Four-wave case.

Four-wave systems are similar to three-wave systems when small scale perturbations on the background of the large scale excitations are considered. Indeed, we show below that the quadratic part of a four-wave Hamiltonian of small scale perturbation has the form (4). We start from a standard four-wave Hamiltonian [10]:

$$H_4=\int\Omega _\textbf{k}\vert a_\textbf{k}\vert^2d\textbf{k}+\f... ...textbf{l}}_{\textbf{m}\textbf{s}} d\textbf{k}d\textbf{l}d\textbf{m}d\textbf{s},$$ (11)

where $T$ is an interaction coefficient. The corresponding equation of motion takes the form

$$i\dot{a}_\textbf{k}=\Omega _{\textbf{k}}a_{\textbf{k}}+\int T^{\t... ...\textbf{k}\textbf{l}}_{\textbf{m}\textbf{s}} d\textbf{l}d\textbf{m}d\textbf{s}.$$ (12)

We then consider a perturbed solution $a_\textbf{k}=C_\textbf{k}+c_\textbf{k}$ where $c_\textbf{k}$ is a small-scale perturbation. Assuming that $C_\textbf{k}$ is an exact solution to the equation of motion with Hamiltonian (13), we obtain the following equation of motion for $c_\textbf{k}$

$$i\dot{c}_\textbf{k} = \Omega _{\textbf{k}}c_{\textbf{k}}+\int T^{\textbf{k}\textbf{l}}_... ...{\textbf{k}\textbf{l}}_{\textbf{m}\textbf{s}}d\textbf{l}d\textbf{m}d\textbf{s}+$$

$$\int T^{\textbf{k}\textbf{l}}_{\textbf{m}\textbf{s}}(C_\textbf{l}... ...{\textbf{k}\textbf{l}}_{\textbf{m}\textbf{s}}d\textbf{l}d\textbf{m}d\textbf{s}.$$

Since $C_\textbf{k}$ is a known large scale solution, we obtain

$$i\dot{c}_\textbf{k} = \int A(\textbf{k},\textbf{s})c_\textbf{s}d\textbf{s}+\frac{1}{2}\int B(\textbf{k},\textbf{s})c_{-\textbf{s}}^* d\textbf{s}$$

$$+ \int \left[ \frac{1}{2} W^{\textbf{k}}_{\textbf{l}\textbf{m}}c_\t... ...xtbf{k}\textbf{m}})^* c_\textbf{l}c_\textbf{m}^* \right] d\textbf{l}d\textbf{m}$$

$$+ \int T^{\textbf{k}\textbf{l}}_{\textbf{m}\textbf{s}} c_\textbf{l}... ...\textbf{k}\textbf{l}}_{\textbf{m}\textbf{s}}d\textbf{l}d\textbf{m}d\textbf{s}.,$$ (13)

where we defined the kernels $A(\textbf{k},\textbf{s})$ and $B(\textbf{k},\textbf{s})$ as

$$A(\textbf{k},\textbf{s}) = \Omega _\textbf{s}\delta_\textbf{s}^\textbf{k}+2\int T^{\textbf{k... ...bf{m}\delta^{\textbf{k}\textbf{l}}_{\textbf{m}\textbf{s}}d\textbf{l}d\textbf{m}$$ (14)

$$B(\textbf{k},\textbf{s}) = 2\int T^{\textbf{k},-\textbf{s}}_{\textbf{m},\textbf{l}}C_\textbf... ...}\delta^{\textbf{k},-\textbf{s}}_{\textbf{m},\textbf{l}}d\textbf{l}d\textbf{m},$$

and

$$W^{\textbf{k}}_{\textbf{l}\textbf{m}} = 2 \int T^{\textbf{k}\text... ..._\textbf{s}^* \delta^{\textbf{k}\textbf{s}}_{\textbf{l}\textbf{m}} d\textbf{s}.$$

The linear part of equation (14) has the same form as the linear part of the corresponding equation obtained for the three-wave case (9). Thus, this linear part corresponds to the same first two terms as in Hamiltonian (11). Note also similarity of the quadratic terms in (9) and (14). Note that (14) correspond to the following Hamiltonian

$$H = \int A(\textbf{k},\textbf{l})c_\textbf{l}c_\textbf{k}^*d\textbf{l... ...k},\textbf{l})c_{-\textbf{l}}^*c_\textbf{k}^*+c.c.\right]d\textbf{k}d\textbf{l}$$

$$+ \frac{1}{2}\int [W^\textbf{k}_{\textbf{l}\textbf{m}}c_\textbf{k}^*c_\textbf{l}c_\textbf{m}+c.c.]d\textbf{k}d\textbf{l}d\textbf{m}+$$

$$+\frac{1}{2}\int T^{\textbf{k}\textbf{l}}_{\textbf{m}\textbf{s}}c... ...textbf{l}}_{\textbf{m}\textbf{s}} d\textbf{k}d\textbf{l}d\textbf{m}d\textbf{s},$$

This appears to be a standard Hamiltonian for the inhomogeneous system with four-wave interactions. Indeed, the quadratic part (first line) of this Hamiltonian is the Hamiltonian (4). Cubic term is the three-wave interactions with the background large scale wave (i.e. four wave interaction where the role of the fourth wave is assumed by the background wave). Notice that unlike traditional three wave interactions in a homogeneous environment, momentum is not conserved by this term. This is the effect of breaking of spatial symmetry by an inhomogeneous background. Lastly, the quartic term (third line) is the standard four wave interactions Hamiltonian. We show in this paper that the quadratic part of this Hamiltonian may be reduced to the novel canonical Hamiltonian for spatially inhomogeneous systems (5).

In this section we have demonstrated that if the general wave system is dominated by three-wave or four-wave interactions, and consists of short scale waves superimposed on known large-scale motion, its quadratic Hamiltonian is given by the (4).