The nonlinear wave equation

The propagation of light in a nonlinear medium is governed by the wave equation, which was derived from Maxwell s equations for an arbitrary homogeneous dielectric medium, 

It is useful to regard (4.19) as a wave equation in which the term S = —p,od2PNL/dt2 acts as a source radiating in a linear medium of refractive index n. Because Pnl (and therefore S) is a nonlinear function of E, Equation (4.19) is a nonlinear partial differential equation in E. This is the basic equation that underlies the theory of nonlinear optics. 

Since S(Eo) is a nonlinear function, new frequencies are created. The source therefore emits an optical field Ei, with frequencies not present in the original wave E0. This leads to numerous interesting phenomena that have been utilized to make useful nonlinear-optics devices. 

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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... 

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)... 

By solving the nonlinear wave equation, under the assumption of undepleted input beams, it is found that the intensity of the output wave at frequency 2w varies as a function of the interaction length inside of the sample, I, and the wave vector mismatch Ak [26] ... 

For the induced nonlinear polarization given by Eq. (91) the nonlinear wave equation can be solved under the assumption of no pump depletion and using the slowly varying approximation [26]. As in the case of THG or EFISGH, the wave equation yields a phase-matching condition on the wave-vectors of the four beams ... 

Third-harmonic generation along the helical axis in a cholesteric medium is governed by the nonlinear wave equation... 

To explore the soliton solution further, Listing 12.17 shows code for setting up two ideal soliton solutions according to Eq. (12.50) but with different spatial locations and different amplitudes. The solution of the wave equation then simulates the time evolution of the solutions according to the nonlinear wave equation. The equations to be solved and the boundary conditions are the same as in the previous listing. In this case two functions are defined on lines 12 and 13, one for the cosh() function and one for the ideal soliton solution of Eq. (12.50). These are then used on lines 30 through 32 to set up an initial solution with an amplitude of 4 for the soliton at x = 4 and an amplitude of 1 for the soliton at x = 20. The pdeivbvtO function then called on line 35 simulates the development of the solu-... 

In a moving co-ordinate system, the traveling wave equations typically reduce to a system of parameterized nonlinear ordinary differential equations. The solutions of this system corresponding to pulses and fronts for the original reaction-diffusion equation are called homoclinic and heteroclinic orbits, correspondingly, or just connecting orbits. 

There are many features of real waves, such as dispersion and nonlinearity, that cannot be described directly in terms of the general wave equation. A brief discussion of such effects is needed to understand solitons. 

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) ... 

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. 

Note that in contrast to the case of the nonlinear Dirac equation, it is not possible to construct the general solutions of the reduced systems (59)-(61). For this reason, we give whenever possible their particular solutions, obtained by reduction of systems of equations in question by the number of components of the dependent function. Let us emphasize that the miraculous efficiency of the t Hooft ansatz [5] for the Yang-Mills equations is a consequence of the fact that it reduces the system of 12 differential equations to a single conformally invariant wave equation. 

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. 

An illustration of this fact comes from the nonlinear Schrodinger equation. This equation describes an electromagnetic wave in a nonlinear medium, where the dispersive effects of the wave in that medium are compensated for by a refocusing property of that nonlinear medium. The result is that this electromagnetic wave is a soliton. Suppose that we have a Fabry-Perot cavity of infinite extend in the x direction that is pumped with a laser [6,7]. The modes allowed in that cavity can be expanded in a Fourier series as follows ... 

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... 

In the case of s linearly damped oscillator, the transformation of the Heisenberg picture into the Schrodinger picture by the method applied in classical quantum theory is impossible because the operator has a time-dependent part due to the dissipative process. Thus, a new way must be found to construct the wave equation of the oscillator. Kostin introduced a supplementary dissipation potential into his wave equation and constructed this dissipation potential by an assumption that the energy eigenvalues of the oscillator decay exponentially over time [39]. In Kostin s version of the wave equation, the operators are time independent, but the dissipation potential is nonlinear with respect to the wave function. In our theory, it is assumed that the abstract wave equation of the linearly damped oscillator has the form... 

Forecast centers in the Baltic area mainly use the WAM model or modified versions, the Swedish Meteorological and Hydrological Institute (SMHI) uses the Hybrid Parametrical Shallow Water model HYPPAS (Gunther et. al., 1979). This model type presumes a char-acterisitc shape of the wind sea spectrum (JONSWAP spectrum, 7.10) as a result of the nonlinear wave interactions, and therefore the difficult coupling source term in the balance equation is omitted. 

II. The Classical Wave Equation for Radiation Fields in Nonlinear Media... 

II. THE CLASSICAL WAVE EQUATION FOR RADIATION FIELDS IN NONLINEAR MEDIA... 

In particular, for what class of intermolecular potentials does this equation have a unique solution for given initial and boundary conditions Is the normal solution of Chapman and Enskog in any sense a good approximation to the actual solution, at least sufficiently far from the boundaries, and for sufficiently long times Can one find explicit solutions to the nonlinear Boltzmann equation with sufficient accuracy so that the specifically nonlinear features of the Boltzmann equation can be tested, in shock wave or sound wave or sound propagation experiments, for example ... 

Only bilinear variables with intermediate wave vector kc are to be included in the nonlinear Langevin equation, while the relations just derived contain sums over all wave vectors. Let us split up the sum in Eq. (67) and use the definition of the diffusion flux = ik /t, to obtain... 

Low-order nonlinear wave model. Let us consider the first-order, convective, nonlinear wave equation for a function u(x,t) satisfying... 

This means that for each incident photon of frequency 0)2 there are two outgoing photons of the same frequency. Simultaneously with the generation of a new wave of frequency co - the incident wave of frequency (O2 is parametrically amplified. If the nonlinear medium is placed between two mirrors reflecting at the frequencies (O2 and (or) ( 3, this parametric effect may be increased. One calls such a device a parametric oscillator (see Figure 3). From this point of view, co = (o corresponds to the so-called pump wave, (jl>2 = (o to the (amplified) signal wave, and (03 = (Oi- CO2) =o)i to the idler wave. Equation [8] may be simplified to... 

The last step is to analyse how the induced nonlinear polarization creates a new wave or interferes with the existing ones to generate the nonlinear Raman signal. The nonlinear wave propagation equation (taking = 0) is... 

Brillouin scatterings are caused by acoustic waves in the material. In analogy to Raman scattering, the nonlinear polarizations responsible for the coupled-wave equations are obtainable from consideration of the appropriate material excitations. 

The physical processes describing the field-matter interaction in an OKE experiment are quite complicated and they are defined properly by the nonlinear optics equations [39-41]. In the next chapter, we will outline the rigorous description of the experiment based on the four-wave mixing phenomena equations. Here we introduce a relatively simple model of the OKE experiment, either continuous or time-resolved, that retains all the principal features. This model is based on few intuitive starting approximations that will be proved later. The first basic approximation is the separation between excitation and the probing processes. 

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