Research Article
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Stability Analysis for Some Nonlinear Fifth-order Equations

Year 2018, Volume: 9, 125 - 134, 28.12.2018

Mustafa Mızrak

Abstract



The main
purpose of this work is to obtain many travelling wave solutions for general Kaup-Kuperschmidt
(KK), general Lax, general Sawada-Kotera (SK) and general Ito equations
with the extended direct algebraic method. The stability of these solutions and
the movement role of the waves by sketching the graphs of the exact solutions
are analyzed.

Keywords

Travelling wave solutions Extended direct algebraic method

References

  • Ablowitz, M.J., Clarkson, P.A. , Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge Univ. Press, New York, 1991.
  • Arshad, M., Seadawy, A. R., Lu, D., Elliptic function and solitary wave solutions of the higher-order nonlinear Schr¨odinger dynamical equation with fourth-order dispersion and cubic-quintic nonlinearity and its stability, Eur. Phys. J. Plus 132(2017), 371.
  • Arshad, M., Seadawy, A. R., Lu, D., Jun, W., Modulation instability analysis of modify unstable nonlinear schrodinger dynamical equation and its optical soliton solutions, Results in Physics, 7 (2017), 4153-4161.
  • Asghar, A., Seadawy, A. R., Lu, D., Soliton solutions of the nonlinear Schr¨odinger equation with the dual power law nonlinearity and resonant nonlinear Schr¨odinger equation and their modulation instability analysis, Optik - International Journal for Light and Electron Optics, 145(2017), 79-88.
  • Ashorman, R., Multi-soliton solutions for a class of fifth-order evolution equations, Int. J. of Hybrid Information Tech. Vol.7 No.4 (2014), 11-18.
  • Baldwin, D., et al., Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs, J. Symb. Comp.37 (2004), 669-705.
  • Ertas, A., Mızrak, M., Explicit Travelling Wave Solutions of Two Nonlinear Evolution Equations, Mathematical Sciences Letters, 3 (2014), 223-228.
  • Fan, E., Hon, Y. C., A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves, Chaos, Solitons and Fract. 15 (2003), 559-566.
  • Gomez S, C. A., New traveling waves solutions to generalized Kaup-Kupershmidt and Ito equations, App. Math. and Comp. 216 (2010), 241-250.
  • Helal,M.A., Seadawy, A.R., Variational method for the derivative nonlinear schrodinger equation with computational applications, Physica Scripta 80 (2009), 350-360.
  • Helal, M. A., Seadawy A. R., Exact soliton solutions of an D-dimensional nonlinear Schrodinger equation with damping and diffusive terms, Z. Angew. Math. Phys.(ZAMP) 62 (2011), 839-847.
  • Hirota, R., Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (1971), 1192-1194.
  • Kangalgil, F., Ayaz, F., New exact travelling wave solutions for the Ostrovsky equation, Physics Letters A 372 (2008), 1831-1835.
  • Kangalgil, F., Ayaz, F., Solitary Wave Solutions for Hirota-Satsuma Coupled KdV Equation and Coupled mKdV Equation by Differential Transform Method, The Arabian Journal for Science and Engineering,Volume 35, Number 2D, November 2010, 203-213.
  • Khater, A.H., Callebaut, D.K., Seadawy, A.R., Nonlinear Dispersive Kelvin-Helmholtz Instabilities in Magnetohydrodynamic Flows, Presented in International Conference on Computational and Applied Mathematics (Belgium) ICCAM 2000 and published in Physica Scripta 67 (2003), 340-349.
  • Lee, C. T., Some remarks on the fifth-order KdV equations, J. Math. Anal. Appl. 425 (2015), 281-294.
  • Liu, S.K., Fu, Z.T., Liu, S.D., Zhao, Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A 289 (2001), 69-74.
  • Lu, D., Seadawy, A.R., Arshad, M., Applications of extended simple equation method on unstable nonlinear Schr¨odinger equations, Optik 140 (2017), 136-144.
  • Malfliet,W., Hereman,W., The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Physica Scripta 54 (1996), 563-568.
  • Mızrak, M., Travelling Wave Solutions of Some Nonlinear Third-Order Equations And Their Stabilities, Journal of Advances in Mathematical Analysis and Applications Vol.2 (2017), No.1, 20-30.
  • Mızrak, M., Stability Analysis For The Kawachara And Modified Kawachara Equations, Middle East Journal of Science (2018) 4(1), 15-22.
  • Saba, F., et al., Modified alternative (G’/G)-expansion method to general Sawada-Kotera equation of fifth-order, J. of the Egyptian Math. Soc. 23 (2015), 416-423.
  • Seadawy, A. R., Exact Solutions of a Two Dimensional Nonlinear Schr¨odinger Equation, Applied Mathematics Letters, Vol. 25 (2012), 687-691.
  • Seadawy, A. R., Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acustic waves in a plasma, Computers and Math. with Appl. 67 (2014), 172-180.
  • Seadawy, A. R., Approximation solutions of derivative nonlinear Schr¨odinger equation with computational applications by variational method, The European Physical Journal-Plus, 130(182) (2015), 1-10.
  • Seadawy, A.R., Nonlinear wave solutions of the three-dimensional Zakharov-Kuznetsov-Burgers equation in dusty plasma, Physica A 439(2015), 124-131.
  • Seadawy, A.R., Stability analysis solutions for nonlinear three-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov equation in a magnetized electron-positron plasma, Physica A 455 (2016), 44-51.
  • Seadawy, A. R., Three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma, Comput. Math.Appl. 71 (2016), 201-212.
  • Seadawy, A.R., Stability analysis of traveling wave solutions for generalized coupled nonlinear KdV equations, Appl. Math. Inf. Sci. 10 (2016), 209-214.
  • Seadawy, A. R., Modulation instability analysis for the generalized derivative higher order nonlinear Schr¨odinger equation and its the bright and dark soliton solutions, J. Electromagn. Waves and Appl. 31 (14) (2017), 1353-1362.
  • Seadawy, A. R., Two-dimensional interaction of a shear flow with a free surface in a stratified fluid and its a solitary wave solutions via mathematical methods, Eur. Phys. J. Plus 132. 518 (2017):12, 1-11.
  • Seadawy, A. R., Exact bright-dark solitary wave solutions of the higher-order cubic-quintic nonlinear Schr¨odinger equation and its stability, Optik - International Journal for Light and Electron Optics 138 (2017), 40-49.
  • Seadawy, A. R., The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schr¨odinger equation and its solutions, Optik - International Journal for Light and Electron Optics, 139 (2017), 31-43.
  • Seadawy, A. R., Three-dimensional weakly nonlinear shallow water waves regime and its travelling wave solutions, Int. J. Comput. Methods 15 (1) (2018).
  • Seadawy, A. R., Sayed, A., Soliton Solutions of Cubic-Quintic Nonlinear Schr¨odinger and Variant Boussinesq Equations by the First Integral Method,Filomat 31:13 (2017), 4199-4208.
  • Seadawy, A.R., El-Kalaawy, O.H., Aldenari, R.B., Water wave solutions of Zufiria’s higher-order Boussinesq type equations and its stability, Appl. Math. Comput. 280 (2016), 57-71.
  • Seadawy, A.R., El-Rashidy, K., Traveling wave solutions for some coupled nonlinear evolution equations, Math. Comput. Model. 57 (2013), 1371-1379.
  • Seadawy, A.R., El-Rashidy, K., Water wave solutions of the coupled system Zakharov-Kuznetsov and generalized coupled KdV equations, Sci. World J.201 (2014) 4, Article ID 724-759.
  • Seadawy, A. R., et al., Travelling wave solutions of the generalized nonlinear fifth-order KdV water wave equations and its stability, J. Taibah Uni. for Sci. 11 (2017), 623-633.
  • Seadawy, A. R., Lu, D., Bright and dark solitary wave soliton solutions for the generalized higher order nonlinear Schr¨odinger equation and its stability, Results in Phy. 7 (2017), 43-48.
  • Tariq, K.U., Seadawy, A. R., Bistable bright-dark solitary wave solutions of the (3 + 1)-dimensional breaking soliton, Boussinesq equation with dual dispersion and modified KdV-KP equations and their applications, Results Phys. 7 (2017), 1143-1149.
  • Wang, M.L., Solitary wave solutions for variant Boussinesq equations, Phys. Lett. A 199 (1995), 169-172.
  • Wazwaz, A. M., The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, Appl. Math.Compu. 184 (2007), 1002-1014.
  • Wazwaz, A. M., Partial Differential Equations and Solitary Waves Theory, Springer-Verlag Berlin Heidelberg, 2009.
  • Wazwaz, A. M., Ebaid, A., A study on couplings of the fifth-order integrable Sawada-Kotera and Lax equations, Rom. J. Phys., Vol.59 Nos. 5-6 (2014), 454-465.
  • Yue, C., Seadawy, A. R., Lu, D., Stability analysis of the soliton solutions for the generalized quintic derivative nonlinear Schr¨odinger equation, Results in Physics 6 (2016), 911-916.
  • Zhibin, L., Mingliang, W., Travelling wave solutions to the two-dimensional Kdv-Burgers equation, J. Phys. A: Math. Gen. 26 (1993), 6027-6031.

Year 2018, Volume: 9, 125 - 134, 28.12.2018

Mustafa Mızrak

Abstract

References

  • Ablowitz, M.J., Clarkson, P.A. , Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge Univ. Press, New York, 1991.
  • Arshad, M., Seadawy, A. R., Lu, D., Elliptic function and solitary wave solutions of the higher-order nonlinear Schr¨odinger dynamical equation with fourth-order dispersion and cubic-quintic nonlinearity and its stability, Eur. Phys. J. Plus 132(2017), 371.
  • Arshad, M., Seadawy, A. R., Lu, D., Jun, W., Modulation instability analysis of modify unstable nonlinear schrodinger dynamical equation and its optical soliton solutions, Results in Physics, 7 (2017), 4153-4161.
  • Asghar, A., Seadawy, A. R., Lu, D., Soliton solutions of the nonlinear Schr¨odinger equation with the dual power law nonlinearity and resonant nonlinear Schr¨odinger equation and their modulation instability analysis, Optik - International Journal for Light and Electron Optics, 145(2017), 79-88.
  • Ashorman, R., Multi-soliton solutions for a class of fifth-order evolution equations, Int. J. of Hybrid Information Tech. Vol.7 No.4 (2014), 11-18.
  • Baldwin, D., et al., Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs, J. Symb. Comp.37 (2004), 669-705.
  • Ertas, A., Mızrak, M., Explicit Travelling Wave Solutions of Two Nonlinear Evolution Equations, Mathematical Sciences Letters, 3 (2014), 223-228.
  • Fan, E., Hon, Y. C., A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves, Chaos, Solitons and Fract. 15 (2003), 559-566.
  • Gomez S, C. A., New traveling waves solutions to generalized Kaup-Kupershmidt and Ito equations, App. Math. and Comp. 216 (2010), 241-250.
  • Helal,M.A., Seadawy, A.R., Variational method for the derivative nonlinear schrodinger equation with computational applications, Physica Scripta 80 (2009), 350-360.
  • Helal, M. A., Seadawy A. R., Exact soliton solutions of an D-dimensional nonlinear Schrodinger equation with damping and diffusive terms, Z. Angew. Math. Phys.(ZAMP) 62 (2011), 839-847.
  • Hirota, R., Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (1971), 1192-1194.
  • Kangalgil, F., Ayaz, F., New exact travelling wave solutions for the Ostrovsky equation, Physics Letters A 372 (2008), 1831-1835.
  • Kangalgil, F., Ayaz, F., Solitary Wave Solutions for Hirota-Satsuma Coupled KdV Equation and Coupled mKdV Equation by Differential Transform Method, The Arabian Journal for Science and Engineering,Volume 35, Number 2D, November 2010, 203-213.
  • Khater, A.H., Callebaut, D.K., Seadawy, A.R., Nonlinear Dispersive Kelvin-Helmholtz Instabilities in Magnetohydrodynamic Flows, Presented in International Conference on Computational and Applied Mathematics (Belgium) ICCAM 2000 and published in Physica Scripta 67 (2003), 340-349.
  • Lee, C. T., Some remarks on the fifth-order KdV equations, J. Math. Anal. Appl. 425 (2015), 281-294.
  • Liu, S.K., Fu, Z.T., Liu, S.D., Zhao, Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A 289 (2001), 69-74.
  • Lu, D., Seadawy, A.R., Arshad, M., Applications of extended simple equation method on unstable nonlinear Schr¨odinger equations, Optik 140 (2017), 136-144.
  • Malfliet,W., Hereman,W., The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Physica Scripta 54 (1996), 563-568.
  • Mızrak, M., Travelling Wave Solutions of Some Nonlinear Third-Order Equations And Their Stabilities, Journal of Advances in Mathematical Analysis and Applications Vol.2 (2017), No.1, 20-30.
  • Mızrak, M., Stability Analysis For The Kawachara And Modified Kawachara Equations, Middle East Journal of Science (2018) 4(1), 15-22.
  • Saba, F., et al., Modified alternative (G’/G)-expansion method to general Sawada-Kotera equation of fifth-order, J. of the Egyptian Math. Soc. 23 (2015), 416-423.
  • Seadawy, A. R., Exact Solutions of a Two Dimensional Nonlinear Schr¨odinger Equation, Applied Mathematics Letters, Vol. 25 (2012), 687-691.
  • Seadawy, A. R., Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acustic waves in a plasma, Computers and Math. with Appl. 67 (2014), 172-180.
  • Seadawy, A. R., Approximation solutions of derivative nonlinear Schr¨odinger equation with computational applications by variational method, The European Physical Journal-Plus, 130(182) (2015), 1-10.
  • Seadawy, A.R., Nonlinear wave solutions of the three-dimensional Zakharov-Kuznetsov-Burgers equation in dusty plasma, Physica A 439(2015), 124-131.
  • Seadawy, A.R., Stability analysis solutions for nonlinear three-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov equation in a magnetized electron-positron plasma, Physica A 455 (2016), 44-51.
  • Seadawy, A. R., Three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma, Comput. Math.Appl. 71 (2016), 201-212.
  • Seadawy, A.R., Stability analysis of traveling wave solutions for generalized coupled nonlinear KdV equations, Appl. Math. Inf. Sci. 10 (2016), 209-214.
  • Seadawy, A. R., Modulation instability analysis for the generalized derivative higher order nonlinear Schr¨odinger equation and its the bright and dark soliton solutions, J. Electromagn. Waves and Appl. 31 (14) (2017), 1353-1362.
  • Seadawy, A. R., Two-dimensional interaction of a shear flow with a free surface in a stratified fluid and its a solitary wave solutions via mathematical methods, Eur. Phys. J. Plus 132. 518 (2017):12, 1-11.
  • Seadawy, A. R., Exact bright-dark solitary wave solutions of the higher-order cubic-quintic nonlinear Schr¨odinger equation and its stability, Optik - International Journal for Light and Electron Optics 138 (2017), 40-49.
  • Seadawy, A. R., The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schr¨odinger equation and its solutions, Optik - International Journal for Light and Electron Optics, 139 (2017), 31-43.
  • Seadawy, A. R., Three-dimensional weakly nonlinear shallow water waves regime and its travelling wave solutions, Int. J. Comput. Methods 15 (1) (2018).
  • Seadawy, A. R., Sayed, A., Soliton Solutions of Cubic-Quintic Nonlinear Schr¨odinger and Variant Boussinesq Equations by the First Integral Method,Filomat 31:13 (2017), 4199-4208.
  • Seadawy, A.R., El-Kalaawy, O.H., Aldenari, R.B., Water wave solutions of Zufiria’s higher-order Boussinesq type equations and its stability, Appl. Math. Comput. 280 (2016), 57-71.
  • Seadawy, A.R., El-Rashidy, K., Traveling wave solutions for some coupled nonlinear evolution equations, Math. Comput. Model. 57 (2013), 1371-1379.
  • Seadawy, A.R., El-Rashidy, K., Water wave solutions of the coupled system Zakharov-Kuznetsov and generalized coupled KdV equations, Sci. World J.201 (2014) 4, Article ID 724-759.
  • Seadawy, A. R., et al., Travelling wave solutions of the generalized nonlinear fifth-order KdV water wave equations and its stability, J. Taibah Uni. for Sci. 11 (2017), 623-633.
  • Seadawy, A. R., Lu, D., Bright and dark solitary wave soliton solutions for the generalized higher order nonlinear Schr¨odinger equation and its stability, Results in Phy. 7 (2017), 43-48.
  • Tariq, K.U., Seadawy, A. R., Bistable bright-dark solitary wave solutions of the (3 + 1)-dimensional breaking soliton, Boussinesq equation with dual dispersion and modified KdV-KP equations and their applications, Results Phys. 7 (2017), 1143-1149.
  • Wang, M.L., Solitary wave solutions for variant Boussinesq equations, Phys. Lett. A 199 (1995), 169-172.
  • Wazwaz, A. M., The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, Appl. Math.Compu. 184 (2007), 1002-1014.
  • Wazwaz, A. M., Partial Differential Equations and Solitary Waves Theory, Springer-Verlag Berlin Heidelberg, 2009.
  • Wazwaz, A. M., Ebaid, A., A study on couplings of the fifth-order integrable Sawada-Kotera and Lax equations, Rom. J. Phys., Vol.59 Nos. 5-6 (2014), 454-465.
  • Yue, C., Seadawy, A. R., Lu, D., Stability analysis of the soliton solutions for the generalized quintic derivative nonlinear Schr¨odinger equation, Results in Physics 6 (2016), 911-916.
  • Zhibin, L., Mingliang, W., Travelling wave solutions to the two-dimensional Kdv-Burgers equation, J. Phys. A: Math. Gen. 26 (1993), 6027-6031.
There are 47 citations in total.

Details

Journal Section Articles
Authors

Mustafa Mızrak

Publication Date December 28, 2018
Published in Issue Year 2018 Volume: 9

Cite

APA Mızrak, M. (2018). Stability Analysis for Some Nonlinear Fifth-order Equations. Turkish Journal of Mathematics and Computer Science, 9, 125-134.
AMA Mızrak M. Stability Analysis for Some Nonlinear Fifth-order Equations. TJMCS. December 2018;9:125-134.
Chicago Mızrak, Mustafa. “Stability Analysis for Some Nonlinear Fifth-Order Equations”. Turkish Journal of Mathematics and Computer Science 9, December (December 2018): 125-34.
EndNote Mızrak M (December 1, 2018) Stability Analysis for Some Nonlinear Fifth-order Equations. Turkish Journal of Mathematics and Computer Science 9 125–134.
IEEE M. Mızrak, “Stability Analysis for Some Nonlinear Fifth-order Equations”, TJMCS, vol. 9, pp. 125–134, 2018.
ISNAD Mızrak, Mustafa. “Stability Analysis for Some Nonlinear Fifth-Order Equations”. Turkish Journal of Mathematics and Computer Science 9 (December 2018), 125-134.
JAMA Mızrak M. Stability Analysis for Some Nonlinear Fifth-order Equations. TJMCS. 2018;9:125–134.
MLA Mızrak, Mustafa. “Stability Analysis for Some Nonlinear Fifth-Order Equations”. Turkish Journal of Mathematics and Computer Science, vol. 9, 2018, pp. 125-34.
Vancouver Mızrak M. Stability Analysis for Some Nonlinear Fifth-order Equations. TJMCS. 2018;9:125-34.