First-order propagation equation

Several types of unidirectional propagation equations are widely used in the nonlinear optics literature. The most important examples are Non-Linear Schrodinger (NLS) equation [32], Nonlinear Envelope Equation [33] (NEE), the First-Order Propagation equation [31] (FOP), Forward Maxwell s equation [34] (FME), and several other equations that are closely related to these. The derivations found in the literature differ from equation to equation, and in some cases the physical meaning of the required approximations may not be readily evident due to a number of neglected terms. 

It frequently happens that we plot or analyze data in terms of quantities that are transformed from the raw experimental variables. The discussion of the propagation of error leads us to ask about the distribution of error in the transformed variables. Consider the first-order rate equation as an important example ... 

We have used various integrators (e.g., Runga-Kutta, velocity verlet, midpoint) to propagate the coupled set of first-order differential equations Eqs. (2.8) and (2.9) for the parameters of the Gaussian basis functions and Eq. (2.11) for the complex amplitudes. The specific choice is guided by the complexity of the problem and/or the stiffness of the differential equations. 

This first order differential equation now governs the evolution of an initial field. For a finite step length the propagation is approximated by... 

The propagation of the wavepacket is thereby reduced to the solution of coupled first-order differential equations for the parameters representing the Gaussian wavepacket, with the true potential being expanded about the instantaneous center of the wavepacket [i2(<),f(<)]. This propagation scheme is very appealing and efficient provided the basic assumptions are fulfilled. The essential prerequisite is that the locally quadratic approximation of the PES is valid over the spread of the wavepacket. This rules out bifurcation of the wavepacket, resonance effects, or strong an-harmonicities. 

Here kp should more appropriately be called an apparent rate constant or overall rate constant since both undissociated (ion pair) and dissociated (free ions) species usually exist and their propagation rate constants are different (discussed later). Integration of this pseudo-first-order rate equation gives the time dependence of the monomer concentration as... 

Let us assume that at r = 0 the wave function If is given in MCTDH form, i.e., given by equation (16). (The question of how to define and generate an initial-state wave function is addressed below.) We want to propagate If while preserving its MCTDH form. As was done above, we derive first-order differential equations (equations of motion) for A and by employing the time-dependent variational principle equation (5). But before doing so, we partition the Hamiltonian H into a separable and residual part ... 

However, the mechanisms by which the initiation and propagation reactions occur are far more complex. Dimeric association of polystyryllithium is reported by Morton, al. ( ) and it is generally accepted that the reactions are first order with respect to monomer concentration. Unfortunately, the existence of associated complexes of initiator and polystyryllithium as well as possible cross association between the two species have negated the determination of the exact polymerization mechanisms (, 10, 11, 12, 13). It is this high degree of complexity which necessitates the use of empirical rate equations. One such empirical rate expression for the auto-catalytic initiation reaction for the anionic polymerization of styrene in benzene solvent as reported by Tanlak (14) is given by ... 

The correction to the relaxing density matrix can be obtained without coupling it to the differential equations for the Hamiltonian equations, and therefore does not require solving coupled equations for slow and fast functions. This procedure has been successfully applied to several collisional phenomena involving both one and several active electrons, where a single TDHF state was suitable, and was observed to show excellent numerical behavior. A simple and yet useful procedure employs the first order correction F (f) = A (f) and an adaptive step size for the quadrature and propagation. The density matrix is then approximated in each interval by... 

In order for the overall rate expression to be 3/2 order in reactant for a first-order initiation process, the chain terminating step must involve a second-order reaction between two of the radicals responsible for the second-order propagation reactions. In terms of our generalized Rice-Herzfeld mechanistic equations, this means that reaction (4a) is the dominant chain breaking process. One may proceed as above to show that the mechanism leads to a 3/2 order rate expression. 

These equations are integrated from some initial conditions. For a specified value of s, the value of x and y shows the location where the solution is u. The equation is semilinear if a and b depend just on x and y (and not u), and the equation is linear if a, b, and/all depend on x and y, but not u. Such equations give rise to shock propagation, and conditions have been derived to deduce the presence of shocks. Courant and Hilbert (1953, 1962) Rhee, H. K., R. Aris, and N. R. Amundson, First-Order Partial Differential Equations, vol. I, Theory and Applications of Single Equations, Prentice-Hall, Englewood Cliffs, N.J. (1986) and LeVeque (1992), ibid. 

This equation is second order in time, and therefore remains invariant under time reversal, that is, the transformation t - — t. A movie of a wave propagating to the left, run backwards therefore pictures a wave propagating to the right. In diffusion or heat conduction, the field equation (for concentration or temperature field) is only first order in time. The equation is not invariant under time reversal supporting the observation that diffusion and heat-flow are irreversible processes. 

Case 2. If the dependence of DP on [P3] shows that the propagation is a second order reaction (equation (12)), and if the whole reaction curves are of first order, it follows that [Pn+] must be constant throughout the reaction, i.e., there is a stationary state. This can be of the First Kind, V = Vt 0, or of the Second Kind, V,- = Vt = 0. The diagnosis... 

This paper is about a reinterpretation of the cationic polymerizations of hydrocarbons (HC) and of alkyl vinyl ethers (VE) by ionizing radiations in bulk and in solution. It is shown first that for both classes of monomer, M, in bulk ([M] = niB) the propagation is unimolecular and not bimolecular as was believed previously. This view is in accord with the fact that for many systems the conversion, Y, depends rectilinearly on the reaction time up to high Y. The growth reaction is an isomerization of a 7t-complex, P +M, between the growing cation PB+ and the double bond of M. Therefore the polymerizations are of zero order with respect to m, with first-order rate constant k p]. The previously reported second-order rate constants kp+ are related to these by the equation... 

In the past two decades, a variety of semiclassical initial-value representations have been developed [105-111], which are equivalent within the semiclassical approximation (i.e., they solve the Schrodinger equation to first order in H), but differ in their accuracy and numerical performance. Most of the applications of initial-value representation methods in recent years have employed the Herman-Kluk (coherent-state) representation of the semiclassical propagator [105, 108, 187, 245, 252-255], which for a general n-dimensional system can be written as... 

First-order error analysis is a method for propagating uncertainty in the random parameters of a model into the model predictions using a fixed-form equation. This method is not a simulation like Monte Carlo but uses statistical theory to develop an equation that can easily be solved on a calculator. The method works well for linear models, but the accuracy of the method decreases as the model becomes more nonlinear. As a general rule, linear models that can be written down on a piece of paper work well with Ist-order error analysis. Complicated models that consist of a large number of pieced equations (like large exposure models) cannot be evaluated using Ist-order analysis. To use the technique, each partial differential equation of each random parameter with respect to the model must be solvable. 

The kinetics of template polymerization depends, in the first place, on the type of polyreaction involved in polymer formation. The polycondensation process description is based on the Flory s assumptions which lead to a simple (in most cases of the second order), classic equation. The kinetics of addition polymerization is based on a well known scheme, in which classical rate equations are applied to the elementary processes (initiation, propagation, and termination), according to the general concept of chain reactions. 

A code has been written to enable the velocities of surface waves in multilayered anisotropic materials, at any orientation and propagation and including piezoelectric effects, to be calculated on a personal computer (Adler et al. 1990). The principle of the calculation is a matrix approach, somewhat along the lines of 10.2 but, because of the additional variables and boundary conditions, and because the wave velocities themselves are being found, it amounts to solving a first-order eight-dimensional vector-matrix equation. A... 

The dependence of the propagation rate on the concentration of growing chains is illustrated in Figures 6 and 7, and is listed in Table II. The first-order rate constant from Table II are plotted as a function of the initiator concentration. Although the kinetics of organolithium polymerization in nonpolar solvents have been subjected for intensive studies, the results were still somewhat controversial. In view of the strong experimental evidence for association between the organolithium species, the kinetic order ascribed to this phenomenon was postulated (30,31) as shown in Equations (5) and (6). 

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