The New Extended KdV Equation for the Case of an Uneven Bottom
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
University of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra, Poland
E-mail: P.Rozmej@if.uz.zgora.pl
Received:
Received: 20 November 2018; revised: 03 December 2018; accepted: 04 December 2018; published online: 20 December 2018
DOI: 10.12921/cmst.2018.0000057
Abstract:
The consistent derivation of the extended KdV equation for an uneven bottom for the case of α = O(β) andδ = O(β2) is presented. This is the only one case when second order KdV type nonlinear wave equation can be derived for arbitrary bounded bottom function.
Key words:
nonlinear equations, second order perturbation approach, surface gravity waves, uneven bottom
References:
[1] A. Karczewska, P. Rozmej, Ł. Rutkowski, A new nonlinear equation in the shallow water wave problem, Physica Scripta,89, 054026, (2014).
[2] A. Karczewska, P. Rozmej, E. Infeld Shallow water soli- ton dynamics beyond KdV, Physical Review E, 90, 012907, (2014).
[3] D.J.Korteweg,G.deVries,Onthechangeofformofthelong waves advancing in a rectangular canal, and on a new type of stationary waves, Phil. Mag. (5), 39, 422 (1895).
[4] T.R.Marchant,N.F.Smyth,TheextendedKorteweg-deVries equation and the resonant flow of a fluid over topography, Journal of Fluid Mechanics, 221, 263-288, (1990).
[5] The review by the anonymous referee of the paper: P. Rozmej, A. Karczewska, Comment on the paper “The third-order perturbed Korteweg-de Vries equation for shal- low water waves with a non-flat bottom” by M. T.C. Fokou Kofané, A. Mohamadou and E. Yomba, Eur. Phys. J. Plus, 132, 410 (2017), arXiv:1804.01940.
[6] P. Rozmej, A. Karczewska, Extended KdV equation for the case of uneven bottom, arXiv:1810.07183. Submitted to Phys. Rev. E.
[7] E. Infeld, A. Karczewska, G. Rowlands, P. Rozmej, Exact cnoidal solutions of the extended KdV equation, Acta Phys. Pol. A, 133, 1191-1199, (2018).
[8] P. Rozmej, A. Karczewska, E. Infeld: Superposition solutions to the extended KdV equation for water surface waves, Non- linear Dynamics 91, 1085-1093, (2018).
[9] P. Rozmej, A. Karczewska, New Exact Superposition Solu- tions to KdV2 Equation, Advances in Mathematical Physics.2018, Article ID 5095482, 1-9, (2018).
[10] 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).
[11] A. Karczewska, P. Rozmej, Shallow water waves – extended Kortewed-de Vries equations, University of Zielona Góra, 2018.
The consistent derivation of the extended KdV equation for an uneven bottom for the case of α = O(β) andδ = O(β2) is presented. This is the only one case when second order KdV type nonlinear wave equation can be derived for arbitrary bounded bottom function.
Key words:
nonlinear equations, second order perturbation approach, surface gravity waves, uneven bottom
References:
[1] A. Karczewska, P. Rozmej, Ł. Rutkowski, A new nonlinear equation in the shallow water wave problem, Physica Scripta,89, 054026, (2014).
[2] A. Karczewska, P. Rozmej, E. Infeld Shallow water soli- ton dynamics beyond KdV, Physical Review E, 90, 012907, (2014).
[3] D.J.Korteweg,G.deVries,Onthechangeofformofthelong waves advancing in a rectangular canal, and on a new type of stationary waves, Phil. Mag. (5), 39, 422 (1895).
[4] T.R.Marchant,N.F.Smyth,TheextendedKorteweg-deVries equation and the resonant flow of a fluid over topography, Journal of Fluid Mechanics, 221, 263-288, (1990).
[5] The review by the anonymous referee of the paper: P. Rozmej, A. Karczewska, Comment on the paper “The third-order perturbed Korteweg-de Vries equation for shal- low water waves with a non-flat bottom” by M. T.C. Fokou Kofané, A. Mohamadou and E. Yomba, Eur. Phys. J. Plus, 132, 410 (2017), arXiv:1804.01940.
[6] P. Rozmej, A. Karczewska, Extended KdV equation for the case of uneven bottom, arXiv:1810.07183. Submitted to Phys. Rev. E.
[7] E. Infeld, A. Karczewska, G. Rowlands, P. Rozmej, Exact cnoidal solutions of the extended KdV equation, Acta Phys. Pol. A, 133, 1191-1199, (2018).
[8] P. Rozmej, A. Karczewska, E. Infeld: Superposition solutions to the extended KdV equation for water surface waves, Non- linear Dynamics 91, 1085-1093, (2018).
[9] P. Rozmej, A. Karczewska, New Exact Superposition Solu- tions to KdV2 Equation, Advances in Mathematical Physics.2018, Article ID 5095482, 1-9, (2018).
[10] 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).
[11] A. Karczewska, P. Rozmej, Shallow water waves – extended Kortewed-de Vries equations, University of Zielona Góra, 2018.
