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Concept
Nonlinear resonance
Summary
In physics, nonlinear resonance is the occurrence of resonance in a nonlinear system. In nonlinear resonance the system behaviour – resonance frequencies and modes – depends on the amplitude of the oscillations, while for linear systems this is independent of amplitude. The mixing of modes in non-linear systems is termed resonant interaction.
Generically two types of resonances have to be distinguished – linear and nonlinear. From the physical point of view, they are defined by whether or not external force coincides with the eigen-frequency of the system (linear and nonlinear resonance correspondingly). Vibrational modes can interact in a resonant interaction when both the energy and momentum of the interacting modes is conserved. The conservation of energy implies that the sum of the frequencies of the modes must sum to zero:
with possibly different being eigen-frequencies of the linear part of some nonlinear partial differential equation. The is the wave vector associated with a mode; the integer subscripts being indexes into Fourier harmonics – or eigenmodes – see Fourier series. Accordingly, the frequency resonance condition is equivalent to a Diophantine equation with many unknowns. The problem of finding their solutions is equivalent to the Hilbert's tenth problem that is proven to be algorithmically unsolvable.
Main notions and results of the theory of nonlinear resonances are:
The use of dispersion relations appearing in various physical applications allows finding the solutions of the frequency resonance condition.
The set of resonances for a given dispersion function and the form of resonance conditions is partitioned into non-intersecting resonance clusters; dynamics of each cluster can be studied independently (at the appropriate time-scale). These are often called "bound waves", which cannot interact, as opposed to the "free waves", which can. A famous example is the soliton of the KdV equation: solitons can move through each other, without interacting.
Official source
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