Nonlinear Schrodinger equation

Inserting the separation ansatz, i.e., U , results in two nonlinearly coupled single particle Schrodinger equations, the so-called time dependent self-consistent field (TDSCF) equations ... 

To describe the orientations of a diatomic or linear polyatomic molecule requires only two angles (usually termed 0 and ([)). For any non-linear molecule, three angles (usually a, P, and y) are needed. Hence the rotational Schrodinger equation for a nonlinear molecule is a differential equation in three-dimensions. 

It is conventional to rewrite eq. (28) as a nonlinear Schrodinger equation with eigenvalue E ... 

Solving the nonlinear Schrodinger equation yields E and T and the desired physical quantity GENP may then be calculated directly from (11) or from... 

F> obtained by solving the equation is consistent with the F> used to calculate the reaction field. Having established an effective nonlinear Hamiltonian, one may solve the Schrodinger equation by any standard (or nonstandard) manner. The common element is that the electrostatic free energy term Gp is combined with the gas-phase Hamiltonian Hq to produce a nonlinear Schrodinger equation... 

In most work reported so far, the solute is treated by the Hartree-Fock method (i.e., Ho is expressed as a Fock operator), in which each electron moves in the self-consistent field (SCF) of the others. The term SCRF, which should refer to the treatment of the reaction field, is used by some workers to refer to a combination of the SCRF nonlinear Schrodinger equation (34) and SCF method to solve it, but in the future, as correlated treatments of the solute becomes more common, it will be necessary to more clearly distinguish the SCRF and SCF approximations. The SCRF method, with or without the additional SCF approximation, was first proposed by Rinaldi and Rivail [87, 88], Yomosa [89, 90], and Tapia and Goscinski [91], A highly recommended review of the foundations of the field was given by Tapia [71],... 

The trial wave functions of a Schrodinger equation are expressed as determinant of the HF orbitals. This will give coupled nonlinear equations. The amplitudes were solved usually by some iteration techniques so the cc energy is computed as... 

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. 

A common way to treat the problem of a picosecond pulse propagation in regular dielectric waveguide with Kerr nonlinearity was to solve the nonlinear Schrodinger equation (NLSE) for the slowly varying temporal amplitude of electrical field ... 

Eq.(3.2) is the mixed type linear/nonlinear Schrodinger equation and belongs to a specific class of equations which are close to, but not exactly, integrable. The lowest-order integrals that remain constant in the system described by Eq.(3.2) as z changes are the dimensionless power ... 

One basic difficulty with the nonlinear equation arises from the following. Consider a physical situation where a source of particles is composed of many emitters, each emitting a particle at a time. If considered alone, each particle would be described by a localized wave /,- solution of the master equation. Now, what happens if, instead of emitting the particles one by one, the source emits many particles at the same time If the master equation were a linear equation, like the usual Schrodinger equation, the answer would be trivial. The general solution would be simply the sum of all particular solutions. 

Looking at this equation, one sees that in free space, where V = 0, the nonlinear master equation transforms into the usual linear Schrodinger equation with a nonnull potential ... 

The computational problem, then, is determination of the cluster amplitudes t for aU of the operators included in tlie particular approximation. In the standard implementation, this task follows the usual procedure of left-multiplying the Schrodinger equation by trial wave functions expressed as dctcnninants of the HF orbitals. This generates a set of coupled, nonlinear equations in the amplitudes which must be solved, usually by some iterative technique. With the amplitudes in hand, the coupled-cluster energy is computed as... 

What is the origin of the nonlinearity introduced in the Schrodinger equation represented by a potential proportional to 2 ... 

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 ... 

The occurrence of the nonlinear Schrodinger equation is then a fairly generic result. For the A potential we have the magnetic field that is easily seen to be... 

The difference this derivation has in comparison to the previous derivation of the nonlinear Schrodinger equation is that the nonlinearity is more fundamentally due to the non-Abelian wavefunction rather than from material coefficients. In effect these material coefficients and phenomenology behave as they do because the variable index of refraction is associated with non-Abelian electrodynamics. Ultimately these two views will merge, for the mechanisms on how photons interact with atoms and molecules will give a more complete picture on how non-Abelian electrodynamics participates in these processes. However, at this stage we can see that we obtain nonlinear terms from a non-Abelian electrodynamics that is fundamentally nonlinear. This is in contrast to the phenomenological approach that imposes these nonlinearities onto a fundamentally linear theory of electrodynamics. 

More recently, Palais [17] showed that the generic cases of soliton—the Korteweg de Vries equation (KdV), the nonlinear Schrodinger equation (NLS), the sine-Gordon equation (SGE)—can be given an SU(2) formulation. In each of the three cases considered below, V is a one-dimensional space that is embedded in the space of off-diagonal complex matrices, ( ) and in each case L(u) at- I u, where u is a potential, i. is a complex parameter, and a is the constant, diagonal, trace zero matrix... 

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