Generalized KdV-type equations versus Boussinesq’s equations for uneven bottom – numerical study
Karczewska Anna 1, Rozmej Piotr 2
1 Faculty of Mathematics, Computer Science and Econometrics
University of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra, Poland
E-mail: a.karczewska@wmie.uz.zgora.pl2 Faculty of Physics and Astronomy, Institute of Physics
University of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra, Poland
E-mail: p.rozmej@if.uz.zgora.pl
Received:
Received: 25 November 2020; accepted: 7 December 2020; published online: 17 December 2020
DOI: 10.12921/cmst.2020.0000036
Abstract:
The paper’s main goal is to compare the motion of solitary surface waves resulting from two similar but slightly different approaches. In the first approach, the numerical evolution of soliton surface waves moving over the uneven bottom is obtained using single wave equations. In the second approach, the numerical evolution of the same initial conditions is obtained by the solution of a coupled set of the Boussinesq equations for the same Euler equations system. We discuss four physically relevant cases of relationships between small parameters α, β, δ. For the flat bottom, these cases imply the Korteweg-de Vries equation (KdV), the extended KdV (KdV2), fifth-order KdV, and the Gardner equation. In all studied cases, the influence of the bottom variations on the amplitude and velocity of a surface wave calculated from the Boussinesq equations is substantially more significant than that obtained from single wave equations.
Key words:
Gardner equation, KdV-type equations, numerical evolution, uneven bottom
References:
[1] C.C. Mei, B. Le Méhauté, Note on the equations of long waves over an uneven bottom, J. Geophys. Research 71, 393–400 (1966).
[2] R. Grimshaw, The solitary wave in water of variable depth, J. Fluid Mech. 42, 639–656 (1970).
[3] V.D. Djordjevic´, L.G. Redekopp, On the development of packets of surface gravity waves moving over an uneven bottom, J. Appl. Math. Phys. (ZAMP) 29, 950–962 (1978).
[4] E.S. Benilov, C.P. Howlin, Evolution of packets of surface gravity waves over strong smooth topography, Stud. Appl. Math. 116, 289–301 (2006).
[5] O. Nakoulima, N. Zahibo, E. Pelinovsky, T. Talipova, A. Kurkin, Solitary wave dynamics in shallow water over periodic topography, Chaos 15, 037107 (2005).
[6] R. Grimshaw, E. Pelinovsky, T. Talipova, Fission of a weakly nonlinear interfacial solitary wave at a step, Geophys. Astrophys. Fluid Dyn. 102, 179–194 (2008).
[7] E. Pelinovsky, B.H. Choi, T. Talipova, S.B. Woo, D.C. Kim, Solitary wave transformation on the underwater step: theory and numerical experiments, Appl. Math. Comput. 217, 1704–1718 (2010).
[8] R.H.J. Grimshaw, N.F. Smyth, Resonant flow of a stratified fluid over topography, J. Fluid Mech. 169, 429–464 (1986).
[9] N.F. Smyth, Modulation theory Sslution for resonant flow over topography, Proc. R. Soc. Lond. A 409, 79–97 (1987).
[10] E.N. Pelinovskii, A.V. Slyunayev, Generation and interaction of large-amplitude solitons, JETP Lett. 67, 655–661 (1998).
[11] A.M. Kamchatnov, Y-H. Kuo, T-C. Lin, T-L. Horng, S-C. Gou, R. Clift, G.A. El, R.H.J. Grimshaw, Undular bore theory for the Gardner equation, Phys. Rev. E 86, 036605 (2012).
[12] E. van Greoesen, S.R. Pudjaprasetya, Uni-directional waves over slowly varying bottom. Part I: Derivation of a KdV-type of equation, Wave Motion 18, 345–370 (1993).
[13] S.R. Pudjaprasetya, E. van Greoesen, Uni-directional waves over slowly varying bottom. Part II: Quasi-homogeneous approximation of distorting waves, Wave Motion 23, 23–38 (1996).
[14] A.E. Green, P.M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech. 78, 237–246 (1976).
[15] J.W. Kim, K.J. Bai, R.C. Ertekin, W.C. Webster, A derivation of the Green-Naghdi equations for irrotational flows, J. Eng. Math. 40, 17–42 (2001).
[16] B.T. Nadiga, L.G. Margolin, P.K. Smolarkiewicz, Different approximations of shallow fluid flow over an obstacle, Phys. Fluids 1996(8), 2066–2077 (1996).
[17] I.T. Selezov, Propagation of unsteady nonlinear surface gravity waves above an irregular bottom, Int. J. Fluid Mech. 27, 146–157 (2000).
[18] X. Niu, X. Yu, Analytic solution of long wave propagation over a submerged hump, Coastal Eng. 58, 143–150 (2011).
[19] H-W. Liu, HJ-J. Xie, Discussion of “Analytic solution of long wave propagation over a submerged hump” by Niu and Yu (2011), Coastal Eng. 58, 948–952 (2011).
[20] S. Israwi, Variable depth KdV equations and generalizations to more nonlinear regimes, ESAIM: M2AN 44, 347–370 (2012).
[21] M. Duruflé, S. Israwi, A numerical study of variable depth KdV quations and generalizations of Camassa-Holm-like equations, J. Comp. Appl. Math. 236, 4149–4265 (2010).
[22] C. Yuan, R. Grimshaw, E. Johnson, X. Chen, The Propagation of internal solitary waves over variable topography in a horizontally two-dimensional framework, J. Phys. Ocean 48, 283–300 (2018).
[23] L. Fan, W. Yan, On the weak solutions and persistence properties for the variable depth KDV general equations, Nonlinear Analysis: Real World Applications 44, 223–245 (2018).
[24] Y. Stepanyants, The effects of interplay between the rotation and shoaling for a solitary wave on variable topography, Stud. Appl. Math. 142, 465–486 (2019).
[25] O. Madsen, C. Mei, The transformation of a solitary wave over an uneven bottom, J. Fluid Mech. 39, 781–791 (1969).
[26] T. Kakutani, Effect of an uneven bottom on gravity waves, J. Phys. Soc. Japan 30, 272–276 (1971).
[27] R.S. Johnson, Some numerical solutions of a variable coefficient Korteweg-de Vries equation (with applications to solitary wave development on a shelf), J. Fluid Mech. 54, 81–91 (1972).
[28] R. Johnson, On the development of a solitary wave moving over an uneven bottom, Math. Proc. Cambridge Phil. Soc. 73, 183–203 (1973).
[29] E.S. Benilov, On the surface waves in a shallow channel with an uneven bottom, Stud. Appl. Math. 87, 1–14 (1992).
[30] R.R. Rosales, G.C. Papanicolau, Gravity waves in a channel with a rough bottom, Stud. Appl. Math. 68(2), 89–102 (1983).
[31] A. Karczewska, P. Rozmej, Can simple KdV-type equations be derived for shallow water problem with bottom bathymetry? Commun Nonlinear Sci Numer Simulat 82, 105073 (2020).
[32] G.I. Burde, A. Sergyeyev, Ordering of two small parameters in the shallow water wave problem, J. Phys. A: Math. Theor. 46, 075501 (2013).
[33] D.J. Korteweg, G. de Vries, On the change of form of the long waves advancing in a rectangular canal, and on a new type of stationary waves, Phil. Mag. (5) 39, 422–443 (1895).
[34] T.R. Marchant, N.F. Smyth, The extended Korteweg-de Vries equation and the resonant flow of a fluid over topography, J. Fluid Mech. 221, 263–288 (1990).
[35] A. Karczewska, P. Rozmej, E. Infeld, Shallow water soliton dynamics beyond Korteweg-de Vries equation, Phys. Rev. E 90, 012907 (2014).
[36] A. Karczewska, P. Rozmej, Shallow water waves – extended Korteweg-de Vries equations, Oficyna Wydawnicza Uniwersytetu Zielonogórskiego, Zielona Góra (2018).
[37] E. Infeld, A. Karczewska, G. Rowlands, P. Rozmej, Exact cnoidal solutions of the extended KdV equation, Acta Phys. Pol. A 133, 1191–1199 (2018).
[38] P. Rozmej, A. Karczewska, E. Infeld, Superposition solutions to the extended KdV equation for water surface waves, Nonlinear Dyn. 91, 1085–1093 (2018).
[39] P. Rozmej, A. Karczewska, New exact superposition solutions to KdV2 equation, Adv. Math. Phys. 2018, 5095482 (2018).
[40] A. Karczewska, P. Rozmej, Remarks on existence/nonexistence of analytic solutions to higher order KdV equations, Acta Phys. Pol. A 136, 910–915 (2019).
[41] J.K. Hunter, J. Scheurle, Existence of perturbed solitary wave solutions to a model equation for water waves, Physica D 32, 253–268 (1988).
[42] B. Dey, A. Khare, C.N. Kumar, Stationary solitons of the fifth order KdV-type. Equations and their stabilization, Phys. Lett. A 223, 449–452 (1996).
[43] T.J. Bridges, G. Derks, G. Gottwald, Stability and instability of solitary waves of the fifth order KdV equation: a numerical framework, Physica D 172, 190–216 (2002).
[44] R. Grimshaw, E. Pelinovsky, T. Talipova, Solitary wave transformation in medium with sign-variable quadratic nonlinearity and cubic nonlinearity, Physica D 132, 40–62 (1999).
[45] L. Ostrovsky, E. Pelinovsky, V. Shrira, Y. Stepanyants, Beyond the KdV: Post-explosion development, CHAOS 25, 097620 (2015).
[46] G. Rowlands, P. Rozmej, E. Infeld, A. Karczewska, Single soliton solution to the extended KdV equation over uneven depth, Eur. Phys. J. E 40, 100 (2017).
The paper’s main goal is to compare the motion of solitary surface waves resulting from two similar but slightly different approaches. In the first approach, the numerical evolution of soliton surface waves moving over the uneven bottom is obtained using single wave equations. In the second approach, the numerical evolution of the same initial conditions is obtained by the solution of a coupled set of the Boussinesq equations for the same Euler equations system. We discuss four physically relevant cases of relationships between small parameters α, β, δ. For the flat bottom, these cases imply the Korteweg-de Vries equation (KdV), the extended KdV (KdV2), fifth-order KdV, and the Gardner equation. In all studied cases, the influence of the bottom variations on the amplitude and velocity of a surface wave calculated from the Boussinesq equations is substantially more significant than that obtained from single wave equations.
Key words:
Gardner equation, KdV-type equations, numerical evolution, uneven bottom
References:
[1] C.C. Mei, B. Le Méhauté, Note on the equations of long waves over an uneven bottom, J. Geophys. Research 71, 393–400 (1966).
[2] R. Grimshaw, The solitary wave in water of variable depth, J. Fluid Mech. 42, 639–656 (1970).
[3] V.D. Djordjevic´, L.G. Redekopp, On the development of packets of surface gravity waves moving over an uneven bottom, J. Appl. Math. Phys. (ZAMP) 29, 950–962 (1978).
[4] E.S. Benilov, C.P. Howlin, Evolution of packets of surface gravity waves over strong smooth topography, Stud. Appl. Math. 116, 289–301 (2006).
[5] O. Nakoulima, N. Zahibo, E. Pelinovsky, T. Talipova, A. Kurkin, Solitary wave dynamics in shallow water over periodic topography, Chaos 15, 037107 (2005).
[6] R. Grimshaw, E. Pelinovsky, T. Talipova, Fission of a weakly nonlinear interfacial solitary wave at a step, Geophys. Astrophys. Fluid Dyn. 102, 179–194 (2008).
[7] E. Pelinovsky, B.H. Choi, T. Talipova, S.B. Woo, D.C. Kim, Solitary wave transformation on the underwater step: theory and numerical experiments, Appl. Math. Comput. 217, 1704–1718 (2010).
[8] R.H.J. Grimshaw, N.F. Smyth, Resonant flow of a stratified fluid over topography, J. Fluid Mech. 169, 429–464 (1986).
[9] N.F. Smyth, Modulation theory Sslution for resonant flow over topography, Proc. R. Soc. Lond. A 409, 79–97 (1987).
[10] E.N. Pelinovskii, A.V. Slyunayev, Generation and interaction of large-amplitude solitons, JETP Lett. 67, 655–661 (1998).
[11] A.M. Kamchatnov, Y-H. Kuo, T-C. Lin, T-L. Horng, S-C. Gou, R. Clift, G.A. El, R.H.J. Grimshaw, Undular bore theory for the Gardner equation, Phys. Rev. E 86, 036605 (2012).
[12] E. van Greoesen, S.R. Pudjaprasetya, Uni-directional waves over slowly varying bottom. Part I: Derivation of a KdV-type of equation, Wave Motion 18, 345–370 (1993).
[13] S.R. Pudjaprasetya, E. van Greoesen, Uni-directional waves over slowly varying bottom. Part II: Quasi-homogeneous approximation of distorting waves, Wave Motion 23, 23–38 (1996).
[14] A.E. Green, P.M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech. 78, 237–246 (1976).
[15] J.W. Kim, K.J. Bai, R.C. Ertekin, W.C. Webster, A derivation of the Green-Naghdi equations for irrotational flows, J. Eng. Math. 40, 17–42 (2001).
[16] B.T. Nadiga, L.G. Margolin, P.K. Smolarkiewicz, Different approximations of shallow fluid flow over an obstacle, Phys. Fluids 1996(8), 2066–2077 (1996).
[17] I.T. Selezov, Propagation of unsteady nonlinear surface gravity waves above an irregular bottom, Int. J. Fluid Mech. 27, 146–157 (2000).
[18] X. Niu, X. Yu, Analytic solution of long wave propagation over a submerged hump, Coastal Eng. 58, 143–150 (2011).
[19] H-W. Liu, HJ-J. Xie, Discussion of “Analytic solution of long wave propagation over a submerged hump” by Niu and Yu (2011), Coastal Eng. 58, 948–952 (2011).
[20] S. Israwi, Variable depth KdV equations and generalizations to more nonlinear regimes, ESAIM: M2AN 44, 347–370 (2012).
[21] M. Duruflé, S. Israwi, A numerical study of variable depth KdV quations and generalizations of Camassa-Holm-like equations, J. Comp. Appl. Math. 236, 4149–4265 (2010).
[22] C. Yuan, R. Grimshaw, E. Johnson, X. Chen, The Propagation of internal solitary waves over variable topography in a horizontally two-dimensional framework, J. Phys. Ocean 48, 283–300 (2018).
[23] L. Fan, W. Yan, On the weak solutions and persistence properties for the variable depth KDV general equations, Nonlinear Analysis: Real World Applications 44, 223–245 (2018).
[24] Y. Stepanyants, The effects of interplay between the rotation and shoaling for a solitary wave on variable topography, Stud. Appl. Math. 142, 465–486 (2019).
[25] O. Madsen, C. Mei, The transformation of a solitary wave over an uneven bottom, J. Fluid Mech. 39, 781–791 (1969).
[26] T. Kakutani, Effect of an uneven bottom on gravity waves, J. Phys. Soc. Japan 30, 272–276 (1971).
[27] R.S. Johnson, Some numerical solutions of a variable coefficient Korteweg-de Vries equation (with applications to solitary wave development on a shelf), J. Fluid Mech. 54, 81–91 (1972).
[28] R. Johnson, On the development of a solitary wave moving over an uneven bottom, Math. Proc. Cambridge Phil. Soc. 73, 183–203 (1973).
[29] E.S. Benilov, On the surface waves in a shallow channel with an uneven bottom, Stud. Appl. Math. 87, 1–14 (1992).
[30] R.R. Rosales, G.C. Papanicolau, Gravity waves in a channel with a rough bottom, Stud. Appl. Math. 68(2), 89–102 (1983).
[31] A. Karczewska, P. Rozmej, Can simple KdV-type equations be derived for shallow water problem with bottom bathymetry? Commun Nonlinear Sci Numer Simulat 82, 105073 (2020).
[32] G.I. Burde, A. Sergyeyev, Ordering of two small parameters in the shallow water wave problem, J. Phys. A: Math. Theor. 46, 075501 (2013).
[33] D.J. Korteweg, G. de Vries, On the change of form of the long waves advancing in a rectangular canal, and on a new type of stationary waves, Phil. Mag. (5) 39, 422–443 (1895).
[34] T.R. Marchant, N.F. Smyth, The extended Korteweg-de Vries equation and the resonant flow of a fluid over topography, J. Fluid Mech. 221, 263–288 (1990).
[35] A. Karczewska, P. Rozmej, E. Infeld, Shallow water soliton dynamics beyond Korteweg-de Vries equation, Phys. Rev. E 90, 012907 (2014).
[36] A. Karczewska, P. Rozmej, Shallow water waves – extended Korteweg-de Vries equations, Oficyna Wydawnicza Uniwersytetu Zielonogórskiego, Zielona Góra (2018).
[37] E. Infeld, A. Karczewska, G. Rowlands, P. Rozmej, Exact cnoidal solutions of the extended KdV equation, Acta Phys. Pol. A 133, 1191–1199 (2018).
[38] P. Rozmej, A. Karczewska, E. Infeld, Superposition solutions to the extended KdV equation for water surface waves, Nonlinear Dyn. 91, 1085–1093 (2018).
[39] P. Rozmej, A. Karczewska, New exact superposition solutions to KdV2 equation, Adv. Math. Phys. 2018, 5095482 (2018).
[40] A. Karczewska, P. Rozmej, Remarks on existence/nonexistence of analytic solutions to higher order KdV equations, Acta Phys. Pol. A 136, 910–915 (2019).
[41] J.K. Hunter, J. Scheurle, Existence of perturbed solitary wave solutions to a model equation for water waves, Physica D 32, 253–268 (1988).
[42] B. Dey, A. Khare, C.N. Kumar, Stationary solitons of the fifth order KdV-type. Equations and their stabilization, Phys. Lett. A 223, 449–452 (1996).
[43] T.J. Bridges, G. Derks, G. Gottwald, Stability and instability of solitary waves of the fifth order KdV equation: a numerical framework, Physica D 172, 190–216 (2002).
[44] R. Grimshaw, E. Pelinovsky, T. Talipova, Solitary wave transformation in medium with sign-variable quadratic nonlinearity and cubic nonlinearity, Physica D 132, 40–62 (1999).
[45] L. Ostrovsky, E. Pelinovsky, V. Shrira, Y. Stepanyants, Beyond the KdV: Post-explosion development, CHAOS 25, 097620 (2015).
[46] G. Rowlands, P. Rozmej, E. Infeld, A. Karczewska, Single soliton solution to the extended KdV equation over uneven depth, Eur. Phys. J. E 40, 100 (2017).
