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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. Google Scholar |
[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. Google Scholar |
[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
References:
[1] |
T. B. Benjamin and J. E. Feir, The disintegration of wavetrains in deep water, J. Fluid Mech., 27 (1967), 417-430. Google Scholar |
[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. Google Scholar |
[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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