A note on KAM for gravity-capillary water waves
Journées équations aux dérivées partielles (2016), Talk no. 7, 18 p.

We present the result and the ideas of the recent paper [8] (obtained in collaboration with M. Berti) concerning the existence of Cantor families of small amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean, with infinite depth, in irrotational regime, under the action of gravity and surface tension at the free boundary. These quasi-periodic solutions are linearly stable.

Published online:
DOI: 10.5802/jedp.648
Classification: 76B15,  37K55,  76D45,  37K50
Keywords: KAM for PDEs, water waves, quasi-periodic solutions.
Montalto, Riccardo 1

1 University of Zürich Winterthurerstrasse 190 CH-8057, Zürich Switzerland
@article{JEDP_2016____A7_0,
     author = {Montalto, Riccardo},
     title = {A note on {KAM} for gravity-capillary water waves},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     note = {talk:7},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2016},
     doi = {10.5802/jedp.648},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jedp.648/}
}
TY  - JOUR
AU  - Montalto, Riccardo
TI  - A note on KAM for gravity-capillary water waves
JO  - Journées équations aux dérivées partielles
N1  - talk:7
PY  - 2016
DA  - 2016///
PB  - Groupement de recherche 2434 du CNRS
UR  - http://archive.numdam.org/articles/10.5802/jedp.648/
UR  - https://doi.org/10.5802/jedp.648
DO  - 10.5802/jedp.648
LA  - en
ID  - JEDP_2016____A7_0
ER  - 
%0 Journal Article
%A Montalto, Riccardo
%T A note on KAM for gravity-capillary water waves
%J Journées équations aux dérivées partielles
%Z talk:7
%D 2016
%I Groupement de recherche 2434 du CNRS
%U https://doi.org/10.5802/jedp.648
%R 10.5802/jedp.648
%G en
%F JEDP_2016____A7_0
Montalto, Riccardo. A note on KAM for gravity-capillary water waves. Journées équations aux dérivées partielles (2016), Talk no. 7, 18 p. doi : 10.5802/jedp.648. http://archive.numdam.org/articles/10.5802/jedp.648/

[1] Alazard T., Baldi P. Gravity capillary standing water waves. Arch. Rat. Mech. Anal., 217, 3, 741-830, 2015.

[2] Baldi P., Berti M., Montalto R. A note on KAM theory for quasi-linear and fully nonlinear KdV. Rend. Lincei Mat. Appl., 24, 437-450, 2013.

[3] Baldi P., Berti M., Montalto R. KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation. Math. Annalen, 359, 1-2, 471-536, 2014.

[4] Baldi P., Berti M., Montalto R. KAM for quasi-linear KdV, C. R. Acad. Sci. Paris, Ser. I 352, 603-607, 2014.

[5] Baldi P., Berti M., Montalto R. KAM for autonomous quasi-linear perturbations of KdV. Ann. I. H. Poincaré (C) Anal. Non Linéaire, 33, 1589-1638, 2016.

[6] Baldi P., Berti M., Montalto R. KAM for autonomous quasi-linear perturbations of mKdV. Bollettino Unione Matematica Italiana, 9, 143-188, 2016.

[7] Bambusi D., Berti M., Magistrelli E. Degenerate KAM theory for partial differential equations. Journal Diff. Equations, 250, 8, 3379-3397, 2011.

[8] Berti M., Montalto R. Quasi-periodic standing wave solutions of gravity-capillary water waves, arXiv:1602.02411 .

[9] Berti M., Biasco L., Procesi M. KAM theory for the Hamiltonian derivative wave equation. Ann. Sci. Éc. Norm. Supér. (4), 46(2):301-373, 2013.

[10] Berti M., Biasco L., Procesi M. KAM for Reversible Derivative Wave Equations. Arch. Rat. Mech. Anal., 212(3):905-955, 2014.

[11] Berti M., Bolle P. A Nash-Moser approach to KAM theory. Fields Institute Communications, 255-284, special volume “Hamiltonian PDEs and Applications”, 2015.

[12] Brezis H., Coron J.-M., Nirenberg L., Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz, Comm. Pure Appl. Math. 33, no. 5, 667–684, 1980.

[13] Craig W., Nicholls D. Travelling two and three dimensional capillary gravity water waves. SIAM J. Math. Anal., 32(2):323–359 (electronic), 2000.

[14] Craig W., Sulem C. Numerical simulation of gravity waves. J. Comput. Phys., 108(1):73-83, 1993.

[15] Craig W., Sulem C. Normal form transformations for capillary-gravity water waves. Field Institute Communications, 73-110, special volume “Hamiltonian PDEs and Applications”, 2015.

[16] Craig W., Wayne C.E., Newton’s method and periodic solutions of nonlinear wave equation, Comm. Pure Appl. Math. 46, 1409-1498, 1993.

[17] Fejoz J. Démonstration du théoréme d’ Arnold sur la stabilité du systéme planétaire (d’ aprés Herman). Ergodic Theory Dynam. Systems 24 (5), 1521-1582, 2004.

[18] Feola R., Procesi M. Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, J. Diff. Eq., 259, no. 7, 3389-3447, 2015.

[19] Iooss G., Plotnikov P. Existence of multimodal standing gravity waves. J. Math. Fluid Mech., 7, 349–364, 2005.

[20] Iooss G., Plotnikov P. Multimodal standing gravity waves: a completely resonant system. J. Math. Fluid Mech., 7(suppl. 1), 110-126, 2005.

[21] Iooss G., Plotnikov P. Small divisor problem in the theory of three-dimensional water gravity waves. Mem. Amer. Math. Soc., 200(940):viii+128, 2009.

[22] Iooss G., Plotnikov P. Asymmetrical tridimensional travelling gravity waves. Arch. Rat. Mech. Anal., 200(3):789-880, 2011.

[23] Iooss G., Plotnikov P., Toland J. Standing waves on an infinitely deep perfect fluid under gravity. Arch. Rat. Mech. Anal., 177(3):367–478, 2005.

[24] Levi-Civita T. Détermination rigoureuse des ondes permanentes d’ampleur finie. Math. Ann., 93 , pp. 264-314, 1925.

[25] Liu J., Yuan X. A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations. Comm. Math. Phys., 307(3), 629–673, 2011.

[26] Kappeler T., Pöschel J. KAM and KdV, Springer, 2003.

[27] Kuksin S., Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum, Funktsional. Anal. i Prilozhen. 21, no. 3, 22–37, 95, 1987.

[28] Kuksin S. Analysis of Hamiltonian PDEs, volume 19 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2000.

[29] Montalto R. Quasi-periodic solutions of forced Kirchhoff equation. NoDEA, Nonlinear Differ. Equ. Appl., in press, DOI: 10.1007/s00030-017-0432-3, 2017.

[30] Plotnikov P., Toland J. Nash-Moser theory for standing water waves. Arch. Rat. Mech. Anal., 159(1):1–83, 2001.

[31] Rabinowitz P., Periodic solutions of nonlinear hyperbolic partial differential equations, Part I. Comm. Pure Appl. Math. 20, 145–205 (1967)

[32] Rabinowitz P., Periodic solutions of nonlinear hyperbolic partial differential equations. Part II. Comm. Pure Appl. Math. 22, 15–39 (1969)

[33] Rüssmann H. Invariant tori in non-degenerate nearly integrable Hamiltonian systems. Regul. Chaotic Dyn. 6 (2), 119-204, 2001.

[34] Wayne E., Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys. 127, 479-528, 1990.

[35] Zakharov V. Stability of periodic waves of finite amplitude on the surface of a deep fluid. Journal of Applied Mechanics and Technical Physics, 9(2):190–194, 1968.

[36] Zhang J., Gao M., Yuan X. KAM tori for reversible partial differential equations. Nonlinearity, 24(4):1189-1228, 2011.

Cited by Sources: