Abstract
In this chapter we present a general method of constructing the effective equation which describes the behavior of small-amplitude solutions for a nonlinear PDE in finite volume, provided that the linear part of the equation is a hamiltonian system with a pure imaginary discrete spectrum. The effective equation is obtained by retaining only the resonant terms of the nonlinearity (which may be hamiltonian, or may be not); the assertion that it describes the limiting behavior of small-amplitude solutions is a rigorous mathematical theorem. In particular, the method applies to the three- and four-wave systems. We demonstrate that different possible types of energy transport are covered by this method, depending on whether the set of resonances splits into finite clusters (this happens, e.g. in case of the Charney-Hasegawa-Mima equation), or is connected (this happens, e.g. in the case of the NLS equation if the space-dimension is at least two). For equations of the first type the energy transition to high frequencies does not hold, while for equations of the second type it may take place. Our method applies to various weakly nonlinear wave systems, appearing in plasma, meteorology and oceanography.
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Notes
- 1.
That is, \(\dot{\boldsymbol{\beta }}_{\mathbf{k}} = (d/dt)\boldsymbol{\beta }_{\mathbf{k}}\), \(\boldsymbol{\beta }_{\mathbf{k}} = \boldsymbol{\beta }_{\mathbf{k}}^{+} + i\boldsymbol{\beta }\mathbf{k}^{-}\), where \(\boldsymbol{\beta }_{\mathbf{k}}^{\pm }\) are standard independent real Wiener processes.
- 2.
- 3.
Note that, due to the fact that the function ψ is real, \(v_{\mathbf{k}} = v_{-\mathbf{k}}^{{\ast}}\).
- 4.
In the case of the non-forced equation the expectations should be taken with respect to the distribution of the initial data, while for the forced equation—with respect to the distribution of the forcing (and, possibly, of the initial data).
- 5.
Notice that, since the equation which we consider has a cubic nonlinearity, equations for moments of even order are decoupled from those for moments of odd order.
- 6.
Note that the relations, defining \(\varSigma _{\mathbf{k}}^{j}\) are not independent.
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Kuksin, S., Maiocchi, A. (2016). The Effective Equation Method. In: Tobisch, E. (eds) New Approaches to Nonlinear Waves. Lecture Notes in Physics, vol 908. Springer, Cham. https://doi.org/10.1007/978-3-319-20690-5_2
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