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

* Corresponding author: Delia Ionescu-Kruse

Received  January 2019 Revised  April 2019 Published  May 2019

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.

Citation: Denys Dutykh, Delia Ionescu-Kruse. Effects of vorticity on the travelling waves of some shallow water two-component systems. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[4]

F. Biesel, Etude théorique de la houle en eau courante, Houille Blanche, 5 (1950), 279-285.   Google Scholar

[5]

J. C. Burns, Long waves in running water, Mathematical Proceedings of the Cambridge Philosophical Society, 49 (1953), 695-706.   Google Scholar

[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.  Google Scholar

[7]

M. Chen, Solitary-wave and multi-pulsed traveling-wave solutions of Boussinesq systems, Applicable Analysis, 72 (2000), 213-240.  doi: 10.1080/00036810008840844.  Google Scholar

[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.  Google Scholar

[9]

W. Choi, Strongly nonlinear long gravity waves in uniform shear flows, Phys. Rev. E, 68 (2003), 026305. doi: 10.1103/PhysRevE.68.026305.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. ConstantinM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

G. A. ElR. 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.  Google Scholar

[22]

J. EscherD. HenryB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

T. Iguchi, A long wave approximation for capillary-gravity waves and the Kawahara equation, Bull. Inst. Math. Acad. Sin., 2 (2007), 179-220.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

A. M. KamchatnovR. 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.  Google Scholar

[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.  Google Scholar

[39]

J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech., 608 (2008), 197-215.  doi: 10.1017/S0022112008002371.  Google Scholar

[40]

B. L. SegalD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[4]

F. Biesel, Etude théorique de la houle en eau courante, Houille Blanche, 5 (1950), 279-285.   Google Scholar

[5]

J. C. Burns, Long waves in running water, Mathematical Proceedings of the Cambridge Philosophical Society, 49 (1953), 695-706.   Google Scholar

[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.  Google Scholar

[7]

M. Chen, Solitary-wave and multi-pulsed traveling-wave solutions of Boussinesq systems, Applicable Analysis, 72 (2000), 213-240.  doi: 10.1080/00036810008840844.  Google Scholar

[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.  Google Scholar

[9]

W. Choi, Strongly nonlinear long gravity waves in uniform shear flows, Phys. Rev. E, 68 (2003), 026305. doi: 10.1103/PhysRevE.68.026305.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. ConstantinM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

G. A. ElR. 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.  Google Scholar

[22]

J. EscherD. HenryB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

T. Iguchi, A long wave approximation for capillary-gravity waves and the Kawahara equation, Bull. Inst. Math. Acad. Sin., 2 (2007), 179-220.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

A. M. KamchatnovR. 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.  Google Scholar

[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.  Google Scholar

[39]

J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech., 608 (2008), 197-215.  doi: 10.1017/S0022112008002371.  Google Scholar

[40]

B. L. SegalD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

Figure 1.  Sketch of the fluid domain: free surface flow on a shear current.
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 vorticityand 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 vorticityand 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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