American Institute of Mathematical Sciences

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July  2016, 36(7): 3927-3959. doi: 10.3934/dcds.2016.36.3927

Solitary gravity-capillary water waves with point vortices

Kristoffer Varholm 1,

1. 

Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway

Received  May 2015 Revised  November 2015 Published  March 2016

We construct small-amplitude solitary traveling gravity-capillary water waves with a finite number of point vortices along a vertical line, on finite depth. This is done using a local bifurcation argument. The properties of the resulting waves are also examined: We find that they depend significantly on the position of the point vortices in the water column.
Keywords: vorticity., Steady water waves, solitary waves, Euler equations, bifurcation theory.
Mathematics Subject Classification: Primary: 35Q31; Secondary: 35C07, 76B2.
Citation: Kristoffer Varholm. Solitary gravity-capillary water waves with point vortices. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3927-3959. doi: 10.3934/dcds.2016.36.3927
References:
[1]

K. I. Babenko, Some remarks on the theory of surface waves of finite amplitude,, Dokl. Akad. Nauk SSSR, 294 (1987), 1033.   Google Scholar

[2]

G. R. Burton and J. F. Toland, Surface waves on steady perfect-fluid flows with vorticity,, Comm. Pure Appl. Math., 64 (2011), 975.  doi: 10.1002/cpa.20365.  Google Scholar

[3]

A. Constantin, W. Strauss and E. Varvaruca, Global bifurcation of steady gravity water waves with critical layers,, preprint, ().   Google Scholar

[4]

A. Constantin, On the deep water wave motion,, J. Phys. A, 34 (2001), 1405.  doi: 10.1088/0305-4470/34/7/313.  Google Scholar

[5]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, vol. 81 of CBMS-NSF Regional Conference Series in Applied Mathematics,, Society for Industrial and Applied Mathematics (SIAM), (2011).  doi: 10.1137/1.9781611971873.  Google Scholar

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A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity,, Duke Math. J., 140 (2007), 591.  doi: 10.1215/S0012-7094-07-14034-1.  Google Scholar

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A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Comm. Pure Appl. Math., 57 (2004), 481.  doi: 10.1002/cpa.3046.  Google Scholar

[8]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity,, Arch. Ration. Mech. Anal., 202 (2011), 133.  doi: 10.1007/s00205-011-0412-4.  Google Scholar

[9]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation,, Arch. Ration. Mech. Anal., 199 (2011), 33.  doi: 10.1007/s00205-010-0314-x.  Google Scholar

[10]

W. Craig and C. Sulem, Numerical simulation of gravity waves,, J. Comput. Phys., 108 (1993), 73.  doi: 10.1006/jcph.1993.1164.  Google Scholar

[11]

W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: A rigorous approach,, Nonlinearity, 5 (1992), 497.  doi: 10.1088/0951-7715/5/2/009.  Google Scholar

[12]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[13]

J. Deny and J. L. Lions, Les espaces du type de Beppo Levi,, Ann. Inst. Fourier, 5 (): 305.   Google Scholar

[14]

M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur, finie., ().   Google Scholar

[15]

M. Ehrnström, J. Escher and G. Villari, Steady water waves with multiple critical layers: Interior dynamics,, J. Math. Fluid Mech., 14 (2012), 407.  doi: 10.1007/s00021-011-0068-8.  Google Scholar

[16]

M. Ehrnström, J. Escher and E. Wahlén, Steady water waves with multiple critical layers,, SIAM J. Math. Anal., 43 (2011), 1436.  doi: 10.1137/100792330.  Google Scholar

[17]

M. Ehrnström, H. Holden and X. Raynaud, Symmetric waves are traveling waves,, Int. Math. Res. Not. IMRN, (2009), 4578.  doi: 10.1093/imrn/rnp100.  Google Scholar

[18]

M. Ehrnström and G. Villari, Linear water waves with vorticity: Rotational features and particle paths,, J. Differential Equations, 244 (2008), 1888.  doi: 10.1016/j.jde.2008.01.012.  Google Scholar

[19]

M. Ehrnström and E. Wahlén, Trimodal steady water waves,, Arch. Ration. Mech. Anal., 216 (2015), 449.  doi: 10.1007/s00205-014-0812-3.  Google Scholar

[20]

J. Escher, A.-V. Matioc and B.-V. Matioc, On stratified steady periodic water waves with linear density distribution and stagnation points,, J. Differential Equations, 251 (2011), 2932.  doi: 10.1016/j.jde.2011.03.023.  Google Scholar

[21]

T. W. Gamelin, Complex Analysis,, Undergraduate Texts in Mathematics, (2001).  doi: 10.1007/978-0-387-21607-2.  Google Scholar

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F. Gerstner, Theorie der wellen,, Annalen Der Physik, 32 (1809), 412.  doi: 10.1002/andp.18090320808.  Google Scholar

[23]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Classics in Mathematics, (2001).   Google Scholar

[24]

M. D. Groves and E. Wahlén, Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity,, Phys. D, 237 (2008), 1530.  doi: 10.1016/j.physd.2008.03.015.  Google Scholar

[25]

D. Henry and A.-V. Matioc, Global bifurcation of capillary-gravity-stratified water waves,, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 775.  doi: 10.1017/S0308210512001990.  Google Scholar

[26]

C. Hirt, S. Claessens, T. Fecher, M. Kuhn, R. Pail and M. Rexer, New ultrahigh-resolution picture of earth's gravity field,, Geophysical Research Letters, 40 (2013), 4279.  doi: 10.1002/grl.50838.  Google Scholar

[27]

V. M. Hur, Global bifurcation theory of deep-water waves with vorticity,, SIAM J. Math. Anal., 37 (2006), 1482.  doi: 10.1137/040621168.  Google Scholar

[28]

V. M. Hur, Symmetry of solitary water waves with vorticity,, Math. Res. Lett., 15 (2008), 491.  doi: 10.4310/MRL.2008.v15.n3.a9.  Google Scholar

[29]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge Texts in Applied Mathematics, (1997).  doi: 10.1017/CBO9780511624056.  Google Scholar

[30]

D. Lannes, The Water Waves Problem, vol. 188 of Mathematical Surveys and Monographs,, American Mathematical Society, (2013).  doi: 10.1090/surv/188.  Google Scholar

[31]

J. Lighthill, Waves in Fluids,, Cambridge University Press, (1978).   Google Scholar

[32]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, vol. 96 of Applied Mathematical Sciences,, Springer-Verlag, (1994).  doi: 10.1007/978-1-4612-4284-0.  Google Scholar

[33]

S. Mardare, On Poincaré and de Rham's theorems,, Rev. Roumaine Math. Pures Appl., 53 (2008), 523.   Google Scholar

[34]

A. I. Markushevich, Theory of Functions of a Complex Variable. Vol. I,, Prentice-Hall, (1965).   Google Scholar

[35]

A. I. Markushevich, Theory of Functions of a Complex Variable. Vol. II,, Prentice-Hall, (1965).   Google Scholar

[36]

B.-V. Matioc, Global bifurcation for water waves with capillary effects and constant vorticity,, Monatsh. Math., 174 (2014), 459.  doi: 10.1007/s00605-013-0583-1.  Google Scholar

[37]

C. C. Mei, The applied dynamics of ocean surface waves,, Ocean Engineering, 11 (1984).  doi: 10.1016/0029-8018(84)90033-7.  Google Scholar

[38]

A. Nekrasov, On steady waves,, Izv. Ivanovo-Voznesensk. Politekhn. In-ta, 3 ().   Google Scholar

[39]

P. I. Plotnikov and J. F. Toland, Convexity of Stokes waves of extreme form,, Arch. Ration. Mech. Anal., 171 (2004), 349.  doi: 10.1007/s00205-003-0292-3.  Google Scholar

[40]

T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, vol. 3 of de Gruyter Series in Nonlinear Analysis and Applications,, Walter de Gruyter & Co., (1996).  doi: 10.1515/9783110812411.  Google Scholar

[41]

J. Shatah, S. Walsh and C. Zeng, Travelling water waves with compactly supported vorticity,, Nonlinearity, 26 (2013), 1529.  doi: 10.1088/0951-7715/26/6/1529.  Google Scholar

[42]

J. Shatah and C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler equation,, Comm. Pure Appl. Math., 61 (2008), 698.  doi: 10.1002/cpa.20213.  Google Scholar

[43]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.   Google Scholar

[44]

J.-M. Vanden-Broeck, Periodic waves with constant vorticity in water of infinite depth,, IMA J. Appl. Math., 56 (1996), 207.   Google Scholar

[45]

K. Varholm, Water Waves with Compactly Supported Vorticity,, Master's thesis, (2014).   Google Scholar

[46]

E. Wahlén, Steady water waves with a critical layer,, J. Differential Equations, 246 (2009), 2468.  doi: 10.1016/j.jde.2008.10.005.  Google Scholar

[47]

S. Walsh, Stratified steady periodic water waves,, SIAM J. Math. Anal., 41 (2009), 1054.  doi: 10.1137/080721583.  Google Scholar

[48]

S. Walsh, Steady stratified periodic gravity waves with surface tension {II}: Global bifurcation,, Discrete Contin. Dyn. Syst., 34 (2014), 3287.  doi: 10.3934/dcds.2014.34.3287.  Google Scholar

[49]

V. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid,, Journal of Applied Mechanics and Technical Physics, 9 (1968), 190.  doi: 10.1007/BF00913182.  Google Scholar

show all references

References:
[1]

K. I. Babenko, Some remarks on the theory of surface waves of finite amplitude,, Dokl. Akad. Nauk SSSR, 294 (1987), 1033.   Google Scholar

[2]

G. R. Burton and J. F. Toland, Surface waves on steady perfect-fluid flows with vorticity,, Comm. Pure Appl. Math., 64 (2011), 975.  doi: 10.1002/cpa.20365.  Google Scholar

[3]

A. Constantin, W. Strauss and E. Varvaruca, Global bifurcation of steady gravity water waves with critical layers,, preprint, ().   Google Scholar

[4]

A. Constantin, On the deep water wave motion,, J. Phys. A, 34 (2001), 1405.  doi: 10.1088/0305-4470/34/7/313.  Google Scholar

[5]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, vol. 81 of CBMS-NSF Regional Conference Series in Applied Mathematics,, Society for Industrial and Applied Mathematics (SIAM), (2011).  doi: 10.1137/1.9781611971873.  Google Scholar

[6]

A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity,, Duke Math. J., 140 (2007), 591.  doi: 10.1215/S0012-7094-07-14034-1.  Google Scholar

[7]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Comm. Pure Appl. Math., 57 (2004), 481.  doi: 10.1002/cpa.3046.  Google Scholar

[8]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity,, Arch. Ration. Mech. Anal., 202 (2011), 133.  doi: 10.1007/s00205-011-0412-4.  Google Scholar

[9]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation,, Arch. Ration. Mech. Anal., 199 (2011), 33.  doi: 10.1007/s00205-010-0314-x.  Google Scholar

[10]

W. Craig and C. Sulem, Numerical simulation of gravity waves,, J. Comput. Phys., 108 (1993), 73.  doi: 10.1006/jcph.1993.1164.  Google Scholar

[11]

W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: A rigorous approach,, Nonlinearity, 5 (1992), 497.  doi: 10.1088/0951-7715/5/2/009.  Google Scholar

[12]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[13]

J. Deny and J. L. Lions, Les espaces du type de Beppo Levi,, Ann. Inst. Fourier, 5 (): 305.   Google Scholar

[14]

M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur, finie., ().   Google Scholar

[15]

M. Ehrnström, J. Escher and G. Villari, Steady water waves with multiple critical layers: Interior dynamics,, J. Math. Fluid Mech., 14 (2012), 407.  doi: 10.1007/s00021-011-0068-8.  Google Scholar

[16]

M. Ehrnström, J. Escher and E. Wahlén, Steady water waves with multiple critical layers,, SIAM J. Math. Anal., 43 (2011), 1436.  doi: 10.1137/100792330.  Google Scholar

[17]

M. Ehrnström, H. Holden and X. Raynaud, Symmetric waves are traveling waves,, Int. Math. Res. Not. IMRN, (2009), 4578.  doi: 10.1093/imrn/rnp100.  Google Scholar

[18]

M. Ehrnström and G. Villari, Linear water waves with vorticity: Rotational features and particle paths,, J. Differential Equations, 244 (2008), 1888.  doi: 10.1016/j.jde.2008.01.012.  Google Scholar

[19]

M. Ehrnström and E. Wahlén, Trimodal steady water waves,, Arch. Ration. Mech. Anal., 216 (2015), 449.  doi: 10.1007/s00205-014-0812-3.  Google Scholar

[20]

J. Escher, A.-V. Matioc and B.-V. Matioc, On stratified steady periodic water waves with linear density distribution and stagnation points,, J. Differential Equations, 251 (2011), 2932.  doi: 10.1016/j.jde.2011.03.023.  Google Scholar

[21]

T. W. Gamelin, Complex Analysis,, Undergraduate Texts in Mathematics, (2001).  doi: 10.1007/978-0-387-21607-2.  Google Scholar

[22]

F. Gerstner, Theorie der wellen,, Annalen Der Physik, 32 (1809), 412.  doi: 10.1002/andp.18090320808.  Google Scholar

[23]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Classics in Mathematics, (2001).   Google Scholar

[24]

M. D. Groves and E. Wahlén, Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity,, Phys. D, 237 (2008), 1530.  doi: 10.1016/j.physd.2008.03.015.  Google Scholar

[25]

D. Henry and A.-V. Matioc, Global bifurcation of capillary-gravity-stratified water waves,, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 775.  doi: 10.1017/S0308210512001990.  Google Scholar

[26]

C. Hirt, S. Claessens, T. Fecher, M. Kuhn, R. Pail and M. Rexer, New ultrahigh-resolution picture of earth's gravity field,, Geophysical Research Letters, 40 (2013), 4279.  doi: 10.1002/grl.50838.  Google Scholar

[27]

V. M. Hur, Global bifurcation theory of deep-water waves with vorticity,, SIAM J. Math. Anal., 37 (2006), 1482.  doi: 10.1137/040621168.  Google Scholar

[28]

V. M. Hur, Symmetry of solitary water waves with vorticity,, Math. Res. Lett., 15 (2008), 491.  doi: 10.4310/MRL.2008.v15.n3.a9.  Google Scholar

[29]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge Texts in Applied Mathematics, (1997).  doi: 10.1017/CBO9780511624056.  Google Scholar

[30]

D. Lannes, The Water Waves Problem, vol. 188 of Mathematical Surveys and Monographs,, American Mathematical Society, (2013).  doi: 10.1090/surv/188.  Google Scholar

[31]

J. Lighthill, Waves in Fluids,, Cambridge University Press, (1978).   Google Scholar

[32]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, vol. 96 of Applied Mathematical Sciences,, Springer-Verlag, (1994).  doi: 10.1007/978-1-4612-4284-0.  Google Scholar

[33]

S. Mardare, On Poincaré and de Rham's theorems,, Rev. Roumaine Math. Pures Appl., 53 (2008), 523.   Google Scholar

[34]

A. I. Markushevich, Theory of Functions of a Complex Variable. Vol. I,, Prentice-Hall, (1965).   Google Scholar

[35]

A. I. Markushevich, Theory of Functions of a Complex Variable. Vol. II,, Prentice-Hall, (1965).   Google Scholar

[36]

B.-V. Matioc, Global bifurcation for water waves with capillary effects and constant vorticity,, Monatsh. Math., 174 (2014), 459.  doi: 10.1007/s00605-013-0583-1.  Google Scholar

[37]

C. C. Mei, The applied dynamics of ocean surface waves,, Ocean Engineering, 11 (1984).  doi: 10.1016/0029-8018(84)90033-7.  Google Scholar

[38]

A. Nekrasov, On steady waves,, Izv. Ivanovo-Voznesensk. Politekhn. In-ta, 3 ().   Google Scholar

[39]

P. I. Plotnikov and J. F. Toland, Convexity of Stokes waves of extreme form,, Arch. Ration. Mech. Anal., 171 (2004), 349.  doi: 10.1007/s00205-003-0292-3.  Google Scholar

[40]

T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, vol. 3 of de Gruyter Series in Nonlinear Analysis and Applications,, Walter de Gruyter & Co., (1996).  doi: 10.1515/9783110812411.  Google Scholar

[41]

J. Shatah, S. Walsh and C. Zeng, Travelling water waves with compactly supported vorticity,, Nonlinearity, 26 (2013), 1529.  doi: 10.1088/0951-7715/26/6/1529.  Google Scholar

[42]

J. Shatah and C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler equation,, Comm. Pure Appl. Math., 61 (2008), 698.  doi: 10.1002/cpa.20213.  Google Scholar

[43]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.   Google Scholar

[44]

J.-M. Vanden-Broeck, Periodic waves with constant vorticity in water of infinite depth,, IMA J. Appl. Math., 56 (1996), 207.   Google Scholar

[45]

K. Varholm, Water Waves with Compactly Supported Vorticity,, Master's thesis, (2014).   Google Scholar

[46]

E. Wahlén, Steady water waves with a critical layer,, J. Differential Equations, 246 (2009), 2468.  doi: 10.1016/j.jde.2008.10.005.  Google Scholar

[47]

S. Walsh, Stratified steady periodic water waves,, SIAM J. Math. Anal., 41 (2009), 1054.  doi: 10.1137/080721583.  Google Scholar

[48]

S. Walsh, Steady stratified periodic gravity waves with surface tension {II}: Global bifurcation,, Discrete Contin. Dyn. Syst., 34 (2014), 3287.  doi: 10.3934/dcds.2014.34.3287.  Google Scholar

[49]

V. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid,, Journal of Applied Mechanics and Technical Physics, 9 (1968), 190.  doi: 10.1007/BF00913182.  Google Scholar

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