Nonlinear coupling terms

To obtain nonlinear coupling terms, we consider two linearly independent, not identical E modes, namely. 

The convective terms also introduce wavelike characteristics into the flow equations. A model for these generally nonlinear, coupled terms is Berger s equation,... 

Next, we approximate the nonlinear coupling term. Unlike in NLS, we preserve the frequency dependence exactly, but neglect the transverse wave-number dependence ... 

Thus, the additional approximations underlying the NEE are paraxiality both in the free propagator and in the nonlinear coupling, and a small error in the chromatic dispersion introduced when the background index of refraction is replaced by a constant, frequency independent value in both the spatio-temporal correction term and in the nonlinear coupling term. Note that the latter approximations are usually not serious at all. 

The first vibrational-anharmonicity term of Equation 18 has been found to dominate over nonlinear coupling in the specific case of N2, [73] and Oxtoby has argued that this term will dominate in general. [72] However, several theories have looked at dephasing due to the second, nonlinear coupling term of Equation 18. [71,74-76] Under reasonable approximations and with a steeply repulsive V, the anharmonic term also produces frequency shifts proportional to the solvent force on the vibrator, just as in Equation 19. [72] Again the issue returns to an accurate treatment of the solvent dynamics and the nature of the solvent-solute coupling. 

By making use of the form of the Landau energy proposed by de Gennes [17], all coefficients in this expression have been evaluated from experimental data [4]. Using these data to calculate U, one obtains a value that is considerably smaller than the one determined from the linearized analysis outlined above. As has been noted, however, a consistent analysis should not only include terms quadratic in the strains and bilinear coupling terms of the order parameter and the strain, but rather also nonlinear effects as well as nonlinear coupling terms between strain and the nematic order parameter [4]. 

The most important coupling to deformations of the network is the one that is linear in both the strain of the network and the nematic order parameter. As has been discussed earlier in this section this leads to the consequence that the strain tensor can be used as an order parameter for the nematic-isotropic transition in nematic sidechain elastomers, just as the dielectric or the diamagnetic tensor are used as macroscopic order parameters to characterize this phase transition in low molecular weight materials. But it has also been stressed that nonlinear elastic effects as well as nonlinear coupling terms between the nematic order parameter and the strain tensor must be taken into account as soon as effects that are nonlinear in the nematic order parameter are studied [4, 25]. So far, no deviation from classical mean field behavior concerning the critical exponents has been detected in the static properties of this transition and correspondingly there are no reports as yet discussing static critical fluctuations. 

Since there are very few dynamic experimental investigations of pretransitional effects [8], not much modeling has been reported to date either. Based on work for the macroscopic dynamics of the nematic-isotropic transition in sidechain polymers [27 -29], it has been suggested [28] that the non-meanfield exponent observed in dynamic stress-optical experiments [8] can be accounted for at least qualitatively by the mode-coupling model [28, 29]. Intuitively this qualitatively new dynamic behavior can be traced back to static nonlinear coupling terms between the nematic order parameter and the strain tensor. 

When Corrsin left Caltech in 1947 he was an acknowledged expert in turbulence research. This field with all its manifestations remained his primary interest throughout his career. He thereby contributed successfully to both theoretical and experimental research. He for instance familiarized with diagram techniques to clarify the sequence of nonlinear coupling terms in wave number space. His quest for clarity and precision had just one negative result He never finished the book he plaimed on fluid mechanics. His papers deal mainly with dimensional analysis, or the interpretation of the viscous terms in the turbulent energy equation. Corrsin s contribution to the Hcmdbuch proof also of his pedagogical interests, which culminated in a total of 25 PhD theses. 

When the nonlinear coupling terms do not depend on the angles between the interacting wave vectors, the Lyapounov functional takes the simple form (Brazovskii s model) ... 

Another subject with important potential application is discussed in Section XIV. There we suggested employing the curl equations (which any Bohr-Oppenheimer-Huang system has to obey for the for the relevant sub-Hilbert space), instead of ab initio calculations, to derive the non-adiabatic coupling terms [113,114]. Whereas these equations yield an analytic solution for any two-state system (the abelian case) they become much more elaborate due to the nonlinear terms that are unavoidable for any realistic system that contains more than two states (the non-abelian case). The solution of these equations is subject to boundary conditions that can be supplied either by ab initio calculations or perturbation theory. 

Finally, we discuss the effect of nonlinear coupling on domain growth, decoherence, and thermalization. As the wave functionals l/o of Ho are easily found, Eq. (16) leads to the wave functional beyond the Hartree approximation. Putting the perturbation terms (19) into Eq. (16), we first find the wave functional of the form... 

Equations 8.5-34 and -35 are nonlinearly coupled through T, since kA depends exponentially on T. The equations cannot therefore be treated independently, and there is no exact analytical solution for cA(r) and T(r). A numerical or approximate analytical solution results in tj expressed in terms of three dimensionless parameters ... 

The first term on the right-hand side corresponds to Eq. (2), whereas the second term describes dissipative effects that are induced in the system due to its coupling to the environment. The latter is modeled, as usual [32, 33], as the thermal (temperature T) ensemble of harmonic oscillators, with nonlinear coupling A Qiq) F( thermal bath, expressed in terms of nonlinear molecular and linear environment coupling operators Q(q) and F( qk )- As shown in Ref. 15, it is important to describe the dissipative term in Eq. (10) by making use of the non-Markovian expression... 

The static JT Hamiltonian with the anharmonic vibration and the nonlinear vibronic coupling terms is expressed by... 

Q-y2(Qxz-y2,Q2xy), IlcyiQx -y Qixy) the nth order polynomials of (Qx2-y2, Q2xy) transforming as (x2 — y2, 2ry), and gn a constant. The first term in equation (14) is the linear coupling, and the second one describes the nonlinear couplings. 

This coupling term kie(xi — Xe)2 is a phenomenological approximation of the lowest-order (cubic) potential energy terms that gives rise to nonlinear effects. These effects can be described by perturbative expansions of x (7,30,32,33) ... 

Eq. [33] according to the assumption of the classical character of this collective mode. Depending on the form of the coupling of the electron donor-acceptor subsystem to the solvent field, one may consider linear or nonlinear solvation models. The coupling term - Si -V in Eq. [32] represents the linear coupling model (L model) that results in a widely used linear response approximation. Some general properties of the bilinear coupling (Q model) are discussed below. 

The mathematical structure of the models is their unifying background systems of nonlinear coupled differential equations with eventually nonlocal terms. Approximate analytic solutions have been calculated for linearized or reduced models, and their asymptotic behaviors have been determined, while various numerical simulations have been performed for the complete models. The structure of the fixed points and their values and stability have been analyzed, and some preliminary correspondence between fixed points and morphological classes of galaxies is evident—for example, the parallelism between low and high gas content with elliptical and spiral galaxies, respectively. 

Let us now compare the mathematical structures of the selection Eqn. (III. 15) and the coupled systems of Eq. (III. 16) and (III. 17) The original equation had rs variables and one conservation relation and was linear apart from the mild nonlinearity caused by E. Equations (III. 16) and (III. 17) contain r + s variables only they fulfil two conservation relations but are highly nonlinear through the coupling terms. We recall from Appendix 9 that, For example,... 

A recent attempt at a direct stochastic theory by Weinberger [94] using the deterministic flow term as an external (precomputed) constraint should be mentioned here. The intractability of a large coupled system of second-order partial differential equations for the generating function is then reduced to a (nonlinearly coupled) system of ordinary differential equations. The price is the loss of proper population regulation and possible extinction. 

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