American Institute of Mathematical Sciences

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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

Denys Dutykh 1,2, and  Delia Ionescu-Kruse 3,,

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.

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 & 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

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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

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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. 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.  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. 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.  Google Scholar

[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.  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. 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.  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. 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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  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. 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.  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.
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Figure 2.  The graph of the inequality $c \mathfrak{c}^{+} > 1 $ against the vorticity Ω.
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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.
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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.
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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$.
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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$.
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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$.
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Figure 8.  Analytical expressions (36) and (35), respectively, for different values of the constant vorticity Ω and of the speed of propagation c.
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Figure 9.  Multi-pulse travelling wave solutions with two troughs.
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Figure 10.  One-trough travelling wave solutions to the KB system based on analytical formulas (42) and (43).
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Figure 11.  Analytical expressions (49) and (47), respectively, for different values of the constant vorticity Ω and of the speed of propagation c.
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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.
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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.
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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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