Abstract
In this paper, the Lie group method is performed on the BKL equation. By reducing the BKL equation to four classes of ODEs, we obtain new power series solutions of it. Furthermore, as an important reduced equation, the traveling wave equation is investigated in detail. Treating it as a singular perturbation system in \({\mathbb {R}}^4\), we apply the geometric singular perturbation theory and dynamical system methods to study its phase space geometry. By using some techniques, such as tracking the unstable manifold of the saddle, discussing transversality of the stable and unstable manifolds and investigating some complicated nonlocal bifurcation, we obtain wave speed conditions to guarantee the existence of various bounded traveling waves of the BKL equation.
Similar content being viewed by others
References
Xie, Y.X.: New explicit and exact solutions of the Benney–Kawahara–Lin equation. Chin. Phys. B 18, 4094–4099 (2009)
Benney, D.J.: Long waves on liquid films. J. Math. Phys. 45, 150–155 (1966)
Lin, S.P.: Finite amplitude side-band stability of a viscous film. J. Fluid Mech. 63, 417–429 (1974)
Hunter, J.K., Scheurle, J.: Existence of perturbed solitary wave solutions to a model equation for water waves. Physica D 32, 253–268 (1988)
Kawahara, T., Takaoka, M.: Chaotic behaviour of soliton lattice in an unstable dissipative-dispersive nonlinear system. Physica D 39, 43–58 (1989)
Kuramoto, Y., Tsuzuki, T.: On the formation of dissipative structures in reaction–diffusion systems. Prog. Theor. Phys. 54, 687–699 (1975)
Kuramoto, Y., Tsuzuki, T.: Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Prog. Theor. Phys. 55, 356–369 (1976)
Sivashinsky, G.I.: Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations. Acta. Astronaut. 4, 1177–1206 (1977)
Sivashinsky, G.I.: On irregular wavy flow of a liquid film down a vertical plane. Prog. Theor. Phys. 63, 2112–2114 (1980)
Babchin, A.J., Frenkel, A.L., Levich, B.G., Sivashinsky, G.I.: Nonlinear saturation of Rayleigh–Taylor instability in thin films. Phys. Fluids 26, 3159–3161 (1983)
Kawahara, T.: Formation of saturated solitons in a nonlinear dispersive system with instability and dissipation. Phys. Rev. Lett. 51, 381–383 (1983)
Sivashinsky, G.I.: Large cells in nonlinear marangoni convection. Physica D 4, 227–235 (1982)
Kudryashov, N.A.: Exact solutions of the generalized Kuramoto–Sivashinsky equation. Phys. Lett. A 147, 287–291 (1990)
Kudryashov, N.A., Zargaryan, E.D.: Solitary waves in active–dissipative dispersive media. J. Phys. A 29, 8067–8077 (1996)
Ruan, S.G., Xiao, D.M.: Stability of steady states and existence of travelling waves in a vector-disease model. Proc. R. Soc. Edinb. A 134, 991–1011 (2004)
Bates, P.W., Chen, F.X.: Spectral analysis of traveling waves for nonlocal evolution equations. SIAM J. Math. Anal. 38, 116–126 (2006)
Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)
Li, J.B., Dai, H.H.: On the Study of Singular Nonlinear Traveling Wave Equation: Dynamical System Approach. Science Press, Beijing (2007)
Hsu, S.B., Zhao, X.Q.: Spreading speeds and traveling waves for nonmonotone integrodifference equations. SIAM J. Math. Anal. 40, 776–789 (2008)
Shatah, J., Walsh, S., Zeng, C.C.: Travelling water waves with compactly supported vorticity. Nonlinearity 26, 1529–1564 (2013)
Ovsiannikov, L.V.: Group Analysis of Differential Equations. Academic, New York (1982)
Olver, P.J.: Application of Lie Groups to Differential Equations. Springer, New York (1986)
Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, New York (1989)
Katzengruber, B., Krupa, M., Szmolyan, P.: Bifurcation of traveling waves in extrinsic semiconductors. Physica D 144, 1–19 (2000)
Li, J.B., Qiao, Z.J.: Bifurcations of traveling wave solutions for an integrable equation. J. Math. Phys. 51, 042703 (2010)
Li, J.B., Zhang, Y.: Homoclinic manifolds, center manifolds and exact solutions of four-dimensional traveling wave systems for two classes of nonlinear wave equations. Int. J. Bifurc. Chaos 21, 527–543 (2011)
Li, J.B., Chen, G.R.: Exact traveling wave solutions and their bifurcations for the Kudryashov–Sinelshchikov equation. Int. J. Bifurc. Chaos 22, 1250118 (2012)
Li, J.B., Qiao, Z.J.: Peakon, pseudo-peakon, and cuspon solutions for two generalized Camassa–Holm equations. J. Math. Phys. 54, 123501 (2013)
Li, J.B.: Bifurcations and exact travelling wave solutions of the generalized two-component hunter-saxton system. Discrete Cont. Dyn. B 19, 1719–1729 (2014)
Zhang, Y., Li, J.B.: Comment on superposition of elliptic functions as solutions for a large number of nonlinear equations. J. Math. Phys. 56, 084101 (2015)
Leta, T.D., Li, J.B.: Exact traveling wave solutions and bifurcations of the generalized derivative nonlinear Schrödinger equation. Nonlinear Dyn. 85, 1031–1037 (2016)
Zhang, T.H., Li, J.B.: Exact solitons, periodic peakons and compactons in an optical soliton model. Nonlinear Dyn. 91, 1371–1381 (2018)
Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31, 53–98 (1979)
Li, C.Z., Zhang, Z.F.: A criterion for determining the monotonicity of the ratio of two Abelian integrals. J. Differ. Equ. 24, 407–424 (1990)
Zhang, Z.F., Ding, T.R., Huang, W.Z., Dong, Z.X.: Qualitative Theory of Differential Equations. American Mathematical Society, Providence (1992)
Chow, S.N., Hale, J.K.: Method of Bifurcation Theory. Springer, New York (1982)
Rottschäfer, V., Wayne, C.E.: Existence and stability of traveling fronts in the extended Fisher–Kolmogorov equation. J. Differ. Equ. 176, 532–560 (2001)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields. Springer, New York (1983)
Acknowledgements
This work is supported by the Natural Science Foundation of China (Nos. 11301043 and 11701480), China Postdoctoral Science Foundation (No. 2016 M602663), Innovative Research Team of the Education Department of Sichuan Province (No. 15TD0050).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have 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
Zhou, Y., Liu, Q. Series solutions and bifurcation of traveling waves in the Benney–Kawahara–Lin equation. Nonlinear Dyn 96, 2055–2067 (2019). https://doi.org/10.1007/s11071-019-04905-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-019-04905-x