Stationary nonlinear

We find below a set of stationary nonlinear modes for the step-index nonlinear waveguide, investigate their stability and global dynamics. The latter is simulated numerically by the FD-BPM as a solution to the Cauchy problem for waveguide junctions under consideration. 

It is very important to make classification of dynamic models and choose an appropriate one to provide similarity between model behavior and real characteristics of the material. The following general classification of the models is proposed for consideration deterministic, stochastic or their combination, linear, nonlinear, stationary or non-stationary, ergodic or non-ergodic. 

It is possible to limit our choice for stochastic modeling by stationary, linear, nonlinear, and ergodic models in combination with deterministic function. In this case the following well studied models can be proposed for the accepted concept [1] ... 

Contemporary development of chromatography theory has tended to concentrate on dispersion in electro-chromatography and the treatment of column overload in preparative columns. Under overload conditions, the adsorption isotherm of the solute with respect to the stationary phase can be grossly nonlinear. One of the prime contributors in this research has been Guiochon and his co-workers, [27-30]. The form of the isotherm must be experimentally determined and, from the equilibrium data, and by the use of appropriate computer programs, it has been shown possible to calculate the theoretical profile of an overloaded peak. 

Y. A. Mitropolsky, Nan-stationary processes in nonlinear oscillatory systems, English translation by Air Technical Intelligence Center, Ohio. 

This enlarges the scope of problems that can be treated by these asymptotic methods. For example, the important problem of nonlinear resonance could otherwise be solved only in the stationary state. With this extension it is possible to determine what happens when the zone of resonance is passed at a certain rate. Likewise, with the additional extension for the slow time it is possible to attack the problem of modulated oscillations, which has previously remained outside the scope of the general theory. 

L. Mandelstam and N. Papalexi performed an interesting experiment of this kind with an electrical oscillatory circuit. If one of the parameters (C or L) is made to oscillate with frequency 2/, the system becomes self-excited with frequency/ this is due to the fact that there are always small residual charges in the condenser, which are sufficient to produce the cumulative phenomenon of self-excitation. It was found that in the case of a linear oscillatory circuit the voltage builds up beyond any limit until the insulation is ultimately punctured if, however, the system is nonlinear, the amplitude reaches a stable stationary value and oscillation acquires a periodic character. In Section 6.23 these two cases are represented by the differential equations (6-126) and (6-127) and the explanation is given in terms of their integration by the stroboscopic method. 

From the standpoint of the classical (analytical) theory with which we were concerned in this review, the situation is obviously absurd since each of these two equations is linear and of a dissipative type (since h > 0) trajectories of both of these equations are convergent spirals tending to approach a stable focus. However, if one carries out a simple analysis (see Reference 6, p. 608), one finds that change of equations for = 0, results in the change of the focus in a quasi-discontinuous manner, so that the trajectory can still be closed owing to the existence of two nonanalytic points on the -axis. If, however, the trajectory is closed, this means that there exists a stationary oscillation and in such a case the system (6-197) is nonlinear, although, from the standpoint of the differential equations, it is linear everywhere except at the two points at which the analyticity is lost. 

The stationary problem. To avoid misunderstanding, we concentrate primarily on the simplest problem, the statement of which is related to the stationary heat conduction problem with nonlinear sources ... 

In most cases the order of elution for a series of isomers on liquid crystalline stationary phases is generally in accord with the solute length-to-breadth ratios with differences in vapor pressure and solute polarity also being of Importance in some cases, leading to an inversion of elution order to that predicted from length-to-breadth ratios [828,829,838]. Long and planar molecules fit better into the ordered structure of the liquid crystal phase whereas nonlinear and nonplanar molecules do not permeate so easily between the liquid crystal molecules of the stationary phase and are more easily eluted from the column. 

When the output vector is nonlinearly related to the state vector (Equation 6.3) then substitution of x<,+l> from Equation 6.74 into the Equation 6.3 followed by substitution of the resulting equation into the objective function (Equation 6.4) yields the following equation after application of the stationary condition (Equation 6.78)... 

The number of binding sites can be determined in this model by a plot of d Ink /dlnm at constant temperature, pH, and ion valency. To do that, it may be assumed that dlny /dlnm is approximately zero. The actual value is -0.04 for 0.1 to 0.5 M sodium chloride and less at lower concentrations. To a first approximation, the stoichiometry of water molecules released by binding protein could be determined from the slope of the plot of dink /dlnm vs. m. However, especially at low salt concentration and near the isoelectric point, the slope of such plots is nonlinear. The nonlinearity may be due to hydrophobic interaction between stationary phase and protein or a large change of ionic hydration on binding.34... 

Clearly, the solutions of nonlinear gap equations are not unique. In numerical calculations we separated the physical solutions by observing the sign of 4>q and that of the effective potential at the stationary point W//( o)- The temperature dependence of these two quantities are presented in Fig. 2. It is seen that 4>q (solid line) is positive in the large range of r and goes to zero when r is close to r = 1. Similarly, the depth of the effective potential at the stationary point, Veff(4>o), becomes shallow when r —> 1 and vanishes at T = Tc. 

An unusual feature of a CSTR is the possibility of multiple stationary states for a reaction with certain nonlinear kinetics (rate law) in operation at a specified T, or for an exothermic reaction which produces a difference in temperature between the inlet and outlet of the reactor, including adiabatic operation. We treat these in turn in the next two sections. 

After several cycles of the compression and expansion, the dynamic jc-A curve becomes a single closed loop, somewhat distorted from a genuine ellipsoid. In order to analyze the forms of the hysteresis loop under stationary conditions, we have measured the time trace of the dynamic surface pressure after five cycles of the compression and expansion, and then Fourier-transformed it to the frequency domain. The Fourier-transformation was adapted to evaluate the nonlinear viscoelasticity in a quantitative manner. The detailed theoretical consideration for the use of the Fourier transformation to evaluate the nonlinearity, are contained in the published articles [8,43]. 

For general nonlinear objective functions, it is usually difficult to ascertain the nature of the stationary points without detailed examination of each point. 

The main purpose of this paper is to consider a two-dimensional non-stationary (2D-I-T) problem of a nonlinear waveguide excitation by a non-stationary light beam and to study spatiotemporal phenomena arising upon propagation of the beam in a step-index waveguide, first, in the quasi-static approximation and, second, with account of MD and SS effects. 

The theoretical approach is based on the solution to the mixed type linear/nonlinear generalized Schrodinger equation for spatiotemporal envelope of electrical field with account of transverse spatial derivatives and the transverse profile of refractive index. In the quasi-static approximation, this equation is reduced to the linear/nonlinear Schrodinger equation for spatiotemporal pulse envelope with temporal coordinate given as a parameter. Then the excitation problem can be formulated for a set of stationary light beams with initial amplitude distribution corresponding to temporal envelope of the initial pulse. 

Propagation of non-stationary light beam in a nonlinear medium with material dispersion is described by the scalar wave equation for the linearly-polarized y-component of electrical field E x,z,t) ... 

Non-stationary self-effects of the light beam depend on the pulse duration with respect to the time Tr of nonlinear response of the medium. When the response is instantaneous, the refractive index at the time t is defined by the value of electrical field at the same moment. If the time of a nonlinear response is finite, the nonlinear part of the refractive index satisfies the... 

In papers , unsteady-state regime arising upon propagation of the stationary fundamental mode from linear to nonlinear section of a single-mode step-index waveguide was studied via numerical modeling. It was shown that the stationary solution to the paraxial nonlinear wave equation (2.9) at some distance from the end of a nonlinear waveguide has the form of a transversely stable distribution ( nonlinear mode ) dependent on the field intensity, with a width smaller than that of the initial linear distribution. 

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