Single-wavenumber representations of nonlinear energies are required to investigate energy budget due to nonlinear interactions among Fourier modes in wave turbulence. While we have reported in a previous paper that the single-wavenumber representations successfully works for the Föppl–von Kármán equation, we will show here that for the Majda–McLaughlin–Tabak model the single-wavenumber representations of nonlinear energies is not necessarily unique. Introducing auxiliary variables composed differently from complex amplitudes, two natural representations of the nonlinear energy are obtained. It is numerically observed that the two kinds of the nonlinear-energy spectra, based on these two representations, are qualitatively similar, but the energy budgets are clearly different. To select the appropriate single-wavenumber representation of the nonlinear energy, the properties which an eligible single-wavenumber representation should have are discussed.
1. Zakharov, V. E., L¢vov, V. S., and Falkovich, G. Kolmogorov Spectra of Turbulence I: Wave Turbulence. Springer-Verlag, Berlin, 1992.
http://dx.doi.org/10.1007/978-3-642-50052-7
2. Dias, F., Guyenne, P., and Zakharov, V. E. Kolmogorov spectra of weak turbulence in media with two types of interacting waves. Phys. Lett. A, 2001, 291, 139–145.
http://dx.doi.org/10.1016/S0375-9601(01)00711-3
3. Miquel, B., Alexakis, A., and Mordant, N. Role of dissipation in flexural wave turbulence: from experimental spectrum to Kolmogorov–Zakharov spectrum. Phys. Rev. E, 2014, 89, 062925.
http://dx.doi.org/10.1103/PhysRevE.89.062925
4. Rumpf, B. and Biven, L. Weak turbulence and collapses in the Majda–McLaughlin–Tabak equation: fluxes in wavenumber and in amplitude space. Physica D, 2005, 204, 188–203.
http://dx.doi.org/10.1016/j.physd.2005.04.012
5. Yokoyama, N. and Takaoka, M. Single-wave-number representation of nonlinear energy spectrum in elastic-wave turbulence of the Föppl–von Kármán equation: energy decomposition analysis and energy budget. Phys. Rev. E, 2014, 90, 063004.
http://dx.doi.org/10.1103/PhysRevE.90.063004
6. Majda, A. J., McLaughlin, D. W., and Tabak, E. G. A one dimensional model for dispersive wave turbulence. J. Nonlinear Sci., 1997, 7, 9–44.
http://dx.doi.org/10.1007/BF02679124
7. Cai, D. and McLaughlin, D. W. Chaotic and turbulent behavior of unstable one-dimensional nonlinear dispersive waves. J. Math. Phys., 2000, 41, 4125–4153.
http://dx.doi.org/10.1063/1.533337
8. Cai, D., Majda, A. J., McLaughlin, D. W., and Tabak, E. G. Spectral bifurcations in dispersive wave turbulence. Proc. Natl. Acad. Sci. USA, 1999, 96, 14216–14221.
http://dx.doi.org/10.1073/pnas.96.25.14216
9. Cai, D., Majda, A. J., McLaughlin, D. W., and Tabak, E. G. Dispersive wave turbulence in one dimension. Physica D, 2001, 152–153, 551–572.
http://dx.doi.org/10.1016/S0167-2789(01)00193-2
10. Zakharov, V. E., Guyenne, P., Pushkarev, A. N., and Dias, F. Wave turbulence in one-dimensional models. Physica D, 2001, 152–153, 573–619.
http://dx.doi.org/10.1016/S0167-2789(01)00194-4
11. Pushkarev, A. and Zakharov, V. E. Quasibreathers in the MMT model. Physica D, 2013, 248, 55–61.
http://dx.doi.org/10.1016/j.physd.2013.01.003
