Реферат: Nonlinear multi-wave coupling and resonance in elastic structures

where and are unknown coefficients which have to be determined.

By substituting the transform (14) into eqs. (12), we obtain the following partial differential equations to define :

(16) .

It is obvious that the eigenvalues of the operator acting on the polynomial components of (i. e. ) are the linear integer-valued combinational values of the operator given at various arguments of the wave vector .

In the lowest-order approximation in eqs. (16) read

.

The polynomial components of are associated with their eigenvalues , i. e. , where

or ,

while in the lower-order approximation in .

So, if at least the one eigenvalue of approaches zero, then the corresponding coefficient of the transform (15) tends to infinity. Otherwise, if , then represents the lowest term of a formal expansion in .

Analogously, in the second-order approximation in :

the eigenvalues of can be written in the same manner, i. e. , where , etc.

By continuing the similar formal iterations one can define the transform (15). Thus, the sets (12) and (13), even in the absence of eigenvalues equal to zeroes, are associated with formally equivalent dynamical systems, since the function can be a divergent function. If is an analytical function, then these systems are analytically equivalent . Otherwise, if the eigenvalue in the -order approximation, then eqs. (12) cannot be simply reduced to eqs. (13), since the system (12) experiences a resonance.

For example, the most important 3-order resonances include

triple-wave resonant processes, when and ;

generation of the second harmonic, as and .

The most important 4-order resonant cases are the following:

four-wave resonant processes, when ; (interaction of two wave couples); or when and (break-up of the high-frequency mode into tree waves);

degenerated triple-wave resonant processes at and ;

generation of the third harmonic, as and .

These resonances are mainly characterized by the amplitude modulation , the depth of which increases as the phase detuning approaches to some constant (e. g. to zero, if consider 3-order resonances). The waves satisfying the phase matching conditions form the so-called resonant ensembles .

Finally, in the second-order approximation, the so-called “non-resonant" interactions always take place. The phase matching conditions read the following degenerated expressions

cross-interactions of a wave pair at and ;

self-action of a single wave as and .

Non-resonant coupling is characterized as a rule by a phase modulation .

The principal proposition of this section is following. If any nonlinear system (12) does not have any resonance, beginning from the order up to the order , then the nonlinearity produces just small corrections to the linear field solutions. These corrections are of the same order that an amount of the nonlinearity up to times .