Nonlinear waves

Figure 1 The frame on the left shows the development of a dispersive wave and the frame in the middle that of a nonlinear wave. When these effects are balanced as in the frame on the right a soliton is formed.

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. 

In this overview we focus on the elastodynamical aspects of the transformation and intentionally exclude phase changes controlled by diffusion of heat or constituent. To emphasize ideas we use a one dimensional model which reduces to a nonlinear wave equation. Following Ericksen (1975) and James (1980), we interpret the behavior of transforming material as associated with the nonconvexity of elastic energy and demonstrate that a simplest initial value problem for the wave equation with a non-monotone stress-strain relation exhibits massive failure of uniqueness associated with the phenomena of nucleation and growth. 

For the purpose of illustration, in this paper we use a viscosity-capillarity model (Truskinovsky, 1982 Slemrod, 1983) as an artificial "micromodel",and investigate how the information about the behavior of solutions at the microscale can be used to narrow the nonuniqueness at the macroscale. The viscosity-capillarity model contains a parameter -Je with a scale of length, and the nonlinear wave equation is viewed as a limit of this "micromodel" obtained when this parameter tends to zero. As we show, the localized perturbations of the form x /-4I) can influence the choice of attractor for this type of perturbation, support (but not amplitude) vanishes as the small parameter goes to zero. Another manifestation of this effect is the essential dependence of the limiting solution on the... 

The absolute instability of the "metastable" states in the framework of classical elasticity manifests itself in dynamics as well. The associated elastodynamical problem reduces to a solution of the nonlinear wave equation = o (uJu . It is convenient to rewrite it as a mixed type first order system... 

In order to investigate the solutions of the nonlinear wave equation (8) in a more extensive manner, it is useful to look at the nonlinear term. The quantum... 

In this section we discuss the nonrelativistic 0(3) b quantum electrodynamics. This discussion covers the basic physics of f/(l) electrodynamics and leads into a discussion of nonrelativistic 0(3)h quantum electrodynamics. This discussion will introduce the quantum picture of the interaction between a fermion and the electromagnetic field with the magnetic field. Here it is demonstrated that the existence of the field implies photon-photon interactions. In nonrelativistic quantum electrodynamics this leads to nonlinear wave equations. Some presentation is given on relativistic quantum electrodynamics and the occurrence of Feynman diagrams that emerge from the B are demonstrated to lead to new subtle corrections. Numerical results with the interaction of a fermion, identical in form to a 2-state atom, with photons in a cavity are discussed. This concludes with a demonstration of the Lamb shift and renormalizability. 

E. Infeld and G. Rowlands, Nonlinear Waves, Solitons and Chaos, 2nd ed., Cambridge Univ. Press, 2000. 

P. D. Lax, in Nonlinear Wave Motion, Lectures in Applied Mathematics, Vol. 15, American Mathematical Society, 1974, pp. 85-96. 

Stegeman, G.I. In Proceedings of Erice Summer School on Nonlinear Waves in Solid State Physics Boardman, A,D. Twardowski, T., Eds. in press... 

The nonlinear wave equation thus obtained is the famous sine-Gordon equation, which is well known from soliton theory (see, for example, Dodd et al. [1982] and Rajaraman [1982]). The long wave approximation used to replace the discrete rotor angle < by continuous variable 4>(x, t)... 

Equilibrium Theory and Nonlinear Waves for Reaction Separation Processes... 

In this section, possible applications of the theory presented in the previous section are highlighted. It was shown that the equilibrium theory and nonlinear waves provide a simple means for understanding the dynamic behavior of many integrated processes. It thereby often directly guides the way to improved process operation and improved process control. Several examples will be discussed subsequently. 

Insights from nonlinear wave theory can also be used for designing new control strategies. A major problem in controlling product purities in separation as well as integrated reaction separation processes is often the lack of a cheap, reliable and fast online concentration measurement. This problem can be solved in two different ways (i) through simple inferential control, or (ii) model-based measurement. 

Finally, nonlinear wave can also be used for nonlinear model reduction for applications in advanced, nonlinear model-based control. Successful applications were reported for nonreactive distillation processes with moderately nonideal mixtures [21]. For this class of mixtures the column dynamics is entirely governed by constant pattern waves, as explained above. The approach is based on a wave function which can be used for the approximation of the concentration profiles inside the column. The wave function can be derived from analytical solutions of the corresponding wave equations for some simple limiting cases. It is given by... 

S. GrtinerandA. Kienle. Equilibrium theory and nonlinear waves for reactive distillation columns and chromatographic reactors. Chem. Engng. Sci., 2004, 59, 901-918. 

---