Home Singularity analysis and analytic solutions for the Benney–Gjevik equations
Article
Licensed
Unlicensed Requires Authentication

Singularity analysis and analytic solutions for the Benney–Gjevik equations

  • Andronikos Paliathanasis ORCID logo EMAIL logo , Genly Leon and P. G. L. Leach
Published/Copyright: March 8, 2021
Become an author with De Gruyter Brill
Author Information
International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 24 Issue 3

Abstract

We apply the Painlevé test for the Benney and the Benney–Gjevik equations, which describe waves in falling liquids. We prove that these two nonlinear 1 + 1 evolution equations pass the singularity test for the travelling-wave solutions. The algebraic solutions in terms of Laurent expansions are presented.

Keywords: liquid film; Painlevé test; singularity analysis
Corresponding author: Andronikos Paliathanasis, Institute of Systems Science, Durban University of Technology, PO Box 1334, Durban 4000, Republic of South Africa, E-mail:

Acknowledgements

AP and GL were funded by Agencia Nacional de Investigación y Desarrollo (ANID) through the program FONDECYT Iniciación grant no. 11180126. GL thanks the Department of Mathematics and the Vicerrectoría de Investigación y Desarrollo Tecnológico at Universidad Cató lica del Norte for financial support.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: FONDECYT Iniciación grant no. 11180126.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

A Coefficients for Eqs. (1) and (2)

The coefficients of Eq. (1) are as follows [1]

(18) A u = 2 u 2

(19) B u = − 8 15 R u 6 + 2 3 u 3 cot a

(20) C u = − 16 5 R u 5 + 2 u 2 cot a

(21) D u = − 2 u 4 − 32 68 R 2 u 10 + 40 63 R u 7 cot a

(22) E u = − 52 3 u 3 + 433 63 R 2 u 9 + 392 45 R u 6 cot a

(23) F u = − 14 u 2 − 29 R 2 u 8 + 64 5 R u 5 cot a ,

where R denotes the Rayleigh’s number, a is the inclination of the plane on which the motion occurs, while the dependent variable u t , x describes the surface of the flow. While for Eq. (2) coefficients D ̄ u and E ̄ u are

(24) D ̄ u = 3 W u 2 , E ̄ u = 3 W u 3

in which W is a parameter of order a −2 [9].

References

[1] D. J. Benney, “Long waves on liquid films,” Stud. Appl. Math., vol. 45, p. 150, 1966. https://doi.org/10.1002/sapm1966451150.Search in Google Scholar

[2] T. B. Benjamin, “Wave formation in laminar flow down an inclined plane,” J. Fluid Mech., vol. 2, p. 554, 1957. https://doi.org/10.1017/s0022112057000373.Search in Google Scholar

[3] C. S. Yih, “Stability of liquid flow down an inclined plane,” Phys. Fluids, vol. 6, p. 321, 1963. https://doi.org/10.1063/1.1706737.Search in Google Scholar

[4] C. C. Mei, “Nonlinear gravity waves in a thin sheet of viscous fluid,” J. Math. Phys., vol. 45, p. 266, 1966. https://doi.org/10.1002/sapm1966451266.Search in Google Scholar

[5] A. D. D Craik, Wave Interactions and Fluid Flows, Cambridge, Cambridge University Press, 1988.Search in Google Scholar

[6] S. Kalliadasis, C. Ruyer-Quil, B. Scheid, and M. G. Velarde, Falling Liquid Films, London, Springer-Verlag, 2012.10.1007/978-1-84882-367-9Search in Google Scholar

[7] R. V. Craster and O. K. Matar, “On viscous beads flowing down a vertical fibre,” J. Fluid Mech., vol. 553, p. 96, 2006. https://doi.org/10.1017/s0022112006008706.Search in Google Scholar

[8] A. Oron, S. H. Davis, and S. G. Bankoff, “Long-scale evolution of thin liquid films,” Rev. Mod. Phys., vol. 69, p. 931, 1997. https://doi.org/10.1103/revmodphys.69.931.Search in Google Scholar

[9] B. Gjevik, “Occurrence of finite-amplitude surface waves on falling liquid films,” Phys. Fluid., vol. 13, p. 1915, 1970. https://doi.org/10.1063/1.1693186.Search in Google Scholar

[10] A. Ramani, B. Grammaticos, and T. Bountis, “The Painlevé property and singularity analysis of integrable and non-integrable systems,” Phys. Rep., vol. 180, p. 159, 1989. https://doi.org/10.1016/0370-1573(89)90024-0.Search in Google Scholar

[11] S. Cotsakis and P. G. L. Leach, Proc. of Institute of Mathematics of NAS of Ukraine, eConf, vol 43, p. 128, 2002, https://www.slac.stanford.edu/econf/C0107094/papers/Cotsakis128-135.pdf.Search in Google Scholar

[12] H. I. Levine, “Applications of singularity theory to gravitational lensing. I. Multiple lens planes,” J. Math. Phys., vol. 34, p. 4781, 1993. https://doi.org/10.1063/1.530321.Search in Google Scholar

[13] S. Cotsakis and P. G. L. Leach, “Painleve analysis of the mixmaster universe,” J. Phys. Math. Gen., vol. 27, p. 1625, 1994. https://doi.org/10.1088/0305-4470/27/5/026.Search in Google Scholar

[14] A. Paliathanasis, P. G. L. Leach, and T. Taves, “Integrability of the Einstein-nonlinear SU(2) σ-model in a nontrivial topological sector,” Eur. Phys. J. C, vol. 77, p. 909, 2017. https://doi.org/10.1140/epjc/s10052-017-5477-2.Search in Google Scholar

[15] W. G. Glöckle, G. Baumann, and T. F. Nonnenmacher, “Painlevé test and exact similarity solutions of a class of nonlinear diffusion equations,” J. Math. Phys., vol. 33, p. 2456, 1992. https://doi.org/10.1063/1.529615.Search in Google Scholar

[16] W-H. Steeb, A. Grauel, M. Kloke, and B. M. Spieker, Phys. Scr., vol. 31, p. 5, 1985, https://iopscience.iop.org/article/10.1088/0031-8949/31/1/001/pdf.10.1088/0031-8949/31/1/001Search in Google Scholar

[17] F. Li and X. Zheng, “Singularity analysis of a reaction–diffusion equation with a solution-dependent Dirac delta source,” Appl. Math. Lett., vol. 25, p. 2179, 2012. https://doi.org/10.1016/j.aml.2012.05.018.Search in Google Scholar

[18] B. Fuchssteiner and S. Carillo, “Soliton structure versus singularity analysis: third-order completely intergrable nonlinear differential equations in 1 + 1-dimensions,” Phys. A, vol. 154, p. 467, 1989. https://doi.org/10.1016/0378-4371(89)90260-4.Search in Google Scholar

[19] O. Costin and S. Tanveer, “Complex singularity analysis for a nonlinear PDE,” Commun. Part. Differ. Equ., vol. 31, p. 593, 2006. https://doi.org/10.1080/03605300500455941.Search in Google Scholar

[20] A. Paliathanasis and P. G. L. Leach, “Nonlinear ordinary differential equations: A discussion on symmetries and singularities,” Int. J. Geomet. Methods Mod. Phys., vol. 13, p. 11630009, 2016. https://doi.org/10.1142/s0219887816300099.Search in Google Scholar

[21] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, New York, Springer, 1989.10.1007/978-1-4757-4307-4Search in Google Scholar

[22] U. Obaidullah and S. Jamal, “A computational procedure for exact solutions of Burgers’ hierarchy of nonlinear partial differential equations,” J. Appl. Math. Comp., vol. 65, p. 541, 2021. https://doi.org/10.1007/s12190-020-01403-x.Search in Google Scholar

[23] S. Jamal, “Approximate conservation laws of nonvariational differential equations,” Mathematica, vol. 7, p. 574, 2019. https://doi.org/10.3390/math7070574.Search in Google Scholar

[24] S. Jamal, “A group theoretical application of SO(4,1) in the de Sitter universe,” Gen. Relat. Gravit., vol. 49, p. 88, 2017. https://doi.org/10.1007/s10714-017-2253-4.Search in Google Scholar

[25] S. Jamal and A. G. Johnpillai, “Fourth-order pattern forming pdes: partial and approximate symmetries,” Math. Model Anal., vol. 25, p. 10115, 2020. https://doi.org/10.3846/mma.2020.10115.Search in Google Scholar

[26] S. Jamal, “Perturbative manifolds and the Noether generators of nth-order Poisson equations,” J. Differ. Equ., vol. 266, p. 4018, 2019. https://doi.org/10.1016/j.jde.2018.09.025.Search in Google Scholar

[27] S. Jamal and N. Mnguni, “Moving front solutions of a time-fractional power-law fluid under gravity,” Quaest. Math., 2020. https://doi.org/10.2989/16073606.2020.1790438.Search in Google Scholar

[28] A. Mathebula and S. Jamal, “Contingent relations for Klein–Gordon equations,” Indian J. Phys., 2020. https://doi.org/10.1007/s12648-020-01810-7.Search in Google Scholar

[29] M. J. Ablowitz, A. Ramani, and H. Segur, “Nonlinear evolution equations and ordinary differential equations of painlevé type,” Lett. al Nuovo Cimento, vol. 23, p. 333, 1978. https://doi.org/10.1007/bf02824479.Search in Google Scholar

[30] M. J. Ablowitz, A. Ramani, and H. Segur, “A connection between nonlinear evolution equations and ordinary differential equations of P-type. I,” J. Math. Phys., vol. 21, p. 715, 1980. https://doi.org/10.1063/1.524491.Search in Google Scholar

[31] M. J. Ablowitz, A. Ramani, and H. Segur, “A connection between nonlinear evolution equations and ordinary differential equations of P-type. II,” J. Math. Phys., vol. 21, p. 1006, 1980. https://doi.org/10.1063/1.524548.Search in Google Scholar
Received: 2021-02-08
Accepted: 2021-02-08
Published Online: 2021-03-08

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. One-dimensional optimal system and similarity transformations for the 3 + 1 Kudryashov–Sinelshchikov equation
  4. Hopf bifurcations in a network of FitzHugh–Nagumo biological neurons
  5. Gravity modulation effect on weakly nonlinear thermal convection in a fluid layer bounded by rigid boundaries
  6. A new numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ equation
  7. Chebyshev spectral method for solving a class of local and nonlocal elliptic boundary value problems
  8. Adaptive extragradient methods for solving variational inequalities in real Hilbert spaces
  9. Elastic trend filtering
  10. A new approach to representations of homothetic motions in Lorentz space
  11. Numerical simulation of variable-order fractional differential equation of nonlinear Lane–Emden type appearing in astrophysics
  12. Stability analysis and numerical simulations of the fractional COVID-19 pandemic model
  13. Numerical solutions of the Bagley–Torvik equation by using generalized functions with fractional powers of Laguerre polynomials
  14. N-fold Darboux transformation and exact solutions for the nonlocal Fokas–Lenells equation on the vanishing and plane wave backgrounds
  15. Closed form soliton solutions to the space-time fractional foam drainage equation and coupled mKdV evolution equations
  16. Master-slave synchronization in the Duffing-van der Pol and Φ6 Duffing oscillators
  17. Singularity analysis and analytic solutions for the Benney–Gjevik equations
  18. Trajectory controllability of nonlinear fractional Langevin systems
  19. Similarity transformations for modified shallow water equations with density dependence on the average temperature
  20. Non-equidistant partition predictor–corrector method for fractional differential equations with exponential memory
  21. Dynamical analysis of fractional-order of IVGTT glucose–insulin interaction
  22. Dynamic response of Mathieu–Duffing oscillator with Caputo derivative
  23. Non-Newtonian fluid flow having fluid–particle interaction through a porous zone in a channel with permeable walls
  24. A class of computationally efficient Newton-like methods with frozen inverse operator for nonlinear systems
https://doi.org/10.1515/ijnsns-2021-0051
Keywords for this article
liquid film; Painlevé test; singularity analysis

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. One-dimensional optimal system and similarity transformations for the 3 + 1 Kudryashov–Sinelshchikov equation
  4. Hopf bifurcations in a network of FitzHugh–Nagumo biological neurons
  5. Gravity modulation effect on weakly nonlinear thermal convection in a fluid layer bounded by rigid boundaries
  6. A new numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ equation
  7. Chebyshev spectral method for solving a class of local and nonlocal elliptic boundary value problems
  8. Adaptive extragradient methods for solving variational inequalities in real Hilbert spaces
  9. Elastic trend filtering
  10. A new approach to representations of homothetic motions in Lorentz space
  11. Numerical simulation of variable-order fractional differential equation of nonlinear Lane–Emden type appearing in astrophysics
  12. Stability analysis and numerical simulations of the fractional COVID-19 pandemic model
  13. Numerical solutions of the Bagley–Torvik equation by using generalized functions with fractional powers of Laguerre polynomials
  14. N-fold Darboux transformation and exact solutions for the nonlocal Fokas–Lenells equation on the vanishing and plane wave backgrounds
  15. Closed form soliton solutions to the space-time fractional foam drainage equation and coupled mKdV evolution equations
  16. Master-slave synchronization in the Duffing-van der Pol and Φ6 Duffing oscillators
  17. Singularity analysis and analytic solutions for the Benney–Gjevik equations
  18. Trajectory controllability of nonlinear fractional Langevin systems
  19. Similarity transformations for modified shallow water equations with density dependence on the average temperature
  20. Non-equidistant partition predictor–corrector method for fractional differential equations with exponential memory
  21. Dynamical analysis of fractional-order of IVGTT glucose–insulin interaction
  22. Dynamic response of Mathieu–Duffing oscillator with Caputo derivative
  23. Non-Newtonian fluid flow having fluid–particle interaction through a porous zone in a channel with permeable walls
  24. A class of computationally efficient Newton-like methods with frozen inverse operator for nonlinear systems
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 23.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ijnsns-2021-0051/html
Scroll to top button