Figure 1.
Sketch of the fluid domain: free surface flow on a shear current.
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Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics
September
2019, 39(9): 5521-5541.
doi: 10.3934/dcds.2019225
Effects of vorticity on the travelling waves of some shallow water two-component systems
| 1. | Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France |
| 2. | LAMA, UMR 5127 CNRS, Université Savoie Mont Blanc, Campus Scientifique, 73376 Le Bourget-du-Lac Cedex, France |
| 3. | Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit No. 6, P.O. Box 1-764, 014700 Bucharest, Romania |
In the present study we consider three two-component (integrable and non-integrable) systems which describe the propagation of shallow water waves on a constant shear current. Namely, we consider the two-component Camassa–Holm equations, the Zakharov–Ito system and the Kaup–Boussinesq equations all including constant vorticity effects. We analyze both solitary and periodic-type travelling waves using the simple and geometrically intuitive phase space analysis. We get the pulse-type solitary wave solutions and the front solitary wave solutions. For the Zakharov–Ito system we underline the occurrence of the pulse and anti-pulse solutions. The front wave solutions decay algebraically in the far field. For the Kaup–Boussinesq system, interesting analytical multi-pulsed travelling wave solutions are found.
Keywords:
Waves on shear flow,
solitary waves,
cnoidal waves,
Zakharov–Ito system,
Camassa–Holm equations,
Kaup–Boussinesq equations,
phase-plane analysis,
analytical solutions,
multi-pulsed solutions.
Mathematics Subject Classification:
74J30, 76F10, 35C07, 76B25, 70K05, 35Q35.
Citation:
Denys Dutykh, Delia Ionescu-Kruse. Effects of vorticity on the travelling waves of some shallow water two-component systems. Discrete and Continuous Dynamical Systems,
2019, 39
(9)
: 5521-5541.
doi: 10.3934/dcds.2019225
![]()
References:
| [1] |
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Reprint of the 1972 edition. Dover Publications, Inc., New York, 1992. |
| [2] |
D. M. Ambrose, J. L. Bona and T. Milgrom, Global solutions and ill-posedness for the Kaup system and related Boussinesq systems, 2017, preprint. |
| [3] |
T. B. Benjamin,
The solitary wave on a stream with an arbitrary distribution of vorticity, J. Fluid Mech, 12 (1962), 97-116.
doi: 10.1017/S0022112062000063. |
| [4] |
F. Biesel,
Etude théorique de la houle en eau courante, Houille Blanche, 5 (1950), 279-285.
|
| [5] |
J. C. Burns,
Long waves in running water, Mathematical Proceedings of the Cambridge Philosophical Society, 49 (1953), 695-706.
|
| [6] |
R. Camassa and D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
| [7] |
M. Chen,
Solitary-wave and multi-pulsed traveling-wave solutions of Boussinesq systems, Applicable Analysis, 72 (2000), 213-240.
doi: 10.1080/00036810008840844. |
| [8] |
A. F. Cheviakov,
Symbolic computation of local symmetries of nonlinear and linear partial and ordinary differential equations, Mathematics in Computer Science, 4 (2010), 203-222.
doi: 10.1007/s11786-010-0051-4. |
| [9] |
W. Choi, Strongly nonlinear long gravity waves in uniform shear flows, Phys. Rev. E, 68 (2003), 026305.
doi: 10.1103/PhysRevE.68.026305. |
| [10] |
D. Clamond, D. Dutykh and A. Galligo, Computer algebra applied to a solitary waves study, Proceedings of the International Symposium on Symbolic and Algebraic Computation, 2015,125–132. |
| [11] |
D. Clamond, D. Dutykh and A. Galligo, Algebraic method for constructing singular steady solitary waves: A case study, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472 (2016), 20160194, 18 pp.
doi: 10.1098/rspa.2016.0194. |
| [12] |
A. Constantin,
Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Eur. J. Mech. B (Fluids), 30 (2011), 12-16.
doi: 10.1016/j.euromechflu.2010.09.008. |
| [13] |
A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Conference series in applied mathematics, Vol. 81, SIAM, Philadelphia, 2011.
doi: 10.1137/1.9781611971873. |
| [14] |
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. |
| [15] |
A. Constantin and R. I. Ivanov,
On an integrable two-component Camassa–Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132.
doi: 10.1016/j.physleta.2008.10.050. |
| [16] |
A. Constantin and W. Strauss,
Exact steady periodic water waves with vorticity, Commun. Pure Appl. Math., 57 (2004), 481-527.
doi: 10.1002/cpa.3046. |
| [17] |
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. |
| [18] |
A. D. D. Craik,
The origins of water wave theory, Ann. Rev. Fluid Mech., 36 (2004), 1-28.
doi: 10.1146/annurev.fluid.36.050802.122118. |
| [19] |
A. F. T. Da Silva and D. H. Peregrine,
Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid Mech., 195 (1988), 281-302.
doi: 10.1017/S0022112088002423. |
| [20] |
D. Dutykh and D. Ionescu-Kruse,
Travelling wave solutions for some two-component shallow water models, Journal of Differential Equations, 261 (2016), 1099-1114.
doi: 10.1016/j.jde.2016.03.035. |
| [21] |
G. A. El, R. H. J. Grimshaw and M. V. Pavlov,
Integrable shallow-water equations and undular bores, Stud. Appl. Math., 106 (2001), 157-186.
doi: 10.1111/1467-9590.00163. |
| [22] |
J. Escher, D. Henry, B. Kolev and T. Lyons,
Two-component equations modelling water waves with constant vorticity, Annali di Matematica, 195 (2016), 249-271.
doi: 10.1007/s10231-014-0461-z. |
| [23] |
N. C. Freeman and R. S. Johnson,
Shallow water waves on shear flows, J. Fluid Mech., 42 (1970), 401-409.
doi: 10.1017/S0022112070001349. |
| [24] |
G. Gui and Y. Liu,
On the global existence and wave–breaking criteria for the two-component Camassa–Holm system, Journal of Functional Analysis, 258 (2010), 4251-4278.
doi: 10.1016/j.jfa.2010.02.008. |
| [25] |
J. Haberlin and T. Lyons, Solitons of shallow water models from energy dependent spectral problems, Eur. Phys. J. Plus, 133 (2018), 16.
doi: 10.1140/epjp/i2018-11848-8. |
| [26] |
D. Holm and R. Ivanov, Two-component CH system: Inverse scattering, peakons and geometry, Inverse Problems, 27 (2011), 045013, 19pp.
doi: 10.1088/0266-5611/27/4/045013. |
| [27] |
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. |
| [28] |
T. Iguchi,
A long wave approximation for capillary-gravity waves and the Kawahara equation, Bull. Inst. Math. Acad. Sin., 2 (2007), 179-220.
|
| [29] |
D. Ionescu-Kruse,
Variational derivation of the Camassa–Holm shallow water equation with non-zero vorticity, Disc. Cont. Dyn. Syst.-A, 19 (2007), 531-543.
doi: 10.3934/dcds.2007.19.531. |
| [30] |
D. Ionescu-Kruse,
Variational derivation of two-component Camassa–Holm shallow water system, Appl. Anal., 92 (2013), 1241-1253.
doi: 10.1080/00036811.2012.667082. |
| [31] |
D. Ionescu-Kruse,
On the small-amplitude long waves in linear shear flows and the Camassa–Holm equation, J. Math. Fluid Mech., 16 (2014), 365-374.
doi: 10.1007/s00021-013-0156-z. |
| [32] |
M. Ito,
Symmetries and conservation laws of a coupled nonlinear wave equation, Phys. Lett. A, 91 (1982), 335-338.
doi: 10.1016/0375-9601(82)90426-1. |
| [33] |
R. Ivanov,
Two-component integrable systems modelling shallow water waves: The constant vorticity case, Wave Motion, 46 (2009), 389-396.
doi: 10.1016/j.wavemoti.2009.06.012. |
| [34] |
R. Ivanov and T. Lyons, Integrable models for shallow water with energy dependent spectral problems, Journal of Nonlinear Mathematical Physics, 19 (2012), 1240008, 17pp.
doi: 10.1142/S1402925112400086. |
| [35] |
R. S. Johnson,
On solutions of the Burns condition (which determines the speed of propagation of linear long waves on a shear flow with or without a critical layer), Geophys. Astrophys. Fluid. Dyn., 57 (1991), 115-133.
doi: 10.1080/03091929108225231. |
| [36] |
R. S. Johnson,
The Camassa–Holm equation for water waves moving over a shear flow, Fluid Dynam. Res., 33 (2003), 97-111.
doi: 10.1016/S0169-5983(03)00036-4. |
| [37] |
A. M. Kamchatnov, R. A. Kraenkel and B. A. Umarov,
Asymptotic soliton train solutions of Kaup–Boussinesq equations, Wave Motion, 38 (2003), 355-365.
doi: 10.1016/S0165-2125(03)00062-3. |
| [38] |
D. J. Kaup,
A higher-order water-wave equation and method for solving it, Prog. Theor. Phys., 54 (1975), 396-408.
doi: 10.1143/PTP.54.396. |
| [39] |
J. Ko and W. Strauss,
Effect of vorticity on steady water waves, J. Fluid Mech., 608 (2008), 197-215.
doi: 10.1017/S0022112008002371. |
| [40] |
B. L. Segal, D. Moldabayev and H. Kalisch,
Explicit solutions for a long-wave model with constant vorticity, Eur. J. Mech. B/Fluids, 65 (2017), 247-256.
doi: 10.1016/j.euromechflu.2017.04.008. |
| [41] |
P. D. Thompson,
The propagation of small surface disturbances through rotational flow, Ann. NY Acad. Sci., 51 (1949), 463-474.
doi: 10.1111/j.1749-6632.1949.tb27285.x. |
| [42] |
J.-M. Vanden-Broeck,
Steep solitary waves in water of finite depth with constant vorticity, J. Fluid Mech., 274 (1994), 339-348.
doi: 10.1017/S0022112094002144. |
| [43] |
E. Wahlén,
Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483.
doi: 10.1016/j.jde.2008.10.005. |
| [44] |
E. Wahlén, Non-existence of three-dimensional travelling water waves with constant non-zero vorticity, J. Fluid Mech., 746 (2014), R2, 7pp.
doi: 10.1017/jfm.2014.131. |
| [45] |
V. E. Zakharov, The inverse scattering method, In: Solitons (Topics in Current Physics, vol 17) ed. R. K. Bullough and P. J. Caudrey (Berlin: Springer, 1980), 1980,243–285. |
show all references
References:
| [1] |
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Reprint of the 1972 edition. Dover Publications, Inc., New York, 1992. |
| [2] |
D. M. Ambrose, J. L. Bona and T. Milgrom, Global solutions and ill-posedness for the Kaup system and related Boussinesq systems, 2017, preprint. |
| [3] |
T. B. Benjamin,
The solitary wave on a stream with an arbitrary distribution of vorticity, J. Fluid Mech, 12 (1962), 97-116.
doi: 10.1017/S0022112062000063. |
| [4] |
F. Biesel,
Etude théorique de la houle en eau courante, Houille Blanche, 5 (1950), 279-285.
|
| [5] |
J. C. Burns,
Long waves in running water, Mathematical Proceedings of the Cambridge Philosophical Society, 49 (1953), 695-706.
|
| [6] |
R. Camassa and D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
| [7] |
M. Chen,
Solitary-wave and multi-pulsed traveling-wave solutions of Boussinesq systems, Applicable Analysis, 72 (2000), 213-240.
doi: 10.1080/00036810008840844. |
| [8] |
A. F. Cheviakov,
Symbolic computation of local symmetries of nonlinear and linear partial and ordinary differential equations, Mathematics in Computer Science, 4 (2010), 203-222.
doi: 10.1007/s11786-010-0051-4. |
| [9] |
W. Choi, Strongly nonlinear long gravity waves in uniform shear flows, Phys. Rev. E, 68 (2003), 026305.
doi: 10.1103/PhysRevE.68.026305. |
| [10] |
D. Clamond, D. Dutykh and A. Galligo, Computer algebra applied to a solitary waves study, Proceedings of the International Symposium on Symbolic and Algebraic Computation, 2015,125–132. |
| [11] |
D. Clamond, D. Dutykh and A. Galligo, Algebraic method for constructing singular steady solitary waves: A case study, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472 (2016), 20160194, 18 pp.
doi: 10.1098/rspa.2016.0194. |
| [12] |
A. Constantin,
Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Eur. J. Mech. B (Fluids), 30 (2011), 12-16.
doi: 10.1016/j.euromechflu.2010.09.008. |
| [13] |
A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Conference series in applied mathematics, Vol. 81, SIAM, Philadelphia, 2011.
doi: 10.1137/1.9781611971873. |
| [14] |
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. |
| [15] |
A. Constantin and R. I. Ivanov,
On an integrable two-component Camassa–Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132.
doi: 10.1016/j.physleta.2008.10.050. |
| [16] |
A. Constantin and W. Strauss,
Exact steady periodic water waves with vorticity, Commun. Pure Appl. Math., 57 (2004), 481-527.
doi: 10.1002/cpa.3046. |
| [17] |
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. |
| [18] |
A. D. D. Craik,
The origins of water wave theory, Ann. Rev. Fluid Mech., 36 (2004), 1-28.
doi: 10.1146/annurev.fluid.36.050802.122118. |
| [19] |
A. F. T. Da Silva and D. H. Peregrine,
Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid Mech., 195 (1988), 281-302.
doi: 10.1017/S0022112088002423. |
| [20] |
D. Dutykh and D. Ionescu-Kruse,
Travelling wave solutions for some two-component shallow water models, Journal of Differential Equations, 261 (2016), 1099-1114.
doi: 10.1016/j.jde.2016.03.035. |
| [21] |
G. A. El, R. H. J. Grimshaw and M. V. Pavlov,
Integrable shallow-water equations and undular bores, Stud. Appl. Math., 106 (2001), 157-186.
doi: 10.1111/1467-9590.00163. |
| [22] |
J. Escher, D. Henry, B. Kolev and T. Lyons,
Two-component equations modelling water waves with constant vorticity, Annali di Matematica, 195 (2016), 249-271.
doi: 10.1007/s10231-014-0461-z. |
| [23] |
N. C. Freeman and R. S. Johnson,
Shallow water waves on shear flows, J. Fluid Mech., 42 (1970), 401-409.
doi: 10.1017/S0022112070001349. |
| [24] |
G. Gui and Y. Liu,
On the global existence and wave–breaking criteria for the two-component Camassa–Holm system, Journal of Functional Analysis, 258 (2010), 4251-4278.
doi: 10.1016/j.jfa.2010.02.008. |
| [25] |
J. Haberlin and T. Lyons, Solitons of shallow water models from energy dependent spectral problems, Eur. Phys. J. Plus, 133 (2018), 16.
doi: 10.1140/epjp/i2018-11848-8. |
| [26] |
D. Holm and R. Ivanov, Two-component CH system: Inverse scattering, peakons and geometry, Inverse Problems, 27 (2011), 045013, 19pp.
doi: 10.1088/0266-5611/27/4/045013. |
| [27] |
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. |
| [28] |
T. Iguchi,
A long wave approximation for capillary-gravity waves and the Kawahara equation, Bull. Inst. Math. Acad. Sin., 2 (2007), 179-220.
|
| [29] |
D. Ionescu-Kruse,
Variational derivation of the Camassa–Holm shallow water equation with non-zero vorticity, Disc. Cont. Dyn. Syst.-A, 19 (2007), 531-543.
doi: 10.3934/dcds.2007.19.531. |
| [30] |
D. Ionescu-Kruse,
Variational derivation of two-component Camassa–Holm shallow water system, Appl. Anal., 92 (2013), 1241-1253.
doi: 10.1080/00036811.2012.667082. |
| [31] |
D. Ionescu-Kruse,
On the small-amplitude long waves in linear shear flows and the Camassa–Holm equation, J. Math. Fluid Mech., 16 (2014), 365-374.
doi: 10.1007/s00021-013-0156-z. |
| [32] |
M. Ito,
Symmetries and conservation laws of a coupled nonlinear wave equation, Phys. Lett. A, 91 (1982), 335-338.
doi: 10.1016/0375-9601(82)90426-1. |
| [33] |
R. Ivanov,
Two-component integrable systems modelling shallow water waves: The constant vorticity case, Wave Motion, 46 (2009), 389-396.
doi: 10.1016/j.wavemoti.2009.06.012. |
| [34] |
R. Ivanov and T. Lyons, Integrable models for shallow water with energy dependent spectral problems, Journal of Nonlinear Mathematical Physics, 19 (2012), 1240008, 17pp.
doi: 10.1142/S1402925112400086. |
| [35] |
R. S. Johnson,
On solutions of the Burns condition (which determines the speed of propagation of linear long waves on a shear flow with or without a critical layer), Geophys. Astrophys. Fluid. Dyn., 57 (1991), 115-133.
doi: 10.1080/03091929108225231. |
| [36] |
R. S. Johnson,
The Camassa–Holm equation for water waves moving over a shear flow, Fluid Dynam. Res., 33 (2003), 97-111.
doi: 10.1016/S0169-5983(03)00036-4. |
| [37] |
A. M. Kamchatnov, R. A. Kraenkel and B. A. Umarov,
Asymptotic soliton train solutions of Kaup–Boussinesq equations, Wave Motion, 38 (2003), 355-365.
doi: 10.1016/S0165-2125(03)00062-3. |
| [38] |
D. J. Kaup,
A higher-order water-wave equation and method for solving it, Prog. Theor. Phys., 54 (1975), 396-408.
doi: 10.1143/PTP.54.396. |
| [39] |
J. Ko and W. Strauss,
Effect of vorticity on steady water waves, J. Fluid Mech., 608 (2008), 197-215.
doi: 10.1017/S0022112008002371. |
| [40] |
B. L. Segal, D. Moldabayev and H. Kalisch,
Explicit solutions for a long-wave model with constant vorticity, Eur. J. Mech. B/Fluids, 65 (2017), 247-256.
doi: 10.1016/j.euromechflu.2017.04.008. |
| [41] |
P. D. Thompson,
The propagation of small surface disturbances through rotational flow, Ann. NY Acad. Sci., 51 (1949), 463-474.
doi: 10.1111/j.1749-6632.1949.tb27285.x. |
| [42] |
J.-M. Vanden-Broeck,
Steep solitary waves in water of finite depth with constant vorticity, J. Fluid Mech., 274 (1994), 339-348.
doi: 10.1017/S0022112094002144. |
| [43] |
E. Wahlén,
Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483.
doi: 10.1016/j.jde.2008.10.005. |
| [44] |
E. Wahlén, Non-existence of three-dimensional travelling water waves with constant non-zero vorticity, J. Fluid Mech., 746 (2014), R2, 7pp.
doi: 10.1017/jfm.2014.131. |
| [45] |
V. E. Zakharov, The inverse scattering method, In: Solitons (Topics in Current Physics, vol 17) ed. R. K. Bullough and P. J. Caudrey (Berlin: Springer, 1980), 1980,243–285. |
Figure 2.
The graph of the inequality $c \mathfrak{c}^{+} > 1 $ against the vorticity Ω.
Figure 3.
The phase-portrait and the solitary wave profiles in the CH2 model with constant vorticity, for: (a) $ c \mathfrak{c}^{+}>0$ ; (b) $ c \mathfrak{c}^{+} = 0$ . We highlight the fact that two fronts tend only algebraically to the equilibrium state H = 1.
Figure 4.
The phase-portrait and the solitary wave profiles in the ZI model with constant vorticity, for: (a) $ c \mathfrak{c}^{+}>1$ ; (b) $c \mathfrak{c}^{+} = 1 $ . We highlight the fact that two fronts tend only algebraically to the equilibrium state H = 1.
Figure 5.
The phase-portrait and the wave profiles for the ZI model with constant vorticity, in the case the polynomial $ \mathcal{R}(H)$ has only two real roots, H1 < 0 and H2 > 0 and the constant $ \mathcal{K}_{1}<0$ .
Figure 6.
The phase-portrait and the wave profiles for the ZI model with constant vorticity in the case the polynomial $ \mathcal{R}(H)$ has four real roots: H1 < 0 and 0 < H2 < H3 < H4 , and the constant: (a)
$ \mathcal{K}_{1}﹥0$ ; (b) $ \mathcal{K}_{1}<0$ .
Figure 7.
The phase-portrait and the wave profiles for the ZI model with constant vorticity in the case the polynomial $ \mathcal{R}(H)$ has four real roots: H1 < H2< H3 < 0 and H4 > 0, and the constant: (a) $ \mathcal{K}_{1}﹥0$ ; (b) $ \mathcal{K}_{1}<0$ .
Figure 8.
Analytical expressions (36) and (35), respectively, for different values of the constant vorticity Ω and of the speed of propagation c.
Figure 9.
Multi-pulse travelling wave solutions with two troughs.
Figure 10.
One-trough travelling wave solutions to the KB system based on analytical formulas (42) and (43).
Figure 11.
Analytical expressions (49) and (47), respectively, for different values of the constant vorticity Ω and of the speed of propagation c.
Figure 12.
The periodic velocity profile for the KB system in the case the polynomial $ \mathcal{P}(H)$ has one positive real root, one negative real root and two complex conjugate roots.
Figure 13.
The periodic velocity profiles for the KB system in the case the polynomial $ \mathcal{P}(H)$ has two positive real roots and two negative real roots.
Figure 14.
The periodic velocity profiles for the KB system in the case the polynomial $ \mathcal{P}(H)$ has three real positive roots and one negative real root.
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