Abstract
This paper deals with the water waves problem for uneven bottom under the influence of surface tension. We consider here an asymptotic limit for the Green–Naghdi equations in KdV scale, that is the Boussinesq system. The derivation of the KdV equation with uneven bottom under the influence of surface tension has been established. Indeed, this derivation is obtained in a formal way by using the Whitham technique, then the analytic solution to this equation has been obtained in case of flat bottom. However, in case of uneven bottom an \(H^s\)-consistent solution has been obtained. Also, an \(H^s\)-consistent solution for the Boussinesq system has been established, taking into consideration the influence of surface tension and uneven bottom. Finally, we confirmed the obtained theoretical results of this paper numerically, by devoting the last section to make a numerical validation. Moreover, analytic solutions to the KdV\(\sigma \) and Boussinesq system were established in case of several bottom parametrization including the linear one.
Similar content being viewed by others
Availability of data and material
Not applicable.
References
Abu Arqub, O.: Application of residual power series method for the solution of time-fractional Schrödinger equations in one-dimensional space. Fundam. Inf. 166(2), 87–110 (2019)
Abu Arqub, O.: Numerical algorithm for the solutions of fractional order systems of Dirichlet function types with comparative analysis. Fundam. Inf. 166(2), 111–137 (2019)
Abu Arqub, O., Banan, M.: Modulation of reproducing kernel Hilbert space method for numerical solutions of Riccati and Bernoulli equations in the Atangana–Baleanu fractional sense. Chaos Solitons Fractals 125, 163–170 (2019)
Abu Arqub, O., Maayah, B.: Fitted fractional reproducing kernel algorithm for the numerical solutions of ABC-Fractional Volterra integro-differential equations. Chaos Solitons Fractals 126, 394–402 (2019)
Aurther, C.H., Granero-Belinchon, R., Shkoller, S., Wilkening, J.: Rigorous asymptotic models of water waves. Water Waves 1(1), 71–130 (2019)
Bayat, M., Paker, I.: Accurate analytical solution for nonlinear free vibration of beams. Struct. Eng. Mech. 43(3), 337–347 (2012)
Bayat, M., Paker, I., Domairry, G.: Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: a review. Latin Am. J. Solids Struct. 9, 1–93 (2012)
Bona, J.L., Chen, M., Saut, J.C.: Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory. J. Nonlinear Sci. 12, 283–318 (2002)
Burden, R.L., Douglas Faires, J.: Numerical Analysis. Brooks/Cole, Pacific Grove (1997)
Capistrano-Filho, R.A., Gallego, F.A., Pazoto, A.F.: On the Well-Posedness and Large-Time Behavior of Higher order Boussinesq System. IOP Publishing Ltd and London Mathematical Society, Bristol (2019)
Craig, W.: An existence theory for water waves and the Boussinesq and the Korteweg–de Vries scaling limits. Commun. Partial Differ. Equ. 10, 787–1003 (1985)
Green, A.E., Naghdi, P.M.: A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78, 237–246 (1976)
Haidar, M., El Arwadi, T., Israwi, S.: Existence of a regular solution for 1D Green–Naghdi equations with surface tension at a large time instant. Bound. Value Probl. 1, 1–20 (2018)
Israwi, S.: Variable depth KDV equations and generalizations to more nonlinear regimes. Math. Model. Numer. Anal. 44(2), 347–370 (2009)
Israwi, S.: Large time existence for 1D Green–Naghdi equations. Nonlinear Anal. 74, 83–93 (2011)
Israwi, S., Mourad, A.: An explicit solution with correctors for the Green–Naghdi equations. Mediterr. J. Math. 18, 519–532 (2013)
Israwi, S., Talhouk, R.: Local well-posedness of a nonlinear KdV-type equation. C. R. Math. 351, 895–899 (2013)
Karczewska, A., Rozmej, P.: Shallow Water Waves-Extended Korteweg–de Vries equations. Second Order Perturbation Approach, pp. 18–19 (2018)
Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Philos. Mag. J. Sci. 39(240), 422–443 (1895)
Lannes, D.: Well-Posedness of the Water Waves Equations, vol. 18, pp. 605–654. American Mathematical Society, Providence (2005)
Lannes, D.: The Water Waves Problem. American Mathematical Society, Providence (2013)
Nalimov, V.I.: The Cauchy–Poison problem (Russian). Dinamika Splosn 254, 104–210 (1974). Sredy Vyp. 18 Dinamika Zidkost. so Svobod. Granicami
Wu, S.: Well-posedness in sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130, 39–72 (1997)
Wu, S.: Well-posedness in sobolev spaces of the full water wave problem in 3-D. J. Am. Math. Soc. 12(2), 445–495 (1999)
Yosihara, H.: Gravity waves on the free surface of an incompressible perfect fluid of finite depth. Publ. Res. Inst. Math. Sci. 18(1), 49–96 (1982)
Acknowledgements
Not applicable.
Funding
This paper is partially supported by Lebanese University.
Author information
Authors and Affiliations
Contributions
M.H. writes the original draft, T.E.A. makes the numerical validation and S.I. contributes to establish the well-posedness of the system. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Haidar, M., El Arwadi, T. & Israwi, S. Explicit solutions and numerical simulations for an asymptotic water waves model with surface tension. J. Appl. Math. Comput. 63, 655–681 (2020). https://doi.org/10.1007/s12190-020-01333-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-020-01333-8