Setting the stage I: Dynamical Equations of motion

where is often called the field variable. It represents the amplitude of the interacting plane wave. The Hamiltonian is represented as an expansion in powers of small amplitude,

where is a term proportional to product of amplitudes ,

where and are wavevectors on a -dimensional Fourier space lattice. Such general -wave Hamiltonian describe the wave-wave interactions where waves collide to create waves. Here represents the amplitude of the process. In this paper we are going to consider expansions of Hamiltonians up to forth order in wave amplitude.

Under rather general conditions the quadratic part of a Hamiltonian, which correspond to a linear equation of motion, can be diagonalised to the form

This form of Hamiltonian correspond to noninteracting (linear) waves. First correction to the quadratic Hamiltonian is a cubic Hamiltonian, which describes the processes of decaying of single wave into two waves or confluence of two waves into a single one. Such a Hamiltonian has the form

where is a formal parameter corresponding to small nonlinearity ( is proportional to the small amplitude whereas is normalised so that .) Most general form of three-wave Hamiltonian would also have terms describing the confluence of three waves or spontaneous appearance of three waves out of vacuum. Such a terms would have a form

The most general four-wave Hamiltonian will have , , , and terms. Nevertheless , ,
and terms can be excluded from Hamiltonian by
appropriate canonical transformations, so that we limit our consideration to
only terms of
, namely

It turns out that generically most of the weakly nonlinear systems can be separated into two major classes: the ones dominated by three-wave interactions, so that describes all the relevant dynamics and can be neglected, and the systems where the three-wave resonance conditions cannot be satisfied, so that the can be eliminated from a Hamiltonian by an appropriate near-identical canonical transformation [25]. Consequently, for the purpose of this paper we are going to neglect either or , and study the case of resonant three-wave or four-wave interactions.

Examples of three-wave system include the water surface capillary waves, internal waves in the ocean and Rossby waves. The most common examples of the four-wave systems are the surface gravity waves and waves in the NLS model of nonlinear optical systems and Bose-Einstein condensates. For reference we will give expressions for the frequencies and the interaction coefficients corresponding to these examples.

For the *capillary waves* we have [1,5],

(4) |

and

(5) |

where

(6) |

and is the surface tension coefficient.

(7) |

and

(8) |

where is the gradient of the Coriolis parameter and is the Rossby deformation radius.

The simplest expressions correspond to the *NLS waves* [15,7],

(9) |

The