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

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

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 postcritical Rayleigh-Benard convection by the space-time tardily varying amplitudes while ZEE equationexplains the evolution of a grove population. Some novel complex and real-valued exact solutions for theequations under consideration are presented. Using a new conservation theorem, we construct conservationlaws for the ZEE equation. The physical expression for some of the solutions is presented to shedmore light on the mechanism of the solutions.

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  • [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.