Setting the stage I: Dynamical Equations of motion

Wave turbulence formulation deals with a many-wave system with dispersion and weak nonlinearity. For systematic derivations one needs to start from Hamiltonian equation of motion. Here we consider a system of weakly interacting waves in a periodic box [1],

$$i\dot c_l = \frac{\partial {\cal H}}{\partial \bar c_l},$$ (1)

where $c_l$ 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,

$${\cal H} = {\cal H}_2 + {\cal H}_3+ {\cal H}_4 + {\cal H}_5 + \dots,$$ (2)

where $H_j$ is a term proportional to product of $j$ amplitudes $c_l$,

$${\cal H}_j = \sum\limits_{q_1,q_2,q_3, \dots q_n, p_1,p_2,\dots p... ...ts \bar c_{q_m} c_{p_1} c_{p_2}\dots c_{p_m}+{\rm c.c}.\right), \ \ \ n+m=j \\$$

where $q_1,q_2,q_3, \dots q_n$ and $p_1,p_2,\dots p_m$ are wavevectors on a $d$-dimensional Fourier space lattice. Such general $j$-wave Hamiltonian describe the wave-wave interactions where $n$ waves collide to create $m$ waves. Here $T^{q_1 q_2 \dots q_n}_{p1,p2 \dots p_m}$ represents the amplitude of the $n\to m$ 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

$${\cal H}_2 = \sum_{n=1}^\infty \omega_n\vert c_n\vert^2.$$ (3)

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

$${\cal H}_3= \epsilon \sum_{l,m,n=1}^\infty V^l_{mn} \bar c_{l} c_m c_n\delta^l_{m+n}+c.c.,$$

where $\epsilon \ll 1$ is a formal parameter corresponding to small nonlinearity ( $\epsilon$ is proportional to the small amplitude whereas $c_n$ is normalised so that $c_n \sim 1$.) 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

$$\sum_{l,m,n=1}^\infty \ U^{lmn} c_{l} c_m c_n\delta_{l+m+n}+c.c.$$

It can be shown however that for systems that are dominated by three-wave resonances such terms do not contribute to long term dynamics of systems. We therefore choose to omit those terms.

The most general four-wave Hamiltonian will have $1\to 3$, $3\to 1$, $2\to 2$, $4\to 0$ and $0\to 4$ terms. Nevertheless $1\to 3$, $3\to 1$, $4\to 0$ and $0\to 4$ terms can be excluded from Hamiltonian by appropriate canonical transformations, so that we limit our consideration to only $2\to 2$ terms of ${\cal H}_4$, namely

$${\cal H}_4= \epsilon^2 \sum_{m,n,\mu,\nu=1}^\infty W^{lm}_{\mu\nu} \bar c_{l} \bar c_m c_\mu c_\nu.$$

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 ${\cal H}_3$ describes all the relevant dynamics and ${\cal H}_4$ can be neglected, and the systems where the three-wave resonance conditions cannot be satisfied, so that the ${\cal H}_3$ 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 ${\cal H}_3$ or ${\cal H}_4$, 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],

$$\omega_j = \sqrt{\sigma k^3},$$ (4)

and

$$V^l_{mn} = {1 \over 8 \pi \sqrt{2 \sigma}} (\omega_{l} \omega_m \... ...r ( k_l k_m)^{1/2} k_n } - {L_{k_l, -k_n} \over ( k_l k_n)^{1/2} k_m } \right],$$ (5)

where

$$L_{k_m, k_n} = ({\bf k}_m \cdot {\bf k}_n) + k_m k_n$$ (6)

and $\sigma$ is the surface tension coefficient.

For the Rossby waves [13,14],

$$\omega_j = {\beta k_{jx} \over 1 + \rho^2 k_j^2},$$ (7)

and

$$V^l_{mn} = -{i \beta \over 4 \pi} \vert k_{lx} k_{mx} k_{nx} \ver... ... - { k_{my} \over 1 + \rho^2 k_m^2} - { k_{ny} \over 1 + \rho^2 k_n^2} \right),$$ (8)

where $\beta$ is the gradient of the Coriolis parameter and $\rho$ is the Rossby deformation radius.

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

$$\omega_j=\vert k_j\vert^2, \hspace{1cm} W^{lm}_{\mu \nu} = 1.$$ (9)

The surface gravity waves are on the other extreme. The frequency is $\omega = \sqrt{gk}$ but the matrix element is given by notoriously long expressions which can be found in [1,17].