Lushnikov, Pavel M.; Zakharov, Vladimir E. On optimal canonical variables in the theory of ideal fluid with free surface. (English) Zbl 1070.35035 Physica D 203, No. 1-2, 9-29 (2005); erratum ibid. 206, No. 3-4, 275 (2005). Summary: Dynamics of ideal fluid with free surface can be effectively solved by perturbing the Hamiltonian in the weak nonlinearity limit. However it is shown that perturbation theory, which includes third and fourth order terms in the Hamiltonian, results in ill-posed equations because of short wavelength instability. To fix that problem we introduce the canonical Hamiltonian transformation from original physical variables to new variables for which instability is absent. Cited in 2 Documents MSC: 35Q35 PDEs in connection with fluid mechanics 76B45 Capillarity (surface tension) for incompressible inviscid fluids 35R25 Ill-posed problems for PDEs Keywords:Hamiltonian; surface waves; ill-posedness; canonical transformtion PDFBibTeX XMLCite \textit{P. M. Lushnikov} and \textit{V. E. Zakharov}, Physica D 203, No. 1--2, 9--29 (2005; Zbl 1070.35035) Full Text: DOI arXiv References: [1] Zakharov, V. E.; Filonenko, N., The energy spectrum for stochastic oscillation of a fluid’s surface, Doklady Akad. Nauk, 170, 1292-1295 (1966) [2] Zakharov, V. E., Stability of periodic waves of finite amplitude on a surface of deep fluid, J. Appl. Mech. Tech. Phys., 2, 190-198 (1968) [3] Zakharov, V., Statistical theory of gravity and capillary waves on the surface of finite-depth fluid, Eur. J. Mech. B/Fluids, 18, 327-344 (1999) · Zbl 0960.76014 [4] Craig, W.; Worfolk, P. A., An integrable normal form for water waves in infinite depth, Physica D, 84, 513-531 (1995) · Zbl 0883.35092 [5] Dyachenko, A. I.; Korotkevich, A. O.; Zakharov, V. E., Decay of the monochromatic capillary wave, JETP Lett., 77, 477-481 (2003) [6] Dyachenko, A. I.; Korotkevich, A. O.; Zakharov, V. E., Weak turbulence of gravity waves, JETP Lett., 77, 546-550 (2003) [7] Dyachenko, A. I.; Korotkevich, A. O.; Zakharov, V. E., Weak turbulent Kolmogorov spectrum for surface gravity waves, Phys. Rev. Lett., 92, 134501 (2004) [8] Onorato, M.; Osborne, A. R.; Serio, M.; Resio, D.; Pushkarev, A.; Zakharov, V. E.; Brandini, C., Freely decaying weak turbulence for sea surface gravity waves, Phys. Rev. Lett., 89, 144501(4) (2002) [9] Pushkarev, A. N., On the Kolmogorov and frozen turbulence in numerical simulation of capillary waves, Eur. J. Mech. B/Fluids, 18, 345-352 (1999) · Zbl 0943.76035 [10] Pushkarev, A. N.; Zakharov, V. E., Turbulence of capillary waves, Phys. Rev. Lett., 76, 3320-3323 (1996) [11] Pushkarev, A. N.; Zakharov, V. E., Turbulence of capillary waves—theory and numerical simulations, Physica D, 135, 98-116 (2000) · Zbl 0960.76039 [12] Tanaka, M., Verification of Hasselmann’s energy transfer among surface gravity waves by direct numerical simulations of primitive equations, J. Fluid Mech., 444, 199-221 (2001) · Zbl 0995.76011 [13] Zakharov, V. E.; Dyachenko, A. I.; Vasilyev, O. A., New method for numerical simulation of a nonstationary potential flow of imcompressible fluid with a free surface, Eur. J. Mech. B/Fluids, 21, 283-291 (2002) · Zbl 1016.76062 [14] Krasitskii, V. P., Canonical transformation in a theory of weakly nonlinear waves with a nondecay dispersion law, Sov. Phys. JETP, 71, 921-927 (1990) [15] Kuznetsov, E. A.; Spector, M. D.; Zakharov, V. E., Surface singularities of ideal fluid, Phys. Lett. A, 182, 387 (1993) [16] Kuznetsov, E. A.; Spector, M. D.; Zakharov, V. E., Formation of singularities on the free surface of an ideal fluid, Phys. Rev. E, 49, 1283 (1994) [17] Dyachenko, A. I.; Zakharov, V. E.; Kuznetsov, E. A., Nonlinear dynamics of the free surface of an ideal fluid, Plasma Phys. Rep., 22, 829 (1996) [18] A.I. Dyachenko, Private communications, 2004.; A.I. Dyachenko, Private communications, 2004. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.