Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

A study of bifurcation parameters in travelling wave solutions of a damped forced Korteweg de Vries-Kuramoto Sivashinsky type equation
Pages: 691 - 705, Issue 4, August 2018

doi:10.3934/dcdss.2018043      Abstract        References        Full text (443.4K)           Related Articles

Mudassar Imran - International Center for Applied Mathematics and Computational Bioengineering, Department of Mathematics and Natural Sciences, Gulf University for Science and Technology, Kuwait (email)
Youssef Raffoul - 300 College Park, Department of Mathematics, University of Dayton, Dayton, Ohio 45469-2316, United States (email)
Muhammad Usman - 300 College Park, Department of Mathematics, University of Dayton, Dayton, Ohio 45469-2316, United States (email)
Chi Zhang - Department of Mechanical Engineering, 300 College Park, University of Dayton, Dayton, Ohio 45469, United States (email)

1 T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. R. Soc. London, Ser. A, 272 (1972), 47-78.       
2 D. J. Benney, Long waves on liquid films, J. Math. Phys., 45 (1966), 150-155.       
3 H. A. Biagioni, J. L. Bona, R. J. Iorio Jr and M. Scialom, On the Korteweg-De Vries -Kuramoto-Sivashinsky Equation, Advances in Differential Equations, 1 (1996), 1-20.       
4 J. Boussinesq, Théorie de l'intumescence liquide appelée "onde solitaire" ou "de translation", se propageant dans un canal rectangulaire; C. R. Acad. Sci. Paris, 72 (1871), 755-759.
5 J. Boussinesq, Théorie générale des mouvements, qui sont propagés dans un canal rectangulaire horizontal; C. R. Acad. Sci. Paris, 73 (1871), 256-260.
6 J. Boussinesq, Théorie des ondes et des remous qui se propagent le long dún canal rectangulaire horizontal, en communiquant au liquide continu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl., 17 (1872), 55-108.       
7 J. Boussinesq, Essai sur la théorie des eaux courantes, Mémoires Présentés Par Divers Savants À L'Acad. des Sci. Inst. Nat. France, 23 (1877), 1-680.
8 B. I. Cohen, J. A. Krommes, W. M. Tang and M. N. Rosenbluth, Non-linear saturation of the dissipative trapped ion mode by mode coupling, Nucl. Fusion, 16 (1976), 971-992.
9 A. T. Cousin and N. A. Larkin, Initial boundary value problem for the Kuramoto-Sivashinsky equation, Mat. Contemp., 18 (2000), 97-109.       
10 P. W. Eloe and M. Usman, Fully Nonlinear Boundary Value Problems with Impulse, E. J. Qualitative Theory of Diff. Equ., 21 (2011), 1-11.       
11 C. S. Gardner, J. M. Greene and M. D. Kruskal, Phys. Rev. Lett., 19 (1967), 1095.
12 J.-M. Ghidaglia, Weakly damped forced korteweg-de vries equations behave as finite dimensional dynamical system in the long time, Journal of Differential Equations, 74 (1988), 369-390.       
13 A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.       
14 R. Grimshaw and X. Tian, Periodic and chaotic behaviour in a reduction of the perturbed Korteweg-de Vries equation, Proc. R. Soc. Lond. A., 445 (1994), 1-21.       
15 J. K. Hale, Oscillations in Nonlinear Systems, McGraw-Hill Book Company, INC., 1963.       
16 J. Henrard and K. R. Meyer, Averaging and bifurcations in symmetric systems, SIAM Journal on Applied Mathematics, 32 (1977), 133-145.       
17 J. Henrard and K. R. Meyer, Averaging and bifurcations in symmetric systems, SIAM Journal on Applied Mathematics, 32 (1977), 133-145.       
18 J. Jones and W. C. Troy, Steady solutions of the Kuramoto-Sivashinsky equation for small wave speed, Journal Of Differential Equations, 96 (1992), 28-55.       
19 P. Kent and J. Elgin, Travelling-waves of the Kuramoto-Sivashinsky equation:periodic multiplying bifurcations, Nonlinearity, 5 (1992), 899-919.       
20 Y. Kuramoto and T. Tsuzuki, On the formation of dissipative structures in reaction-diffusion systems, Prog. Theor. Phys., 54 (1975), 687-699.
21 D. J. Korteweg and G. de. Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39 (1895), 422-443.       
22 Y. Kuramoto, Diffusion-induced chaos in reactions systems, Suppl. Prog. Theor. Phys., 64 (1978), 346-367.
23 Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys., 55 (1976), 356-369.
24 Y. Kuramoto and T. Yamada, Turbulent state in chemical reaction, Prog. Theor. Phys., 56 (1976), 724-740.       
25 C.-p. Li, Computing bifurcation diagrams of steady state Kuramoto-Sivashinsky equation by difference method, Journal of Shanghai University, 3 (1999), 248-250.       
26 S. P. Lin, Finite amplitude side-band stability of a viscous film, J. Fluid Mech., 63 (1974), 417-429.
27 A. Maccari, Bifurcation control in the Burgers-KdV equation, Phys. Scr., 77 (2008), 035003.
28 A. Maccari, The nonlocal oscillator, Il Nuovo Cimento, 111 (1996), 917-930.       
29 A. Maccari, The dissipative nonlocal oscillator in resonance with a periodic excitation, Il Nuovo Cimento, 111 (1996), 1173-1186.
30 A. Maccari, Dissipative bidimensional systems and resonant excitations, International Journal of Nonlinear Mechanics, 33 (1998), 713-726.       
31 A. Maccari, Approximate solution of a class of nonlinear oscillators in resonance with a periodic excitation, Nonlinear Dynamics, 15 (1998), 329-343.       
32 M. B. A. Mansour, Traveling wave solutions of the Burgers-KdV equation with a fourth order term, Reports on Mathematical Physics, 63 (2009), 153-161.       
33 K. R. Meyer and D. S. Schmidt, Entrainment Domains, Funkcialaj Ekvacioj, 20 (1977), 171-192.       
34 A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics. Analytical, Computational, and Experimental Methods, Wiley Series in Nonlinear Science, John Wiley & Sons, Inc., New York, 1995.       
35 T. Ogawa, Travelling wave solutions to a perturbed Korteweg-de Vries equation, Hiroshima Math. J., 24 (1994), 401-422.       
36 D. T. Papageorgiou, The route to chaos for the Kuramoto-Sivashinsky equation, NASA Contractor Report 187461, ICASE Report 90-78, 1990.
37 H. Qiong-Wei and T. Jia-Shi, Dynamic bifurcation of a modified Kuramoto-Sivashinsky equation with higher order nonlinearity, Chin. Phys. B., 20 (2011), 094701, 5pp.
38 J. S. Russell, Report on waves, Rept. 14th Meeting of the British Association for the Advancement of Science, John Murray, London, (1844), 311-390.
39 S. S. Shen, A Course on Nonlinear Waves, Nonlinear Topics in the Mathematical Sciences, Kluwer Academic Publishers, Dordrecht, 1993.       
40 G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames, Acta Astronautica, 4 (1977), 1177-1206.       
41 G. I. Sivashinsky, Self-turbulence in the motion of a free particle, Foundations of Physica, 8 (1978), 735-744.
42 Y. Smyrlis and D. Papageorgiou, Predicting chaos for infinite dimensional dynamical systems: The Kuramoto-Sivashinsky equation, a case study, Applied Mathematics, Proc. Natl. Acad. Sci. USA, 88 (1991), 11129-11132.       
43 E. Tadmor, The well-posedness of the Kuramoto-Sivashinsky equation, SIAM J. Math. Anal., 17 (1986), 884-893.       

Go to top