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Symmetry analysis and some new exact solutions of the Newell-Whitehead-Segel and Zeldovich equations

Year 2019, Volume: 2 Issue: 4, 182 - 192, 25.12.2019

Abdullahi Yusuf , Behzad Ghanbari , Sania Qureshi Mustafa Inc Dumitru Baleanu

Abstract

The present study provides an investigation of the Newel-Whitehead-Segel (NWS) and Zeldovich equations

(ZEE) equation via Lie symmetry analysis and generalize exponential rational function methods.

The NWS equation exhibits the relation between a continuous nite bandwidth of modes and a post

critical Rayleigh-Benard convection by the space-time tardily varying amplitudes while ZEE equation

explains the evolution of a grove population. Some novel complex and real-valued exact solutions for the

equations under consideration are presented. Using a new conservation theorem, we construct conservation

laws for the ZEE equation. The physical expression for some of the solutions is presented to shed

more light on the mechanism of the solutions.

Keywords

Newell-Whitehead-Segel equation Zeldovich equation symmetries

References

  • [1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse ScatteringTransform (Cambridge University Press, Cambridge, 1990).
  • [2] F. Tchier, A. I. Aliyu, A. Yusuf, and M. Inc. Dynamics of solitons to the ill-posed Boussinesq equation.Eur. Phys. J. Plus 132, 136 (2017).
  • [3] F. Tchier, A. Yusuf, A. I. Aliyu, and M. Inc. Soliton solutions and conservation laws for lossy nonlineartransmission line equation. Superlattices Microstruct 107, 320 (2017).
  • [4] W. X. Ma. A soliton hierarchy associated with so(3,R). Appl. Math. Comput. 220, 117 (2013).
  • [5] R. Dodd, J. Eilbeck, J. Gibbon, and H. Morris, Solitons and Nonlinear Wave Equations (Academic Press,1988).
  • [6] N. Zabusky, A Synergetic Approach to Problems of Nonlinear Dispersive Wave Propagation and Interaction(Academic Press, 1967).
  • [7] J. H. He. Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitonsand Fractals 26(3), (2005) 695-700.
  • [8] J. H. He, Variational principles for some nonlinear partial di erential equations with variable coecients,Chaos, Solitons and Fractals, 19, (2004) 4.
  • [9] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers,Boston, 1994.
  • [10] K. Khan, M. A. Akbar, Exact and solitary wave solutions for the Tzitzeica-Dodd-Bullough and themodi ed KdV-Zakharov-Kuznetsov equations using the modi ed simple equation method, Ain ShamsEngr. J. 4(4) (2013) 903-909.
  • [11] K. Khan, M. A. Akbar. Traveling wave solutions of the (2+1)-dimensional Zoomeron equation and theBurgers equations via the MSE method and the Exp-function method. Ain Shams Engr. J. 5(1) (2014)247-256.
  • [12] H. I. Abdel-Gawad, M. Tantawy, Mustafa Inc and Abdullahi Yusuf. On multi-fusion solitons inducedby inelastic collision for quasi-periodic propagation with nonlinear refractive index and stability analysis.Modern Physics Letters B Vol. 32, No. 29 (2018) 1850353.
  • [13] H. O. Roshid, N. Rahman, M.A. Akbar. Traveling waves solutions of nonlinear Klein Gordon equationby extended (G/G)-expasion method. Annals of Pure and Appl. Math.3, (2013) 10-16.
  • [14] U. Khan, R. Ellahi, R. Khan, S. T. Mohyud-Din, Extracting new solitary wave solutions of Benny{Lukeequation and Phi-4 equation of fractional order by using (G0=G)-expansion method, Opt Quant Electron(2017) 49:362.
  • [15] Behzad Ghanbari, Abdullahi Yusuf and Mustafa Inc. Dark optical solitons and modulation instabilityanalysis of nonlinear Schrodinger equation with higher order dispersion and cubic-quintic. J. CoupledSyst. Multiscale Dyn. 6, 217-227 (2018) nonlinearity
  • [16] Abdullahi Yusuf, Mustafa Inc, and Mustafa Bayram. Stability Analysis and Conservation Laws viaMultiplier Approach for the Perturbed Kaup-Newell. J. Adv. Phys. 7, 451-0453 (2018)
  • [17] B.H. Gilding, R. Kersner, Traveling Waves in Nonlinear Di usion-convection-reaction, University ofTwente, Memorandum, 1585 (2001).
  • [18] A. Korkmaz, Complex wave solutions to mathematical biology models I:Newell-Whitehead-Segel andZeldovich equations, journal of computational and nonlinear dynamics, 13(8), 081004.
  • [19] B. Ghanbari, M. Inc, A new generalized exponential rational function method to nd exact specialsolutions for the resonance nonlinear Schrodinger equation, Eur. Phys. J. Plus (2018) 133: 142.

Year 2019, Volume: 2 Issue: 4, 182 - 192, 25.12.2019

Abdullahi Yusuf , Behzad Ghanbari , Sania Qureshi Mustafa Inc Dumitru Baleanu

Abstract

References

  • [1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse ScatteringTransform (Cambridge University Press, Cambridge, 1990).
  • [2] F. Tchier, A. I. Aliyu, A. Yusuf, and M. Inc. Dynamics of solitons to the ill-posed Boussinesq equation.Eur. Phys. J. Plus 132, 136 (2017).
  • [3] F. Tchier, A. Yusuf, A. I. Aliyu, and M. Inc. Soliton solutions and conservation laws for lossy nonlineartransmission line equation. Superlattices Microstruct 107, 320 (2017).
  • [4] W. X. Ma. A soliton hierarchy associated with so(3,R). Appl. Math. Comput. 220, 117 (2013).
  • [5] R. Dodd, J. Eilbeck, J. Gibbon, and H. Morris, Solitons and Nonlinear Wave Equations (Academic Press,1988).
  • [6] N. Zabusky, A Synergetic Approach to Problems of Nonlinear Dispersive Wave Propagation and Interaction(Academic Press, 1967).
  • [7] J. H. He. Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitonsand Fractals 26(3), (2005) 695-700.
  • [8] J. H. He, Variational principles for some nonlinear partial di erential equations with variable coecients,Chaos, Solitons and Fractals, 19, (2004) 4.
  • [9] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers,Boston, 1994.
  • [10] K. Khan, M. A. Akbar, Exact and solitary wave solutions for the Tzitzeica-Dodd-Bullough and themodi ed KdV-Zakharov-Kuznetsov equations using the modi ed simple equation method, Ain ShamsEngr. J. 4(4) (2013) 903-909.
  • [11] K. Khan, M. A. Akbar. Traveling wave solutions of the (2+1)-dimensional Zoomeron equation and theBurgers equations via the MSE method and the Exp-function method. Ain Shams Engr. J. 5(1) (2014)247-256.
  • [12] H. I. Abdel-Gawad, M. Tantawy, Mustafa Inc and Abdullahi Yusuf. On multi-fusion solitons inducedby inelastic collision for quasi-periodic propagation with nonlinear refractive index and stability analysis.Modern Physics Letters B Vol. 32, No. 29 (2018) 1850353.
  • [13] H. O. Roshid, N. Rahman, M.A. Akbar. Traveling waves solutions of nonlinear Klein Gordon equationby extended (G/G)-expasion method. Annals of Pure and Appl. Math.3, (2013) 10-16.
  • [14] U. Khan, R. Ellahi, R. Khan, S. T. Mohyud-Din, Extracting new solitary wave solutions of Benny{Lukeequation and Phi-4 equation of fractional order by using (G0=G)-expansion method, Opt Quant Electron(2017) 49:362.
  • [15] Behzad Ghanbari, Abdullahi Yusuf and Mustafa Inc. Dark optical solitons and modulation instabilityanalysis of nonlinear Schrodinger equation with higher order dispersion and cubic-quintic. J. CoupledSyst. Multiscale Dyn. 6, 217-227 (2018) nonlinearity
  • [16] Abdullahi Yusuf, Mustafa Inc, and Mustafa Bayram. Stability Analysis and Conservation Laws viaMultiplier Approach for the Perturbed Kaup-Newell. J. Adv. Phys. 7, 451-0453 (2018)
  • [17] B.H. Gilding, R. Kersner, Traveling Waves in Nonlinear Di usion-convection-reaction, University ofTwente, Memorandum, 1585 (2001).
  • [18] A. Korkmaz, Complex wave solutions to mathematical biology models I:Newell-Whitehead-Segel andZeldovich equations, journal of computational and nonlinear dynamics, 13(8), 081004.
  • [19] B. Ghanbari, M. Inc, A new generalized exponential rational function method to nd exact specialsolutions for the resonance nonlinear Schrodinger equation, Eur. Phys. J. Plus (2018) 133: 142.
There are 19 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Abdullahi Yusuf

Behzad Ghanbari

Sania Qureshi This is me

Mustafa Inc This is me

Dumitru Baleanu

Publication Date December 25, 2019
Published in Issue Year 2019 Volume: 2 Issue: 4

Cite

APA Yusuf, A., Ghanbari, B., Qureshi, S., Inc, M., et al. (2019). Symmetry analysis and some new exact solutions of the Newell-Whitehead-Segel and Zeldovich equations. Results in Nonlinear Analysis, 2(4), 182-192.
AMA Yusuf A, Ghanbari B, Qureshi S, Inc M, Baleanu D. Symmetry analysis and some new exact solutions of the Newell-Whitehead-Segel and Zeldovich equations. RNA. December 2019;2(4):182-192.
Chicago Yusuf, Abdullahi, Behzad Ghanbari, Sania Qureshi, Mustafa Inc, and Dumitru Baleanu. “Symmetry Analysis and Some New Exact Solutions of the Newell-Whitehead-Segel and Zeldovich Equations”. Results in Nonlinear Analysis 2, no. 4 (December 2019): 182-92.
EndNote Yusuf A, Ghanbari B, Qureshi S, Inc M, Baleanu D (December 1, 2019) Symmetry analysis and some new exact solutions of the Newell-Whitehead-Segel and Zeldovich equations. Results in Nonlinear Analysis 2 4 182–192.
IEEE A. Yusuf, B. Ghanbari, S. Qureshi, M. Inc, and D. Baleanu, “Symmetry analysis and some new exact solutions of the Newell-Whitehead-Segel and Zeldovich equations”, RNA, vol. 2, no. 4, pp. 182–192, 2019.
ISNAD Yusuf, Abdullahi et al. “Symmetry Analysis and Some New Exact Solutions of the Newell-Whitehead-Segel and Zeldovich Equations”. Results in Nonlinear Analysis 2/4 (December 2019), 182-192.
JAMA Yusuf A, Ghanbari B, Qureshi S, Inc M, Baleanu D. Symmetry analysis and some new exact solutions of the Newell-Whitehead-Segel and Zeldovich equations. RNA. 2019;2:182–192.
MLA Yusuf, Abdullahi et al. “Symmetry Analysis and Some New Exact Solutions of the Newell-Whitehead-Segel and Zeldovich Equations”. Results in Nonlinear Analysis, vol. 2, no. 4, 2019, pp. 182-9.
Vancouver Yusuf A, Ghanbari B, Qureshi S, Inc M, Baleanu D. Symmetry analysis and some new exact solutions of the Newell-Whitehead-Segel and Zeldovich equations. RNA. 2019;2(4):182-9.