# American Institute of Mathematical Sciences

August  2018, 11(4): 691-705. doi: 10.3934/dcdss.2018043

## A study of bifurcation parameters in travelling wave solutions of a damped forced Korteweg de Vries-Kuramoto Sivashinsky type equation

 International Center for Applied Mathematics and Computational Bioengineering, Department of Mathematics and Natural Sciences, Gulf University for Science and Technology, Kuwait 300 College Park, Department of Mathematics, University of Dayton, Dayton, Ohio 45469-2316, USA Department of Mechanical Engineering, 300 College Park, University of Dayton, Dayton, Ohio 45469, USA

Authors name are in alphabetical order.

Received  January 2017 Revised  June 2017 Published  November 2017

In this work, we consider an ordinary differential equation obtained from a damped externally excited Korteweg de Vries-Kuramoto Sivashinsky (KdV-KS) type equation using traveling coordinates. We also include controls and delays and use an asymptotic perturbation method to analyze the stability of the traveling wave solutions. The existence of bounded solutions is presented as well. We consider the primary resonance defined by the detuning parameter. External-excitation and frequency-response curves are shown to exhibit jump and hysteresis phenomena (discontinuous transitions between two stable solutions) for the KdV-KS type equation. We have obtained the existence of the bounded solutions of the system obtained from an ordinary differential equation associated with the KdV-KS equation and also show the global stability for a special case when there is no external force.

Citation: Mudassar Imran, Youssef Raffoul, Muhammad Usman, Chi Zhang. A study of bifurcation parameters in travelling wave solutions of a damped forced Korteweg de Vries-Kuramoto Sivashinsky type equation. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 691-705. doi: 10.3934/dcdss.2018043
##### References:
 [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.  doi: 10.1098/rsta.1972.0032. [2] D. J. Benney, Long waves on liquid films, J. Math. Phys., 45 (1966), 150-155.  doi: 10.1002/sapm1966451150. [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 dèn 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.  doi: 10.1088/0029-5515/16/6/009. [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.  doi: 10.1016/0022-0396(88)90010-1. [13] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21593-8. [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.  doi: 10.1098/rspa.1994.0045. [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.  doi: 10.1137/0132011. [17] J. Henrard and K. R. Meyer, Averaging and bifurcations in symmetric systems, SIAM Journal on Applied Mathematics, 32 (1977), 133-145.  doi: 10.1137/0132011. [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.  doi: 10.1016/0022-0396(92)90143-B. [19] P. Kent and J. Elgin, Travelling-waves of the Kuramoto-Sivashinsky equation:periodic multiplying bifurcations, Nonlinearity, 5 (1992), 899-919.  doi: 10.1088/0951-7715/5/4/004. [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.  doi: 10.1080/14786449508620739. [22] Y. Kuramoto, Diffusion-induced chaos in reactions systems, Suppl. Prog. Theor. Phys., 64 (1978), 346-367.  doi: 10.1143/PTPS.64.346. [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.  doi: 10.1143/PTP.55.356. [24] Y. Kuramoto and T. Yamada, Turbulent state in chemical reaction, Prog. Theor. Phys., 56 (1976), 724-740.  doi: 10.1143/PTP.56.724. [25] C.-p. Li, Computing bifurcation diagrams of steady state Kuramoto-Sivashinsky equation by difference method, Journal of Shanghai University, 3 (1999), 248-250.  doi: 10.1007/s11741-999-0067-7. [26] S. P. Lin, Finite amplitude side-band stability of a viscous film, J. Fluid Mech., 63 (1974), 417-429.  doi: 10.1017/S0022112074001704. [27] A. Maccari, Bifurcation control in the Burgers-KdV equation, Phys. Scr. , 77 (2008), 035003. doi: 10.1088/0031-8949/77/03/035003. [28] A. Maccari, The nonlocal oscillator, Il Nuovo Cimento, 111 (1996), 917-930.  doi: 10.1007/BF02743288. [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.  doi: 10.1016/S0020-7462(97)00045-0. [31] A. Maccari, Approximate solution of a class of nonlinear oscillators in resonance with a periodic excitation, Nonlinear Dynamics, 15 (1998), 329-343.  doi: 10.1023/A:1008235820302. [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.  doi: 10.1016/S0034-4877(09)00010-X. [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. doi: 10.1002/9783527617548. [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. doi: 10.1007/978-94-011-2102-6. [40] G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames, Acta Astronautica, 4 (1977), 1177-1206.  doi: 10.1016/0094-5765(77)90096-0. [41] G. I. Sivashinsky, Self-turbulence in the motion of a free particle, Foundations of Physica, 8 (1978), 735-744.  doi: 10.1007/BF00717503. [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.  doi: 10.1073/pnas.88.24.11129. [43] E. Tadmor, The well-posedness of the Kuramoto-Sivashinsky equation, SIAM J. Math. Anal., 17 (1986), 884-893.  doi: 10.1137/0517063.

show all references

Authors name are in alphabetical order.

##### References:
 [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.  doi: 10.1098/rsta.1972.0032. [2] D. J. Benney, Long waves on liquid films, J. Math. Phys., 45 (1966), 150-155.  doi: 10.1002/sapm1966451150. [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 dèn 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.  doi: 10.1088/0029-5515/16/6/009. [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.  doi: 10.1016/0022-0396(88)90010-1. [13] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21593-8. [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.  doi: 10.1098/rspa.1994.0045. [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.  doi: 10.1137/0132011. [17] J. Henrard and K. R. Meyer, Averaging and bifurcations in symmetric systems, SIAM Journal on Applied Mathematics, 32 (1977), 133-145.  doi: 10.1137/0132011. [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.  doi: 10.1016/0022-0396(92)90143-B. [19] P. Kent and J. Elgin, Travelling-waves of the Kuramoto-Sivashinsky equation:periodic multiplying bifurcations, Nonlinearity, 5 (1992), 899-919.  doi: 10.1088/0951-7715/5/4/004. [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.  doi: 10.1080/14786449508620739. [22] Y. Kuramoto, Diffusion-induced chaos in reactions systems, Suppl. Prog. Theor. Phys., 64 (1978), 346-367.  doi: 10.1143/PTPS.64.346. [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.  doi: 10.1143/PTP.55.356. [24] Y. Kuramoto and T. Yamada, Turbulent state in chemical reaction, Prog. Theor. Phys., 56 (1976), 724-740.  doi: 10.1143/PTP.56.724. [25] C.-p. Li, Computing bifurcation diagrams of steady state Kuramoto-Sivashinsky equation by difference method, Journal of Shanghai University, 3 (1999), 248-250.  doi: 10.1007/s11741-999-0067-7. [26] S. P. Lin, Finite amplitude side-band stability of a viscous film, J. Fluid Mech., 63 (1974), 417-429.  doi: 10.1017/S0022112074001704. [27] A. Maccari, Bifurcation control in the Burgers-KdV equation, Phys. Scr. , 77 (2008), 035003. doi: 10.1088/0031-8949/77/03/035003. [28] A. Maccari, The nonlocal oscillator, Il Nuovo Cimento, 111 (1996), 917-930.  doi: 10.1007/BF02743288. [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.  doi: 10.1016/S0020-7462(97)00045-0. [31] A. Maccari, Approximate solution of a class of nonlinear oscillators in resonance with a periodic excitation, Nonlinear Dynamics, 15 (1998), 329-343.  doi: 10.1023/A:1008235820302. [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.  doi: 10.1016/S0034-4877(09)00010-X. [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. doi: 10.1002/9783527617548. [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. doi: 10.1007/978-94-011-2102-6. [40] G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames, Acta Astronautica, 4 (1977), 1177-1206.  doi: 10.1016/0094-5765(77)90096-0. [41] G. I. Sivashinsky, Self-turbulence in the motion of a free particle, Foundations of Physica, 8 (1978), 735-744.  doi: 10.1007/BF00717503. [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.  doi: 10.1073/pnas.88.24.11129. [43] E. Tadmor, The well-posedness of the Kuramoto-Sivashinsky equation, SIAM J. Math. Anal., 17 (1986), 884-893.  doi: 10.1137/0517063.
Three solutions corresponding to $f=2.5$
External Excitation Response Curve
Frequency-Response Curve without delay
Frequency-Response Curve with varying B
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