# American Institute of Mathematical Sciences

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. [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.

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##### 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.
Sketch of the fluid domain: free surface flow on a shear current.
The graph of the inequality $c \mathfrak{c}^{+} > 1$ against the vorticity Ω.
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.
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.
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 $\mathcal{K}_{1}<0$.
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) $\mathcal{K}_{1}<0$.
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) $\mathcal{K}_{1}<0$.
Analytical expressions (36) and (35), respectively, for different values of the constant vorticityand of the speed of propagation c.
Multi-pulse travelling wave solutions with two troughs.
One-trough travelling wave solutions to the KB system based on analytical formulas (42) and (43).
Analytical expressions (49) and (47), respectively, for different values of the constant vorticityand of the speed of propagation c.
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.
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.
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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