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On the universality of the incompressible Euler equation on compact manifolds
Traveling wave solutions of a highly nonlinear shallow water equation
1. | Delft University of Technology, Delft Institute of Applied Mathematics, Faculty of EEMCS, Mekelweg 4,2628 CD Delft, The Netherlands |
2. | KTH Royal Institute of Technology, Department of Mathematics, Lindstedtsvägen 25,100 44 Stockholm, Sweden |
Motivated by the question whether higher-order nonlinear model equations, which go beyond the Camassa-Holm regime of moderate amplitude waves, could point us to new types of waves profiles, we study the traveling wave solutions of a quasilinear evolution equation which models the propagation of shallow water waves of large amplitude. The aim of this paper is a complete classification of its traveling wave solutions. Apart from symmetric smooth, peaked and cusped solitary and periodic traveling waves, whose existence is well-known for moderate amplitude equations like Camassa-Holm, we obtain entirely new types of singular traveling waves: periodic waves which exhibit singularities on both crests and troughs simultaneously, waves with asymmetric peaks, as well as multi-crested smooth and multi-peaked waves with decay. Our approach uses qualitative tools for dynamical systems and methods for integrable planar systems.
References:
[1] |
T. B. Benjamin and J. E. Feir,
The disintegration of wavetrains in deep water, J. Fluid Mech., 27 (1967), 417-430.
|
[2] |
J. L. Bona, P. E. Souganidis and W. Strauss,
Stability and instability of solitary waves of Korteweg-de Vries type, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 411 (1987), 395-412.
doi: 10.1098/rspa.1987.0073. |
[3] |
G. Brüll, M. Ehrnström, A. Geyer and L. Pei,
Symmetric solutions of evolutionary partial differential equations, Nonlinearity, 30 (2017), 3932-3950.
|
[4] |
R. Camassa and D. D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[5] |
A. Constantin,
The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[6] |
A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res. : Oceans, 117 (2012), C05029.
doi: 10.1029/2012JC007879. |
[7] |
A. Constantin,
Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44 (2014), 781-789.
doi: 10.1175/JPO-D-13-0174.1. |
[8] |
A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, 81 SIAM Philadelphia, 2011. |
[9] |
A. Constantin and J. Escher,
Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math., 173 (2011), 559-568.
doi: 10.4007/annals.2011.173.1.12. |
[10] |
A. Constantin and D. Lannes,
The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.
doi: 10.1007/s00205-008-0128-2. |
[11] |
A. Constantin and W. Strauss,
Stability of peakons, Commun. Pure Appl. Math., 53 (2000), 603-610.
doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. |
[12] |
A. Constantin and W. Strauss,
Stability of the Camassa-Holm solitons, J. Nonlinear Sci., 12 (2002), 415-422.
doi: 10.1007/s00332-002-0517-x. |
[13] |
A. Constantin and W. Strauss,
Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math., 60 (2007), 911-950.
doi: 10.1002/cpa.20165. |
[14] |
B. Deconinck and T. Kapitula,
The orbital stability of the cnoidal waves of the Korteweg-de Vries equation, Phys. Lett. Sect. A Gen. At. Solid State Phys., 374 (2010), 4018-4022.
doi: 10.1016/j.physleta.2010.08.007. |
[15] |
A. Degasperis and M. Procesi, Asymptotic integrability, In A. Degasperis and G. Gaeta, editors, Symmetry and Perturbation Theory, pages 23–37, World Scientific, Singapore, 1999. |
[16] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer, Berlin, 2006. |
[17] |
N. Duruk Mutlubas and A. Geyer,
Orbital stability of solitary waves of moderate amplitude in shallow water, J. Differ. Equations, 255 (2013), 254-263.
doi: 10.1016/j.jde.2013.04.010. |
[18] |
M. Ehrnström, H. Holden and X. Raynaud,
Symmetric waves are traveling waves, Int. Math. Res. Not., 2009 (2009), 4578-4596.
|
[19] |
A. Gasull and A. Geyer,
Traveling surface waves of moderate amplitude in shallow water, Nonlinear Anal. Theory, Methods Appl., 102 (2014), 105-119.
doi: 10.1016/j.na.2014.02.005. |
[20] |
A. Geyer,
Symmetric waves are traveling waves for a shallow water equation modeling surface waves of moderate amplitude, J. Nonlinear Math. Phys., 22 (2015), 545-551.
doi: 10.1080/14029251.2015.1129492. |
[21] |
A. Geyer and V. Mañosa,
Singular solutions for a class of traveling wave equations, Nonlinear Anal. Real World Appl., 31 (2016), 57-76.
doi: 10.1016/j.nonrwa.2016.01.009. |
[22] |
J. Guggenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[23] |
D. Henry, Equatorially trapped nonlinear water waves in a $β$-plane approximation with centripetal forces, J. Fluid Mech. , 804 (2016), R1, 11 pp. |
[24] |
E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, New York-Heidelberg, 1975. |
[25] |
V. M. Hur,
Analyticity of rotational flows beneath solitary water waves, Int. Math. Res. Not. IMRN, 2012 (2012), 2550-2570.
|
[26] |
R. S. Johnson,
Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.
|
[27] |
D. J. Korteweg and G. de Vries,
On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Phil. Mag., 39 (1895), 422-443.
doi: 10.1080/14786449508620739. |
[28] |
J. Lenells,
A variational approach to the stability of periodic peakons, J. Nonlinear Math. Phys., 11 (2004), 151-163.
doi: 10.2991/jnmp.2004.11.2.2. |
[29] |
J. Lenells,
Stability of periodic peakons, Int. Math. Res. Not., 10 (2004), 485-499.
|
[30] |
J. Lenells,
Stability for the periodic Camassa-Holm equation, Math. Scand., 97 (2005), 188-200.
doi: 10.7146/math.scand.a-14971. |
[31] |
J. Lenells,
Traveling wave solutions of the Camassa-Holm equation, J. Differ. Equ., 217 (2005), 393-430.
doi: 10.1016/j.jde.2004.09.007. |
[32] |
J. Lenells,
Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl., 306 (2005), 72-82.
doi: 10.1016/j.jmaa.2004.11.038. |
[33] |
Z. Lin and Y. Liu,
Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math., 62 (2009), 125-146.
|
[34] |
T. Lyons,
The pressure in a deep-water Stokes wave of greatest height, J. Math. Fluid Mech., 18 (2016), 209-218.
doi: 10.1007/s00021-016-0249-6. |
[35] |
R. Quirchmayr,
A new highly nonlinear shallow water wave equation, J. Evol. Equations, 16 (2016), 539-567.
doi: 10.1007/s00028-015-0312-4. |
[36] |
H. Segur, D. Henderson, J. Carter, J. Hammack, C.-M. Li, D. Pheiff and K. Socha,
Stabilizing the Benjamin-Feir instability, J. Fluid Mech., 539 (2005), 229-271.
doi: 10.1017/S002211200500563X. |
[37] |
G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics 140 AMS, Providence, RI, 2012. |
[38] |
J. F. Toland,
Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.
doi: 10.12775/TMNA.1996.001. |
[39] |
E. Varvaruca and G. S. Weiss,
The Stokes conjecture for waves with vorticity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 861-885.
doi: 10.1016/j.anihpc.2012.05.001. |
show all references
A. Geyer acknowledges the support of the Austrian Science Fund (FWF) project J3452 "Dynamical Systems Methods in Hydrodynamics". R. Quirchmayr acknowledges the support of the Austrian Science Fund (FWF), Grant W1245.
References:
[1] |
T. B. Benjamin and J. E. Feir,
The disintegration of wavetrains in deep water, J. Fluid Mech., 27 (1967), 417-430.
|
[2] |
J. L. Bona, P. E. Souganidis and W. Strauss,
Stability and instability of solitary waves of Korteweg-de Vries type, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 411 (1987), 395-412.
doi: 10.1098/rspa.1987.0073. |
[3] |
G. Brüll, M. Ehrnström, A. Geyer and L. Pei,
Symmetric solutions of evolutionary partial differential equations, Nonlinearity, 30 (2017), 3932-3950.
|
[4] |
R. Camassa and D. D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[5] |
A. Constantin,
The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[6] |
A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res. : Oceans, 117 (2012), C05029.
doi: 10.1029/2012JC007879. |
[7] |
A. Constantin,
Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44 (2014), 781-789.
doi: 10.1175/JPO-D-13-0174.1. |
[8] |
A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, 81 SIAM Philadelphia, 2011. |
[9] |
A. Constantin and J. Escher,
Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math., 173 (2011), 559-568.
doi: 10.4007/annals.2011.173.1.12. |
[10] |
A. Constantin and D. Lannes,
The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.
doi: 10.1007/s00205-008-0128-2. |
[11] |
A. Constantin and W. Strauss,
Stability of peakons, Commun. Pure Appl. Math., 53 (2000), 603-610.
doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. |
[12] |
A. Constantin and W. Strauss,
Stability of the Camassa-Holm solitons, J. Nonlinear Sci., 12 (2002), 415-422.
doi: 10.1007/s00332-002-0517-x. |
[13] |
A. Constantin and W. Strauss,
Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math., 60 (2007), 911-950.
doi: 10.1002/cpa.20165. |
[14] |
B. Deconinck and T. Kapitula,
The orbital stability of the cnoidal waves of the Korteweg-de Vries equation, Phys. Lett. Sect. A Gen. At. Solid State Phys., 374 (2010), 4018-4022.
doi: 10.1016/j.physleta.2010.08.007. |
[15] |
A. Degasperis and M. Procesi, Asymptotic integrability, In A. Degasperis and G. Gaeta, editors, Symmetry and Perturbation Theory, pages 23–37, World Scientific, Singapore, 1999. |
[16] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer, Berlin, 2006. |
[17] |
N. Duruk Mutlubas and A. Geyer,
Orbital stability of solitary waves of moderate amplitude in shallow water, J. Differ. Equations, 255 (2013), 254-263.
doi: 10.1016/j.jde.2013.04.010. |
[18] |
M. Ehrnström, H. Holden and X. Raynaud,
Symmetric waves are traveling waves, Int. Math. Res. Not., 2009 (2009), 4578-4596.
|
[19] |
A. Gasull and A. Geyer,
Traveling surface waves of moderate amplitude in shallow water, Nonlinear Anal. Theory, Methods Appl., 102 (2014), 105-119.
doi: 10.1016/j.na.2014.02.005. |
[20] |
A. Geyer,
Symmetric waves are traveling waves for a shallow water equation modeling surface waves of moderate amplitude, J. Nonlinear Math. Phys., 22 (2015), 545-551.
doi: 10.1080/14029251.2015.1129492. |
[21] |
A. Geyer and V. Mañosa,
Singular solutions for a class of traveling wave equations, Nonlinear Anal. Real World Appl., 31 (2016), 57-76.
doi: 10.1016/j.nonrwa.2016.01.009. |
[22] |
J. Guggenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[23] |
D. Henry, Equatorially trapped nonlinear water waves in a $β$-plane approximation with centripetal forces, J. Fluid Mech. , 804 (2016), R1, 11 pp. |
[24] |
E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, New York-Heidelberg, 1975. |
[25] |
V. M. Hur,
Analyticity of rotational flows beneath solitary water waves, Int. Math. Res. Not. IMRN, 2012 (2012), 2550-2570.
|
[26] |
R. S. Johnson,
Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.
|
[27] |
D. J. Korteweg and G. de Vries,
On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Phil. Mag., 39 (1895), 422-443.
doi: 10.1080/14786449508620739. |
[28] |
J. Lenells,
A variational approach to the stability of periodic peakons, J. Nonlinear Math. Phys., 11 (2004), 151-163.
doi: 10.2991/jnmp.2004.11.2.2. |
[29] |
J. Lenells,
Stability of periodic peakons, Int. Math. Res. Not., 10 (2004), 485-499.
|
[30] |
J. Lenells,
Stability for the periodic Camassa-Holm equation, Math. Scand., 97 (2005), 188-200.
doi: 10.7146/math.scand.a-14971. |
[31] |
J. Lenells,
Traveling wave solutions of the Camassa-Holm equation, J. Differ. Equ., 217 (2005), 393-430.
doi: 10.1016/j.jde.2004.09.007. |
[32] |
J. Lenells,
Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl., 306 (2005), 72-82.
doi: 10.1016/j.jmaa.2004.11.038. |
[33] |
Z. Lin and Y. Liu,
Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math., 62 (2009), 125-146.
|
[34] |
T. Lyons,
The pressure in a deep-water Stokes wave of greatest height, J. Math. Fluid Mech., 18 (2016), 209-218.
doi: 10.1007/s00021-016-0249-6. |
[35] |
R. Quirchmayr,
A new highly nonlinear shallow water wave equation, J. Evol. Equations, 16 (2016), 539-567.
doi: 10.1007/s00028-015-0312-4. |
[36] |
H. Segur, D. Henderson, J. Carter, J. Hammack, C.-M. Li, D. Pheiff and K. Socha,
Stabilizing the Benjamin-Feir instability, J. Fluid Mech., 539 (2005), 229-271.
doi: 10.1017/S002211200500563X. |
[37] |
G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics 140 AMS, Providence, RI, 2012. |
[38] |
J. F. Toland,
Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.
doi: 10.12775/TMNA.1996.001. |
[39] |
E. Varvaruca and G. S. Weiss,
The Stokes conjecture for waves with vorticity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 861-885.
doi: 10.1016/j.anihpc.2012.05.001. |


















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