August  2014, 34(8): 3241-3285. doi: 10.3934/dcds.2014.34.3241

Steady stratified periodic gravity waves with surface tension I: Local bifurcation

1. 

University of Missouri, Columbia, MO 65201, United States

Received  July 2013 Revised  September 2013 Published  January 2014

In this paper we consider two-dimensional, stratified, steady water waves propagating over an impermeable flat bed and with a free surface. The motion is assumed to be driven by capillarity (that is, surface tension) on the surface and a gravitational force acting on the body of the fluid. We prove the existence of small-amplitude solutions. This is accomplished by first constructing a 1-parameter family of laminar flow solutions, $\mathcal{T}$, then applying bifurcation theory methods to obtain local curves of small amplitude solutions branching from $\mathcal{T}$ at an eigenvalue of the linearized problem.
Citation: Samuel Walsh. Steady stratified periodic gravity waves with surface tension I: Local bifurcation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3241-3285. doi: 10.3934/dcds.2014.34.3241
References:
[1]

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

[2]

A. Constantin and W. Strauss, Rotational steady water waves near stagnation,, Philos. Trans. Roy. Soc. London Ser. A, 365 (2007), 2227.  doi: 10.1098/rsta.2007.2004.  Google Scholar

[3]

A. Constantin, Nonlinear water waves with applications to wave-current interactions and tsunamis,, vol. 81 of CBMS-NSF Regional Conference Series in Applied Mathematics, (2011).  doi: 10.1137/1.9781611971873.  Google Scholar

[4]

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

[5]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity,, J. Fluid Mech., 498 (2004), 171.  doi: 10.1017/S0022112003006773.  Google Scholar

[6]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, submitted., ().  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[7]

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

[8]

M. Dubreil-Jacotin, Sur la determination rigoureuse des ondes permanentes periodiques d'ampleur finie,, J. Math. Pures Appl., 13 (1934), 217.   Google Scholar

[9]

M. Dubreil-Jacotin, Sur les theoremes d'existence relatifs aux ondes permanentes periodiques a deux dimensions dans les liquides heterogenes,, J. Math. Pures Appl., 16 (1937), 43.   Google Scholar

[10]

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

[11]

D. Henry and A.-V. Matioc, Global bifurcation of capillary-gravity stratified water waves,, Proc. Roy. Soc. Edinburgh Sect. A, ().   Google Scholar

[12]

D. Henry and B.-V. Matioc, On the existence of steady periodic capillary-gravity stratified water waves,, Ann. Sc. Norm. Super. Pisa Cl. Sci., ().   Google Scholar

[13]

D. Henry and B.-V. Matioc, On the regularity of steady periodic stratified water waves,, Commun. Pure Appl. Anal., 11 (2012), 1453.  doi: 10.3934/cpaa.2012.11.1453.  Google Scholar

[14]

M. Jones and J. Toland, Symmetry and the bifurcation of capillary-gravity waves,, Arch. Rational Mech. Anal., 96 (1986), 29.  doi: 10.1007/BF00251412.  Google Scholar

[15]

B. Kinsman, Wind Waves,, Prentice Hall, (1965).  doi: 10.1029/JZ066i008p02411.  Google Scholar

[16]

T. Levi-Civita, Détermination rigoureuse de ondes permanentes d'ampleur finie,, Ann. Math., 93 (1925), 264.  doi: 10.1007/BF01449965.  Google Scholar

[17]

A.-V. Matioc and B.-V. Matioc, Regularity and symmetry properties of rotational solitary water waves,, J. Evol. Equ., 12 (2012), 481.  doi: 10.1007/s00028-012-0141-7.  Google Scholar

[18]

A.-V. Matioc and B.-V. Matioc, On the symmetry of periodic gravity water waves with vorticity,, Differential Integral Equations, 26 (2013), 129.   Google Scholar

[19]

C. Mei, The applied dynamics of ocean surface waves,, World Scientific Pub. Co. Inc., 11 (1984).  doi: 10.1016/0029-8018(84)90033-7.  Google Scholar

[20]

A. I. Nekrasov, The exact theory of steady waves on the surface of a heavy fluid,, Izdat. Akad. Nauk SSSR, (1951).   Google Scholar

[21]

H. Okamoto, On the problem of water waves of permanent configuration,, Nonlinear Anal., 14 (1990), 469.  doi: 10.1016/0362-546X(90)90035-F.  Google Scholar

[22]

H. Okamoto and M. Shōji, The resonance of modes in the problem of two-dimensional capillary-gravity waves,, Physica D: Nonlinear Phenomena, 95 (1996), 336.  doi: 10.1016/0167-2789(96)00071-1.  Google Scholar

[23]

L. Schwartz and L. Vanden-Broeck, Numerical solution of the exact equations for capillary gravity waves,, J. Fluid Mech., 95 (1979), 119.  doi: 10.1017/S0022112079001373.  Google Scholar

[24]

M. Shōji, New bifurcation diagrams in the problem of permanent progressive waves,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), 571.   Google Scholar

[25]

J. Toland and M. Jones, The bifurcation and secondary bifurcation of capillary-gravity waves,, Proc. Roy. Soc. London Ser. A, 399 (1985), 391.  doi: 10.1098/rspa.1985.0063.  Google Scholar

[26]

R. E. L. Turner, Traveling waves in natural systems,, in Variational and topological methods in the study of nonlinear phenomena (Pisa, (2000), 115.   Google Scholar

[27]

E. Wahlén, Steady periodic capillary-gravity waves with vorticity,, SIAM J. Math. Anal., 38 (2006), 921.  doi: 10.1137/050630465.  Google Scholar

[28]

E. Wahlén, Steady periodic capillary waves with vorticity,, Ark. Mat., 44 (2006), 367.  doi: 10.1007/s11512-006-0024-7.  Google Scholar

[29]

E. Wahlén, On rotational water waves with surface tension,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2215.  doi: 10.1098/rsta.2007.2003.  Google Scholar

[30]

E. Wahlén, On Some Nonlinear Aspects of Wave Motion,, PhD thesis, (2008).   Google Scholar

[31]

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

[32]

S. Walsh, Steady stratified periodic gravity waves with surface tension ii: Global bifurcation,, Preprint., ().   Google Scholar

[33]

J. Wilton, On ripples,, Phil. Mag., 29 (1915), 688.  doi: 10.1080/14786440508635350.  Google Scholar

[34]

C.-S. Yih, Dynamics of Nonhomogeneous Fluids,, The Macmillan Co., (1965).   Google Scholar

show all references

References:
[1]

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

[2]

A. Constantin and W. Strauss, Rotational steady water waves near stagnation,, Philos. Trans. Roy. Soc. London Ser. A, 365 (2007), 2227.  doi: 10.1098/rsta.2007.2004.  Google Scholar

[3]

A. Constantin, Nonlinear water waves with applications to wave-current interactions and tsunamis,, vol. 81 of CBMS-NSF Regional Conference Series in Applied Mathematics, (2011).  doi: 10.1137/1.9781611971873.  Google Scholar

[4]

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

[5]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity,, J. Fluid Mech., 498 (2004), 171.  doi: 10.1017/S0022112003006773.  Google Scholar

[6]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, submitted., ().  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[7]

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

[8]

M. Dubreil-Jacotin, Sur la determination rigoureuse des ondes permanentes periodiques d'ampleur finie,, J. Math. Pures Appl., 13 (1934), 217.   Google Scholar

[9]

M. Dubreil-Jacotin, Sur les theoremes d'existence relatifs aux ondes permanentes periodiques a deux dimensions dans les liquides heterogenes,, J. Math. Pures Appl., 16 (1937), 43.   Google Scholar

[10]

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

[11]

D. Henry and A.-V. Matioc, Global bifurcation of capillary-gravity stratified water waves,, Proc. Roy. Soc. Edinburgh Sect. A, ().   Google Scholar

[12]

D. Henry and B.-V. Matioc, On the existence of steady periodic capillary-gravity stratified water waves,, Ann. Sc. Norm. Super. Pisa Cl. Sci., ().   Google Scholar

[13]

D. Henry and B.-V. Matioc, On the regularity of steady periodic stratified water waves,, Commun. Pure Appl. Anal., 11 (2012), 1453.  doi: 10.3934/cpaa.2012.11.1453.  Google Scholar

[14]

M. Jones and J. Toland, Symmetry and the bifurcation of capillary-gravity waves,, Arch. Rational Mech. Anal., 96 (1986), 29.  doi: 10.1007/BF00251412.  Google Scholar

[15]

B. Kinsman, Wind Waves,, Prentice Hall, (1965).  doi: 10.1029/JZ066i008p02411.  Google Scholar

[16]

T. Levi-Civita, Détermination rigoureuse de ondes permanentes d'ampleur finie,, Ann. Math., 93 (1925), 264.  doi: 10.1007/BF01449965.  Google Scholar

[17]

A.-V. Matioc and B.-V. Matioc, Regularity and symmetry properties of rotational solitary water waves,, J. Evol. Equ., 12 (2012), 481.  doi: 10.1007/s00028-012-0141-7.  Google Scholar

[18]

A.-V. Matioc and B.-V. Matioc, On the symmetry of periodic gravity water waves with vorticity,, Differential Integral Equations, 26 (2013), 129.   Google Scholar

[19]

C. Mei, The applied dynamics of ocean surface waves,, World Scientific Pub. Co. Inc., 11 (1984).  doi: 10.1016/0029-8018(84)90033-7.  Google Scholar

[20]

A. I. Nekrasov, The exact theory of steady waves on the surface of a heavy fluid,, Izdat. Akad. Nauk SSSR, (1951).   Google Scholar

[21]

H. Okamoto, On the problem of water waves of permanent configuration,, Nonlinear Anal., 14 (1990), 469.  doi: 10.1016/0362-546X(90)90035-F.  Google Scholar

[22]

H. Okamoto and M. Shōji, The resonance of modes in the problem of two-dimensional capillary-gravity waves,, Physica D: Nonlinear Phenomena, 95 (1996), 336.  doi: 10.1016/0167-2789(96)00071-1.  Google Scholar

[23]

L. Schwartz and L. Vanden-Broeck, Numerical solution of the exact equations for capillary gravity waves,, J. Fluid Mech., 95 (1979), 119.  doi: 10.1017/S0022112079001373.  Google Scholar

[24]

M. Shōji, New bifurcation diagrams in the problem of permanent progressive waves,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), 571.   Google Scholar

[25]

J. Toland and M. Jones, The bifurcation and secondary bifurcation of capillary-gravity waves,, Proc. Roy. Soc. London Ser. A, 399 (1985), 391.  doi: 10.1098/rspa.1985.0063.  Google Scholar

[26]

R. E. L. Turner, Traveling waves in natural systems,, in Variational and topological methods in the study of nonlinear phenomena (Pisa, (2000), 115.   Google Scholar

[27]

E. Wahlén, Steady periodic capillary-gravity waves with vorticity,, SIAM J. Math. Anal., 38 (2006), 921.  doi: 10.1137/050630465.  Google Scholar

[28]

E. Wahlén, Steady periodic capillary waves with vorticity,, Ark. Mat., 44 (2006), 367.  doi: 10.1007/s11512-006-0024-7.  Google Scholar

[29]

E. Wahlén, On rotational water waves with surface tension,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2215.  doi: 10.1098/rsta.2007.2003.  Google Scholar

[30]

E. Wahlén, On Some Nonlinear Aspects of Wave Motion,, PhD thesis, (2008).   Google Scholar

[31]

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

[32]

S. Walsh, Steady stratified periodic gravity waves with surface tension ii: Global bifurcation,, Preprint., ().   Google Scholar

[33]

J. Wilton, On ripples,, Phil. Mag., 29 (1915), 688.  doi: 10.1080/14786440508635350.  Google Scholar

[34]

C.-S. Yih, Dynamics of Nonhomogeneous Fluids,, The Macmillan Co., (1965).   Google Scholar

[1]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344

[2]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[3]

Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020350

[4]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336

[5]

Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342

[6]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[7]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

[8]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[9]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051

[10]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075

[11]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

[12]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[13]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215

[14]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466

[15]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]