the minimization of the Hamiltonian K, Eq. (5.1), may be replaced by the minimization of a specified nonlinear functional 6(0) of the molecular states 0 alone. In the following we refer to either formulation as seems convenient. This argument also enables one to connect these field theoretical models with the earlier suggestion of mine that molecular structure states can be associated with those solutions of the Schrodinger equation for the full molecular Hamiltonian ft that satisfy certain subsidiary conditions3,35), if the latter are associated with the nonlinearity in the functional 6(0). As we shall see, it may happen that 6(0) has two degenerate minima and it is in this sense that the dynamics gives rise to a double-well structure. 

In Section II, we demonstrate this approach with an equation that incorporates a nonlinear Hill function to model genetic control representing a mutually inhibitory network of two elements [26], and an inhibitory loop of three elements [27]. Although theoretical models of these types of networks have been known for at least 30 years [20, 32, 33], they took on new life in 2000 with the construction of genetic regulatory circuits in bacteria that were well described by the equations. 

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