Solutions to Exercises

Abstract

In this Chapter, solutions for the exercises of Part II are given, and an abstract with the interaction coefficient for the gravity water waves is presented.

Appendix: Interaction Coefficient for the Deep Water Surface Waves

Dispersion relation for the deep water surface waves, $\omega = \sqrt{g k}$, does not allow three-wave resonances. Thus the leading process is four-wave, and by a quasi-identity canonical transformation the cubic interaction Hamiltonian (corresponding to quadratic nonlinear terms in the dynamical equation) can be removed from to the description, which transforms the Hamiltonian to the form (6.71). This task appears to be quite cumbersome, and the correct canonical transformation and the resulting Hamiltonian was obtained only in 1990 (24 years after finding the Zakharov-Filonenko spectrum for the gravity WT!) by Krasitskii [2]. Krasitskii’s derivation is quite complicated, and a more efficient procedure was described in ZLF book [3] in Appendix 3 for general four-wave systems. This method uses a trick of finding the canonical transformation as a infinitesimal-time evolution operator with an auxiliary Hamiltonian. Interested reader should read this place in ZLF, as it gives a very clear and detailed explanation of the method. The only remark we add here is that the resulting expression (A3.7) in ZLF can be significantly simplified by removing the non-resonant part (the second and the third lines in (A3.7)) by an appropriate choice of function $\tilde{W}$.

The most compact and, therefore, practically useful expression for the interaction coefficient for the deep water surface waves was obtained by Zakharov in [4]. Here, we reproduce it for reference purposes:

$$ \begin{aligned} W_{12}^{34} & = -\frac{1}{32{\pi^2}({k_1}{k_2}{k_3}{k_4})^{1/4}}\left\{-12{k_1}{k_2}{k_3}{k_4}\right.\\ &\quad - \frac{2}{g^2} ({\omega_1} +{\omega _2})^2 \left[{\omega _3}{\omega_4}(({\bf{k}}_1 \cdot{\bf k}_2) - k_1 k_2) + {\omega_1}{\omega _2}(( {\bf k}_3 \cdot {\bf k}_4) - k_3k_4) \right] \\ & \quad -\frac{2}{g^2} (\omega_1 - \omega_3)^2\left[{\omega _2}{\omega _4}(({\bf k}_1 \cdot {\bf k}_3) + {k_1}{k_3}) + {\omega _1}{\omega_3}(({\bf k}_2 \cdot{\bf k}_4) + k_2 k_4) \right]\\ & \quad - \frac{2}{g^2} (\omega_1- \omega_4)^2 \left[{\omega _2}{\omega_3}(({\bf k}_1 \cdot{\bf k}_4) + {k_1}{k_4}) +{\omega _1}{\omega _4}(({\bf k}_2 \cdot {\bf k}_3) +k_2 k_3) \right] \\& \quad + (({\bf k}_1\cdot {\bf k}_2) + k_1 k_2)(({\bf k}_3 \cdot {\bf k}_4) +k_3 k_4) \\&\quad + ( - ({\bf k}_1 \cdot {\bf k}_3) + k_1 k_3)( - ({\bf k}_2 \cdot {\bf k}_4) +{k_2}{k_4}) \\ & \quad + ( - ({{\bf k}_1}\cdot {\bf k}_4) + {k_1}{k_4})( -({{\bf k}_2}\cdot {\bf k}_3) + {k_2}{k_3}) \\ &\quad + 4{({\omega _1} + {\omega _2})^2}\frac{(({\bf {k}_1}\cdot{\bf k}_2) -{k_1}{k_2})(({{\bf k}_3}\cdot{\bf k}_4) -{k_3}{k_4})} {{\omega^2_{1 + 2}} -{{({\omega _1} + {\omega _2})}^2}} \\& \quad + 4{({\omega _1} - {\omega_3})^2} \frac{(({{\bf k}_1}\cdot {\bf k}_3) +{k_1}{k_3})(({{\bf k}_2}\cdot{\bf k}_4) +{k_2}{k_4})} {{\omega^2_{1 - 3}} -{{({\omega _1} - {\omega _3})}^2}} \\& \quad + 4{{({\omega _1} - {\omega_4})}^2}\frac{(({{\bf k}_1}\cdot{\bf k}_4) +{k_1}{k_4})(({{\bf k}_2}\cdot{\bf k}_3) +{k_2}{k_3})}{{\omega^2_{1 - 4}} -{{({\omega _1} - {\omega _4})}^2}}\}, \\ \end{aligned}$$

(7.16)

where we have used a shorthand notation ω₁₊₂ ≡ ω(k ₁ + k ₂), etc.

From (7.16), one can see that the four-wave interaction coefficient $W_{12}^{34}$ is zero if all four wavevectors are collinear. Thus, in 1D the gravity wave system is five-wave [5], see Hamiltonian (6.82).