Skip to main content Accessibility help

Dynamics of poles in two-dimensional hydrodynamics with free surface: new constants of motion

  • A. I. Dyachenko (a1) (a2), S. A. Dyachenko (a3) (a4), P. M. Lushnikov (a1) (a5) and V. E. Zakharov (a1) (a2) (a6)


We address the problem of the potential motion of an ideal incompressible fluid with a free surface and infinite depth in a two-dimensional geometry. We admit the presence of gravity forces and surface tension. A time-dependent conformal mapping $z(w,t)$ of the lower complex half-plane of the variable $w$ into the area filled with fluid is performed with the real line of $w$ mapped into the free fluid’s surface. We study the dynamics of singularities of both $z(w,t)$ and the complex fluid potential $\unicode[STIX]{x1D6F1}(w,t)$ in the upper complex half-plane of $w$ . We show the existence of solutions with an arbitrary finite number $N$ of complex poles in $z_{w}(w,t)$ and $\unicode[STIX]{x1D6F1}_{w}(w,t)$ which are the derivatives of $z(w,t)$ and $\unicode[STIX]{x1D6F1}(w,t)$ over $w$ . We stress that these solutions are not purely rational because they generally have branch points at other positions of the upper complex half-plane. The orders of poles can be arbitrary for zero surface tension while all orders are even for non-zero surface tension. We find that the residues of $z_{w}(w,t)$ at these $N$ points are new, previously unknown, constants of motion, see also Zakharov & Dyachenko (2012, authors’ unpublished observations, arXiv:1206.2046) for the preliminary results. All these constants of motion commute with each other in the sense of the underlying Hamiltonian dynamics. In the absence of both gravity and surface tension, the residues of $\unicode[STIX]{x1D6F1}_{w}(w,t)$ are also the constants of motion while non-zero gravity $g$ ensures a trivial linear dependence of these residues on time. A Laurent series expansion of both $z_{w}(w,t)$ and $\unicode[STIX]{x1D6F1}_{w}(w,t)$ at each poles position reveals the existence of additional integrals of motion for poles of the second order. If all poles are simple then the number of independent real integrals of motion is $4N$ for zero gravity and $4N-1$ for non-zero gravity. For the second-order poles we found $6N$ motion integrals for zero gravity and $6N-1$ for non-zero gravity. We suggest that the existence of these non-trivial constants of motion provides an argument in support of the conjecture of complete integrability of free surface hydrodynamics in deep water. Analytical results are solidly supported by high precision numerics.


Corresponding author

Email address for correspondence:


Hide All
Alpert, B., Greengard, L. & Hagstrom, T. 2000 Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation. SIAM J. Numer. Anal. 37, 11381164.
Arnold, V. I. 1989 Mathematical Methods of Classical Mechanics. Springer.
Baker, G., Caflisch, R. E. & Siegel, M. 1993 Singularity formation during Rayleigh–Taylor instability. J. Fluid Mech. 252, 5178.
Baker, G. R., Meiron, D. I. & Orszag, S. A. 1982 Generalized vortex methods for free-surface flow problems. J. Fluid Mech. 123, 477501.
Baker, G. R. & Shelley, M. J. 1990 On the connection between thin vortex layers and vortex sheets. J. Fluid Mech. 215, 161194.
Baker, G. R. & Xie, C. 2011 Singularities in the complex physical plane for deep water waves. J. Fluid Mech. 685, 83116.
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods: Second Revised Edition. Dover Publications.
Caflisch, R. & Orellana, O. 1989 Singular solutions and ill–posedness for the evolution of vortex sheets. SIAM J. Math. Anal. 20 (2), 293307.
Caflisch, R., Orellana, O. & Siegel, M. 1990 A localized approximation method for vortical flows. SIAM J. Appl. Maths 50 (6), 15171532.
Caflisch, R. E., Ercolani, N., Hou, T. Y. & Landis, Y. 1993 Multi-valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems. Commun. Pure Appl. Maths 46 (4), 453499.
Chalikov, D. & Sheinin, D. 1998 Direct modeling of one-dimensional nonlinear potential waves. Adv. Fluid Mech. 17, 207258.
Chalikov, D. & Sheinin, D. 2005 Modeling of extreme waves based on equation of potential flow with a free surface. J. Comput. Phys. 210, 247273.
Chalikov, D. V. 2016 Numerical Modeling of Sea Waves. Springer.
Cowley, S. J., Baker, G. R. & Tanveer, S. 1999 On the formation of Moore curvature singularities in vortex sheets. J. Fluid Mech. 378, 233267.
Crowdy, D. G. 2002 On a class of geometry-driven free boundary problems. SIAM. J. Appl. Maths 62, 945954.
Dubrovin, B. A., Fomenko, A. T. & Novikov, S. P. 1985 Modern Geometry: Methods and Applications: Part II: The Geometry and Topology of Manifolds. Springer.
Dyachenko, A. I. 2001 On the dynamics of an ideal fluid with a free surface. Dokl. Math. 63 (1), 115117.
Dyachenko, A. I., Kachulin, D. I. & Zakharov, V. E. 2013a On the nonintegrability of the free surface hydrodynamics. JETP Lett. 98, 4347.
Dyachenko, A. I., Kuznetsov, E. A., Spector, M. & Zakharov, V. E. 1996 Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A 221, 7379.
Dyachenko, A. I., Lushnikov, P. M. & Zakharov, V. E. 2019 Non-canonical Hamiltonian structure and Poisson bracket for two-dimensional hydrodynamics with free surface. J. Fluid Mech. 869, 526552.
Dyachenko, A. I. & Zakharov, V. E. 1994 Is free surface hydrodynamics an integrable system? Phys. Lett. A 190 (2), 144148.
Dyachenko, S. & Newell, A. C. 2016 Whitecapping. Stud. Appl. Maths 137, 199213.
Dyachenko, S. A., Lushnikov, P. M. & Korotkevich, A. O. 2013b The complex singularity of a Stokes wave. JETP Lett. 98 (11), 675679.
Dyachenko, S. A., Lushnikov, P. M. & Korotkevich, A. O. 2016 Branch cuts of Stokes wave on deep water. Part I. Numerical solution and Padé approximation. Stud. Appl. Maths 137, 419472.
Baker, G. A. Jr & Graves-Morris, P. R. 1996 Padé Approximants, 2nd edn. Cambridge University Press.
Gardner, C. S., Greene, J. M., Kruskal, M. D. & Miura, R. M. 1967 Method for solving the Korteweg-deVries equation. Phys. Rev. Lett. 19, 10951097.
Gonnet, P., Pachon, R. & Trefethen, L. N. 2011 Robust rational interpolation and least-squares. Elec. Trans. Numer. Anal. 1388, 146167.
Grant, M. A. 1973 The singularity at the crest of a finite amplitude progressive Stokes wave. J. Fluid Mech. 59 (2), 257262.
Karabut, E. A. & Zhuravleva, E. N. 2014 Unsteady flows with a zero acceleration on the free boundary. J. Fluid Mech. 754, 308331.
Krasny, R. 1986 A study of singularity formation in a vortex sheet by the point–vortex approximation. J. Fluid Mech. 167, 6593.
Kuznetsov, E. A., Spector, M. D. & Zakharov, V. E. 1993 Surface singularities of ideal fluid. Phys. Lett. A 182 (4–6), 387393.
Kuznetsov, E. A., Spector, M. D. & Zakharov, V. E. 1994 Formation of singularities on the free surface of an ideal fluid. Phys. Rev. E 49, 12831290.
Lamb, H. 1945 Hydrodynamics. Dover Books on Physics.
Landau, L. D. & Lifshitz, E. M. 1989 Fluid Mechanics, 3rd edn. vol. 6. Pergamon.
Lushnikov, P. M. & Zubarev, N. M. 2018 Exact solutions for nonlinear development of a Kelvin–Helmholtz instability for the counterflow of superfluid and normal components of Helium II. Phys. Rev. Lett. 120, 204504.
Lushnikov, P. M. 2004 Exactly integrable dynamics of interface between ideal fluid and light viscous fluid. Phys. Lett. A 329, 4954.
Lushnikov, P. M. 2016 Structure and location of branch point singularities for Stokes waves on deep water. J. Fluid Mech. 800, 557594.
Lushnikov, P. M., Dyachenko, S. A. & Silantyev, D. A. 2017 New conformal mapping for adaptive resolving of the complex singularities of Stokes wave. Proc. R. Soc. Lond. A 473, 20170198.
Meiron, D. I., Baker, G. R. & Orszag, S. A. 1982 Analytic structure of vortex sheet dynamics. Part 1. Kelvin–Helmholtz instability. J. Fluid Mech. 114, 283298.
Meison, D., Orzag, S. & Izraely, M. 1981 Applications of numerical conformal mapping. J. Comput. Phys. 40, 345360.
Mineev-Weinstein, M., Wiegmann, P. B. & Zabrodin, A. 2000 Integrable structure of interface dynamics. Phys. Rev. Lett. 84 (22), 51065109.
Moore, D. W. 1979 The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. R. Soc. Lond. A 365 (1720), 105119.
Novikov, S., Manakov, S. V., Pitaevskii, L. P. & Zakharov, V. E. 1984 Theory of Solitons: The Inverse Scattering Method. Springer.
Ovsyannikov, L. V. 1973 Dynamics of a fluid. M.A. Lavrent’ev Institute of Hydrodynamics Sib. Branch USSR Ac. Sci. 15, 104125.
Richardson, S. 1972 Hele Shaw flows with a free boundary produced by the injection of fluid into a narrow channel. J. Fluid Mech. 56, 609618.
Shelley, M. J. 1992 A study of singularity formation in vortex–sheet motion by a spectrally accurate vortex method. J. Fluid Mech. 244, 493526.
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441455.
Stokes, G. G. 1880 On the theory of oscillatory waves. Math. Phys. Papers 1, 197229.
Tanveer, S. 1991 Singularities in water waves and Rayleigh–Taylor instability. Proc. R. Soc. Lond. A 435, 137158.
Tanveer, S. 1993 Singularities in the classical Rayleigh–Taylor flow: formation and subsequent motion. Proc. R. Soc. Lond. A 441, 501525.
Weinstein, A. 1983 The local structure of Poisson manifolds. J. Differ. Geom. 18, 523557.
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on a surface. J. Appl. Mech. Tech. Phys. 9 (2), 190194.
Zakharov, V. E. & Dyachenko, A. I.2012 Free-surface hydrodynamics in the conformal variables, authors’ unpublished observations, arXiv:1206.2046.
Zakharov, V. E., Dyachenko, A. I. & Vasiliev, O. A. 2002 New method for numerical simulation of nonstationary potential flow of incompressible fluid with a free surface. Eur. J. Mech. (B/Fluids) 21, 283291.
Zakharov, V. E. & Faddeev, L. D. 1971 Korteweg-de Vries equation: A completely integrable Hamiltonian system. Funct. Anal. Applics 5, 280287.
Zakharov, V. E. & Shabat, A. B. 1972 Exact theory of 2-dimensional sef-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62.
Zubarev, N. M. 2000 Charged-surface instability development in liquid helium: an exact solution. JETP Lett. 71, 367369.
Zubarev, N. M. 2002 Exact solutions of the equations of motion of liquid helium with a charged free surface. J. Expl Theor. Phys. 94, 534544.
Zubarev, N. M. 2008 Formation of singularities on the charged surface of a liquid-helium layer with a finite depth. J. Expl Theor. Phys. 107, 668678.
Zubarev, N. M. & Karabut, E. A. 2018 Exact local solutions for the formation of singularities on the free surface of an ideal fluid. JETP Lett. 107, 412417.
Zubarev, N. M. & Kuznetsov, E. A. 2014 Singularity formation on a fluid interface during the Kelvin–Helmholtz instability development. J. Expl Theor. Phys. 119, 169178.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed