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 (eg 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.
To obtain a formal transform (15) in the resonant case, one should revise a structure of the set (13) by modifying its right-hand side:
(16) ; ,
where the nonlinear terms. Here are the uniform-th order polynomials. These should consist of the resonant terms only. In this case the eqs. (16) are associated with the so-called normal forms .
Remarks
In practice the series are usually truncated up to first - or second-order terms in.
The theory of normal forms can be simply generalized in the case of the so-called essentially nonlinear systems, since the small parameter can be omitted in the expressions (12) - (16) without changes in the main result. The operator can depend also upon the spatial variables. p> Formally, the eigenvalues ​​of operator can be arbitrary complex numbers. This means that the resonances can be defined and classified even in appropriate nonlinear systems that should not be oscillatory one (eg in the case of evolution equations).
Resonance in multi-frequency systems
The resonance plays a principal role in the dynamical behavior of most physical systems. Intuitively, the resonance is associated with a particular case of a forced excitation of a linear oscillatory system. The excitation is accompanied with a more or less fast amplitude growth, as the natural frequency of the oscillatory system coincides with (or sufficiently close to) that of external harmonic force. In turn, in the case of the so-called parametric resonance one should refer to some kind of comparativeness between the natural frequency and the frequency of the parametric excitation. So that, the resonances can be simply classified, according to the above outlined scheme, by their orde...
