>
whose analytical solutions can be written immediately as a superposition of harmonic waves
,
where are constant complex amplitudes; is the number of normal waves of the-th type, so that (for instance, if the operator is a polynomial, then, where is a scalar, is a constant vector, is some differentiable function. For more detail see [6]). p> A question is following. What is the difference between these two systems, or in other words, how the small nonlinearity is effective ?
According to a method of normal forms (see for example [7,8]), we look for a solution to eqs. (12) in the form of a quasi-automorphism, i. e.
(14)
where denotes an unknown-dimensional vector function, whose components can be represented as formal power series in, i. e. a quasi-bilinear form:
(15),
for example
В
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
В
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:
В
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. p> 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 ; p> generation of...
