Nonlinear wave equation

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. 

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... 

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

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-... 

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