Wave equation asymptotic solution

Here the distortion (diagonal) and back coupling matrix elements in the two-level equations (section B2.2.8.4) are ignored so that = exp(ik.-R) remains an imdistorted plane wave. The asymptotic solution for ij-when compared with the asymptotic boundary condition then provides the Bom elastic ( =f) or inelastic scattering amplitudes... 

We call this a partial M/ave expansion. To detennine tire coefficients one matches asymptotic solutions to the radial Scln-ddinger equation with the corresponding partial wave expansion of equation (A3.11.106). It is customary to write the asymptotic radial Scln-ddinger equation solution as... 

Figure 12.3b shows the widths of the Bragg layers as a function of the layer number of the CBNL depicted in Fig. 12.3a. There are two notable properties of the Bragg layers (1) the width of the high-index layers is smaller than the width of the low-index layers, and (2) the width of the layers decreases exponentially as a function of the radius, converging asymptotically to a constant value. The first property exists in conventional DBRs as well and stems from the dependence of the spatial oscillation period, or the wavelength, on the index of refraction. The second property is unique to the cylindrical geometry and arises from the nonperiodic nature of the solutions of the wave equation (Bessel or Hankel functions) in this geometry.

This is the constant-pattern simplification that enables many solutions to be obtained from what might otherwise be complex rate equations. It represents a condition that is approached as the wave becomes fully developed and leads to what are termed asymptotic solutions. 

The solution of the first kind is stable and arises as the limit, t —> oo, of the non-stationary kinetic equations. Contrary, the solution of the second kind is unstable, i.e., the solution of non-stationary kinetic equations oscillates periodically in time. The joint density of similar particles remains monotonously increasing with coordinate r, unlike that for dissimilar particles. The autowave motion observed could be classified as the non-linear standing waves. Note however, that by nature these waves are not standing waves of concentrations in a real 3d space, but these are more the waves of the joint correlation functions, whose oscillation period does not coincide with that for concentrations. Speaking of the auto-oscillatory regime, we mean first of all the asymptotic solution, as t —> oo. For small t the transient regime holds depending on the initial conditions. 

If there are n0 open channels at energy E, there are n linearly independent degenerate solutions of the Schrodinger equation. Each solution is characterized by a vector of coefficients aips, for i = 0,1, defined by the asymptotic form of the multichannel wave function in Eq. (8.1). The rectangular column matrix a consists of the two n0 x n0 coefficient matrices ao, < i Any nonsingular linear combination of the column vectors of a produces a physically equivalent set of solutions. When multiplied on the right by the inverse of the original matrix a0, the transformed a-matrix takes the canonical form... 

The first step is the asymptotic solution of the wave equation when i is very large. For any value of the energy constant W, a value of x can be found such that for it and all larger values... 

We now proceed to obtain an accurate solution of the wave equation throughout configuration space (— °° asymptotic solution, by introducing as a factor a power series in x and determining its coefficients by substitution in the wave equation. 

Problem 17-1. The equation for the free particle is separable in many coordinate systems. Using cylindrical polar coordinates, set up and separate the wave equation, obtain the solutions in

recursion formula for the coefficients in the series solution of the p equation. Hint In applying the polynomial method, omit the step of finding the asymptotic solution. 

The wave functions (64.11) represent the exact asymptotical solutions of the Schrbdinger equation (6.1) for x x and x>>X2 f where the potential energy V(x) is constant therefore, and 1 2 are plane waves. We may use instead the quasiclassical wave functions... 

Wentzel-Kramers-Brillouin (WKB) asymptotic solutions of the wave equations... 

We have shown that the zeroth-order geometric optics approximation can be used to describe the propagation of normally incident, elliptically polarized light in an inhomogeneous, locally uniaxial medium. The approximation corresponds to finding an asymptotic solution of the wave equation in the short-wavelength limit. It is found that a set of pseudo-Stokes parameters, linearly related to the usual Stokes parameters, can be defined to characterize the propa-... 

In order to compare the signal induced in a pickup coil with the continuous wave signal detected in a resonator, one needs to find a common route towards the solution of the Bloch equations. The asymptotic solutions described in the preceding section are not applicable to the case in which the cw-EPR resonator is used. As an alternative, one may use a matrix technique. 

Then it became apparent that certain physical principles could be used to simplify the complete equations so they could be solved relatively easily. Such a simplification was first carried out by von Karman and Penner [9], Their approach was considered one of the more significant advances in laminar flame propagation, but it could not have been developed and verified if it were not for the extensive work of Hirschfelder and his collaborators. The major simplification that von Karman and Penner introduced is the fact that the eigenvalue solution of the equations is the same for all ignition temperatures, whether it be near T or not. More recently, asymptotic analyses have been developed that provide formulas with greater accuracy and further clarification of the wave structure. These developments are described in detail in three books [10-12],... 

As the positron approaches the target system it interacts with and distorts it, so that the total wave function no longer has the separable form of equation (3.3). Nevertheless, an equivalent Schrodinger equation can be derived for the positron, the solution to which is a function of the positron coordinate rq only, with the correct asymptotic form but at the cost of introducing a non-local optical potential. 

Each I>(R, r E, n) with n = 0,1,..., nmax is a degenerate, yet independent, solution of the full nuclear Schrodinger equation with energy E. They are distinguished by the one and only particular vibrational channel that is associated with an outgoing free wave in the asymptotic region. 

Modeling EM solitary waves in a plasma is quite a challenging problem due to the intrinsic nonlinearity of these objects. Most of the theories have been developed for one-dimensional quasi-stationary EM energy distributions, which represent the asymptotic equilibrium states that are achieved by the radiation-plasma system after long interaction times. The analytical modeling of the phase of formation of an EM soliton, which we qualitatively described in the previous section, is still an open problem. What are usually called solitons are asymptotic quasi-stationary solutions of the Maxwell equations that is, the amplitude of the associated vector potential is either an harmonic function of time (for example, for linear polarization) or it is a constant (circular polarization). Let s briefly review the theory of one-dimensional RES. 

Above investigation for the behaviour of the solution at the free stream can be extended to other equations of (2.4.15) to check their usefulness in obtaining the eigen function. At the free stream the characteristic roots for Eqn. (2.4.15a) are —a, —Q. Equation (2.4.15b) being a third order equation has three roots given by [—a, —Q, a- -Q ]. Thus, this equation is also violently unstable, even for low wave numbers and Reynolds numbers. Equation (2.4.15c) has the asymptotic behaviour for large y s as dictated by the characteristic roots given by [—a, —Q, Finally, the characteristic... 

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