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

Solitary gravity-capillary water waves with point vortices

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.
Citation: Kristoffer Varholm. Solitary gravity-capillary water waves with point vortices. Discrete and Continuous Dynamical Systems, 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), 1033-1037.

[2]

G. R. Burton and J. F. Toland, Surface waves on steady perfect-fluid flows with vorticity, Comm. Pure Appl. Math., 64 (2011), 975-1007. doi: 10.1002/cpa.20365.

[3]

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

[4]

A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417. doi: 10.1088/0305-4470/34/7/313.

[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), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971873.

[6]

A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603. doi: 10.1215/S0012-7094-07-14034-1.

[7]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046.

[8]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Ration. Mech. Anal., 202 (2011), 133-175. doi: 10.1007/s00205-011-0412-4.

[9]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation, Arch. Ration. Mech. Anal., 199 (2011), 33-67. doi: 10.1007/s00205-010-0314-x.

[10]

W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73-83. doi: 10.1006/jcph.1993.1164.

[11]

W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: A rigorous approach, Nonlinearity, 5 (1992), 497-522. doi: 10.1088/0951-7715/5/2/009.

[12]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[13]

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

[14]

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

[15]

M. Ehrnström, J. Escher and G. Villari, Steady water waves with multiple critical layers: Interior dynamics, J. Math. Fluid Mech., 14 (2012), 407-419. doi: 10.1007/s00021-011-0068-8.

[16]

M. Ehrnström, J. Escher and E. Wahlén, Steady water waves with multiple critical layers, SIAM J. Math. Anal., 43 (2011), 1436-1456. doi: 10.1137/100792330.

[17]

M. Ehrnström, H. Holden and X. Raynaud, Symmetric waves are traveling waves, Int. Math. Res. Not. IMRN, (2009), 4578-4596. doi: 10.1093/imrn/rnp100.

[18]

M. Ehrnström and G. Villari, Linear water waves with vorticity: Rotational features and particle paths, J. Differential Equations, 244 (2008), 1888-1909. doi: 10.1016/j.jde.2008.01.012.

[19]

M. Ehrnström and E. Wahlén, Trimodal steady water waves, Arch. Ration. Mech. Anal., 216 (2015), 449-471. doi: 10.1007/s00205-014-0812-3.

[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), 2932-2949. doi: 10.1016/j.jde.2011.03.023.

[21]

T. W. Gamelin, Complex Analysis, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2001. doi: 10.1007/978-0-387-21607-2.

[22]

F. Gerstner, Theorie der wellen, Annalen Der Physik, 32 (1809), 412-445. doi: 10.1002/andp.18090320808.

[23]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[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), 1530-1538. doi: 10.1016/j.physd.2008.03.015.

[25]

D. Henry and A.-V. Matioc, Global bifurcation of capillary-gravity-stratified water waves, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 775-786. doi: 10.1017/S0308210512001990.

[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), 4279-4283. doi: 10.1002/grl.50838.

[27]

V. M. Hur, Global bifurcation theory of deep-water waves with vorticity, SIAM J. Math. Anal., 37 (2006), 1482-1521 (electronic). doi: 10.1137/040621168.

[28]

V. M. Hur, Symmetry of solitary water waves with vorticity, Math. Res. Lett., 15 (2008), 491-509. doi: 10.4310/MRL.2008.v15.n3.a9.

[29]

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

[30]

D. Lannes, The Water Waves Problem, vol. 188 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/188.

[31]

J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge-New York, 1978.

[32]

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

[33]

S. Mardare, On Poincaré and de Rham's theorems, Rev. Roumaine Math. Pures Appl., 53 (2008), 523-541.

[34]

A. I. Markushevich, Theory of Functions of a Complex Variable. Vol. I, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965.

[35]

A. I. Markushevich, Theory of Functions of a Complex Variable. Vol. II, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965.

[36]

B.-V. Matioc, Global bifurcation for water waves with capillary effects and constant vorticity, Monatsh. Math., 174 (2014), 459-475. doi: 10.1007/s00605-013-0583-1.

[37]

C. C. Mei, The applied dynamics of ocean surface waves, Ocean Engineering, 11 (1984), p321. doi: 10.1016/0029-8018(84)90033-7.

[38]

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

[39]

P. I. Plotnikov and J. F. Toland, Convexity of Stokes waves of extreme form, Arch. Ration. Mech. Anal., 171 (2004), 349-416. doi: 10.1007/s00205-003-0292-3.

[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., Berlin, 1996. doi: 10.1515/9783110812411.

[41]

J. Shatah, S. Walsh and C. Zeng, Travelling water waves with compactly supported vorticity, Nonlinearity, 26 (2013), 1529-1564. doi: 10.1088/0951-7715/26/6/1529.

[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), 698-744. doi: 10.1002/cpa.20213.

[43]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.

[44]

J.-M. Vanden-Broeck, Periodic waves with constant vorticity in water of infinite depth, IMA J. Appl. Math., 56 (1996), 207-217.

[45]

K. Varholm, Water Waves with Compactly Supported Vorticity, Master's thesis, Norwegian University of Science and Technology, 2014.

[46]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005.

[47]

S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105. doi: 10.1137/080721583.

[48]

S. Walsh, Steady stratified periodic gravity waves with surface tension {II}: Global bifurcation, Discrete Contin. Dyn. Syst., 34 (2014), 3287-3315. doi: 10.3934/dcds.2014.34.3287.

[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), 190-194. doi: 10.1007/BF00913182.

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), 1033-1037.

[2]

G. R. Burton and J. F. Toland, Surface waves on steady perfect-fluid flows with vorticity, Comm. Pure Appl. Math., 64 (2011), 975-1007. doi: 10.1002/cpa.20365.

[3]

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

[4]

A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417. doi: 10.1088/0305-4470/34/7/313.

[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), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971873.

[6]

A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603. doi: 10.1215/S0012-7094-07-14034-1.

[7]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046.

[8]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Ration. Mech. Anal., 202 (2011), 133-175. doi: 10.1007/s00205-011-0412-4.

[9]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation, Arch. Ration. Mech. Anal., 199 (2011), 33-67. doi: 10.1007/s00205-010-0314-x.

[10]

W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73-83. doi: 10.1006/jcph.1993.1164.

[11]

W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: A rigorous approach, Nonlinearity, 5 (1992), 497-522. doi: 10.1088/0951-7715/5/2/009.

[12]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[13]

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

[14]

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

[15]

M. Ehrnström, J. Escher and G. Villari, Steady water waves with multiple critical layers: Interior dynamics, J. Math. Fluid Mech., 14 (2012), 407-419. doi: 10.1007/s00021-011-0068-8.

[16]

M. Ehrnström, J. Escher and E. Wahlén, Steady water waves with multiple critical layers, SIAM J. Math. Anal., 43 (2011), 1436-1456. doi: 10.1137/100792330.

[17]

M. Ehrnström, H. Holden and X. Raynaud, Symmetric waves are traveling waves, Int. Math. Res. Not. IMRN, (2009), 4578-4596. doi: 10.1093/imrn/rnp100.

[18]

M. Ehrnström and G. Villari, Linear water waves with vorticity: Rotational features and particle paths, J. Differential Equations, 244 (2008), 1888-1909. doi: 10.1016/j.jde.2008.01.012.

[19]

M. Ehrnström and E. Wahlén, Trimodal steady water waves, Arch. Ration. Mech. Anal., 216 (2015), 449-471. doi: 10.1007/s00205-014-0812-3.

[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), 2932-2949. doi: 10.1016/j.jde.2011.03.023.

[21]

T. W. Gamelin, Complex Analysis, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2001. doi: 10.1007/978-0-387-21607-2.

[22]

F. Gerstner, Theorie der wellen, Annalen Der Physik, 32 (1809), 412-445. doi: 10.1002/andp.18090320808.

[23]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[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), 1530-1538. doi: 10.1016/j.physd.2008.03.015.

[25]

D. Henry and A.-V. Matioc, Global bifurcation of capillary-gravity-stratified water waves, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 775-786. doi: 10.1017/S0308210512001990.

[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), 4279-4283. doi: 10.1002/grl.50838.

[27]

V. M. Hur, Global bifurcation theory of deep-water waves with vorticity, SIAM J. Math. Anal., 37 (2006), 1482-1521 (electronic). doi: 10.1137/040621168.

[28]

V. M. Hur, Symmetry of solitary water waves with vorticity, Math. Res. Lett., 15 (2008), 491-509. doi: 10.4310/MRL.2008.v15.n3.a9.

[29]

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

[30]

D. Lannes, The Water Waves Problem, vol. 188 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/188.

[31]

J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge-New York, 1978.

[32]

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

[33]

S. Mardare, On Poincaré and de Rham's theorems, Rev. Roumaine Math. Pures Appl., 53 (2008), 523-541.

[34]

A. I. Markushevich, Theory of Functions of a Complex Variable. Vol. I, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965.

[35]

A. I. Markushevich, Theory of Functions of a Complex Variable. Vol. II, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965.

[36]

B.-V. Matioc, Global bifurcation for water waves with capillary effects and constant vorticity, Monatsh. Math., 174 (2014), 459-475. doi: 10.1007/s00605-013-0583-1.

[37]

C. C. Mei, The applied dynamics of ocean surface waves, Ocean Engineering, 11 (1984), p321. doi: 10.1016/0029-8018(84)90033-7.

[38]

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

[39]

P. I. Plotnikov and J. F. Toland, Convexity of Stokes waves of extreme form, Arch. Ration. Mech. Anal., 171 (2004), 349-416. doi: 10.1007/s00205-003-0292-3.

[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., Berlin, 1996. doi: 10.1515/9783110812411.

[41]

J. Shatah, S. Walsh and C. Zeng, Travelling water waves with compactly supported vorticity, Nonlinearity, 26 (2013), 1529-1564. doi: 10.1088/0951-7715/26/6/1529.

[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), 698-744. doi: 10.1002/cpa.20213.

[43]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.

[44]

J.-M. Vanden-Broeck, Periodic waves with constant vorticity in water of infinite depth, IMA J. Appl. Math., 56 (1996), 207-217.

[45]

K. Varholm, Water Waves with Compactly Supported Vorticity, Master's thesis, Norwegian University of Science and Technology, 2014.

[46]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005.

[47]

S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105. doi: 10.1137/080721583.

[48]

S. Walsh, Steady stratified periodic gravity waves with surface tension {II}: Global bifurcation, Discrete Contin. Dyn. Syst., 34 (2014), 3287-3315. doi: 10.3934/dcds.2014.34.3287.

[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), 190-194. doi: 10.1007/BF00913182.

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