Time-periodic standing waves are demonstrated to be low-dimensional by use of the proper orthogonal decomposition (POD). Moreover, the nonlinear dynamics of the system restricted to this low-dimensional linear subspace are shown to accurately recover the spatio-temporal full PDE dynamics. A global set of modes, generated with sequential POD, are then used to produce time-periodic standing wave branches as a function of the period. This representation quantitatively reproduces the entire branch, including both large- and small amplitude solutions, using only a few POD modes. This technique offers a new direction of exploration for this challenging problem, including an efficient way to characterize the bifurcation structure and stability of these solutions.


  1. proper orthogonal decomposition
  2. periodic orbits
  3. standing water waves

MSC codes

  1. 37C27
  2. 62H25
  3. 65M70

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C. J. Amick, L. E. Fraenkel, and J. F. Toland, On the Stokes conjecture for the wave of extreme form, Acta Math., 148 (1982), pp. 192--214.
G. R. Baker, D. I. Meiron, and S. A. Orszag, Generalized vortex methods for free-surface flow problems, J. Fluid Mech., 123 (1982), pp. 477--501.
T. J. Bridges, Wave breaking and the surface velocity field for three-dimensional water waves, Nonlinearity, 22 (2009), pp. 947--953.
D. S. Broomhead and G. P. King, Extracting qualitative dynamics from experimental data, Phys. D, 20 (1986), pp. 217--236.
P. J. Bryant and M. Stiassnie, Different forms for nonlinear standing waves in deep water, J. Fluid Mech., 272 (1994), pp. 135--156.
J. Chandezon, D. Maystre, and G. Raoult, A new theoretical method for diffraction gratings and its numerical application, J. Optics, 11 (1980), pp. 235--241.
E. A. Christensen, M. Brøns, and J. N. Sørensen, Evaluation of proper orthogonal decomposition-based decomposition techniques applied to parameter-dependent nonturbulent flows, SIAM J. Sci. Comput., 21 (2000), pp. 1419--1434.
P. Concus, Standing capillary-gravity waves of finite amplitude, J. Fluid Mech., 14 (1962), pp. 568--576.
W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), pp. 73--83.
A. D. D. Craik, George Gabriel Stokes on water wave theory, in Ann. Rev. Fluid Mech. 37, Annual Reviews, Palo Alto, CA, 2005, pp. 23--42.
J. Cruz, Ocean Wave Energy: Current Status and Future Perspectives, Springer-Verlag, Berlin, 2008.
P. del Sastre and R. Bermejo, The POD technique for computing bifurcation diagrams: A comparison among different models in fluids, in Numerical Mathematics and Advanced Applications, A. B. de Castro, D. Gómez, P. Quintela, and P. Salgado, eds., Springer, Berlin, Heidelberg, 2006, pp. 880--888.
F. Dias and T. J. Bridges, The numerical computation of freely propagating time-dependent irrotational water waves, Fluid Dynam. Res., 38 (2006), p. 803.
E. Ding, E. Shlizerman, and J. N. Kutz, Modeling multipulsing transition in ring cavity lasers with proper orthogonal decomposition, Phys. Rev. A, 82 (2010), 23823.
P. Druault and C. Chaillou, Use of proper orthogonal decomposition for reconstructing the 3D in-cylinder mean-flow field from PIV data, C. R. Mécanique, 335 (2007), pp. 42--47.
A. Dyachenko and V. Zakharov, Modulation instability of Stokes wave $\rightarrow$ freak wave, J. Exper. Theor. Phys. Lett., 81 (2005), pp. 255--259.
A. L. Dychenko, V. E. Zakharov, and E. A. Kuznetsov, Nonlinear dynamics on the free surface of an ideal fluid, Plasma Phys. Rep., 22 (1996), pp. 916--928.
J. Eggers and M. A. Fontelos, The role of self-similarity in singularities of partial differential equations, Nonlinearity, 22 (2009), pp. R1--R44.
J. D. Fenton and M. M. Rienecker, A Fourier method for solving nonlinear water-wave problems: Application to solitary-wave interactions, J. Fluid Mech., 118 (1982), pp. 411--443.
G. Haller, Chaos Near Resonance, Appl. Math. Sci. 138, Springer-Verlag, New York, 1999.
J. L. Hammack, A note on tsunamis: Their generation and propagation in an ocean of uniform depth, J. Fluid Mech., 60 (1973), pp. 769--799.
M. Hinze and S. Volkwein, Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: Error estimates and suboptimal control, in Dimension Reduction of Large-Scale Systems, Lecture Notes in Comput. Appl. Math., D. Sorensen, P. Benner, and V. Mehrmann, eds., Springer, Berlin, Heidelberg, 2005, pp. 261--306.
P. Holmes, J. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, UK, 1996.
M. Ilak and C. W. Rowley, Reduced-order modeling of channel flow using traveling POD and balanced POD, in Proceedings of the 3rd AIAA Flow Control Conference, San Francisco, 2006, 2006-3194.
M. Ilak and C. W. Rowley, Modeling of transitional channel flow using balanced proper orthogonal decomposition, Phys. Fluids, 20 (2008), 034103.
G. Iooss, P. I. Plotnikov, and J. F. Toland, Standing waves on an infinitely deep perfect fluid under gravity, Arch. Ration. Mech. Anal., 177 (2005), pp. 367--478.
L. Jiang, C. Ting, M. Perlin, and W. W. Schultz, Moderate and steep Faraday waves: Instabilities, modulation and temporal asymmetries, J. Fluid Mech., 329 (1996), pp. 275--307.
R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge, UK, 1997.
Y. Kervella, D. Dutykh, and F. Dias, Comparison between three-dimensional linear and nonlinear tsunami generation models, Theor. Comput. Fluid Dyn., 21 (2007), pp. 245--269.
K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems, Numer. Math., 90 (2001), pp. 117--148.
H. Lamb, Hydrodynamics, Cambridge University Press, Cambridge, UK, 1895.
J. A. Lee and M. Verleysen, Nonlinear Dimensionality Reduction, Springer, New York, 2007.
A. Liakopoulos, P. A. Blythe, and H. Gunes, A reduced dynamical model of convective flows in tall laterally heated cavities, Proc. Roy. Soc. Ser. A, 453 (1997), pp. 663--672.
M. S. Longuet-Higgins and M. J. H. Fox, Theory of the almost-highest wave: The inner solution, J. Fluid Mech., 80 (1977), pp. 721--741.
G. N. Mercer and A. J. Roberts, Standing waves in deep water: Their stability and extreme form, Phys. Fluids A, 4 (1992), pp. 259--269.
G. N. Mercer and A. J. Roberts, The form of standing waves on finite depth water, Wave Motion, 19 (1994), pp. 233--244.
P. A. Milewski, J.-M. Vanden-Broeck, and Z. Wang, Dynamics of steep two-dimensional gravity capillary solitary waves, J. Fluid Mech., 664 (2010), pp. 466--477.
D. Nicholls, Boundary perturbation methods for water waves, GAMM-Mitt., 30 (2007), pp. 44--74.
J. Nocedal and S. J. Wright, Numerical Optimization, Springer, New York, 1999.
M. Okamura, Standing gravity waves of large amplitude on deep water, Wave Motion, 37 (2003), pp. 173--182.
M. Okamura, Almost limiting short-crested gravity waves in deep water, J. Fluid Mech., 646 (2010), pp. 481--503.
W. G. Penney and A. T. Price, Finite periodic stationary gravity waves in a perfect liquid, Part II, Phil. Trans. R. Soc. London A, 244 (1952), pp. 254--284.
N. Périnet, D. Juric, and L. S. Tuckerman, Numerical simulation of Faraday waves, J. Fluid Mech., 635 (2009), pp. 1--26.
N. Phillips, A coordinate system having some special advantages for numerical forecasting, J. Meteor., 14 (1957), pp. 184--185.
B. Podvin and P. Le Quéré, Low-order models for flow in a differentially heated cavity, Phys. Fluids, 13 (2011), pp. 3204--3214.
J. Rayleigh, Deep water waves, progressive or stationary, to the third order approximation, Proc. R. Soc. A, 91 (1915), pp. 345--353.
P. J. Schmid, K. E. Meye, and O. Pust, Dynamic mode decomposition and proper orthogonal decomposition of flow in a lid-driven cylindrical cavity, in Proceedings of the 8th International Symposium on Particle Image Velocimetry, Melbourne, Victoria, Australia, 2009, 0186.
W. Schultz, J. M. Vanden-Broeck, L. Jiang, and M. Perlin, Highly nonlinear standing water waves with small capillary effect, J. Fluid Mech., 369 (1998), pp. 253--272.
L. Sirovich, Turbulence and the dynamics of coherent structures. I. Coherent structures, Quart. Appl. Math., 45 (1987), pp. 561--571.
L. Sirovich, Modeling the functional organization of the visual cortex, Phys. D, 96 (1996), pp. 355--366.
D. H. Smith and A. J. Roberts, Branching behavior of standing waves---the signatures of resonance, Phys. Fluids, 11 (1999), pp. 1051--1064.
G. G. Stokes, Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form, in Mathematical and Physical Papers, Vol. 1, Cambridge University Press, Cambridge, UK, 1880, pp. 225--228.
I. Tadjbakhsh and J. B. Keller, Standing surface waves of finite amplitude, J. Fluid Mech., 8 (1960), pp. 442--451.
G. I. Taylor, An experimental study of standing waves, Proc. Roy. Soc. A, 218 (1953), pp. 44--59.
L. Trefethen, Spectral Methods in MATLAB, Software Environ. Tools 10, SIAM, Philadelphia, 2000.
C. P. Tsai and D. S. Jeng, Numerical Fourier solutions of standing waves in finite water depth, Appl. Ocean Res., 16 (1994), pp. 185--193.
K. S. Turitsyn, L. Lai, and W. W. Zhang, Asymmetric disconnection of an underwater air bubble: Persistent neck vibrations evolve into a smooth contact, Phys. Rev. Lett., 103 (2009), 124501.
J. M. Vanden-Broeck, Nonlinear gravity-capillary standing waves in water of arbitrary uniform depth, J. Fluid Mech., 139 (1984), pp. 97--104.
J. Wilkening, Breakdown of self-similarity at the crests of large amplitude standing water waves, Phys. Rev. Lett., 107 (2011), 184501.
J. Wilkening and J. Yu, Overdetermined shooting methods for computing time-periodic water waves with spectral accuracy, submitted.
M. O. Williams, E. Shlizerman, and J. N. Kutz, The multi-pulsing transition in mode-locked lasers: A low-dimensional approach using waveguide arrays, J. Opt. Soc. Amer. B Opt. Phys., 27 (2010), pp. 2471--2481.
V. E. Zakharov, Stability of periodic waves on finite amplitude on the surface of a deep fluid, Zhurnal Prildadnoi Mekhaniki i Tekhnicheskoi Fiziki, 9 (1968), pp. 86--94.

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Published In

cover image SIAM Journal on Applied Dynamical Systems
SIAM Journal on Applied Dynamical Systems
Pages: 1033 - 1061
ISSN (online): 1536-0040


Submitted: 30 August 2011
Accepted: 31 May 2012
Published online: 13 September 2012


  1. proper orthogonal decomposition
  2. periodic orbits
  3. standing water waves

MSC codes

  1. 37C27
  2. 62H25
  3. 65M70



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