New analytical solutions and modulation instability analysis for the nonlinear (1+1)-dimensional Phi-four model

Gasmi, Boubekeur; Ciancio, Armando; Moussa, Alaaeddin; Alhakim, Lama; Mati, Yazid

doi:10.2478/ijmce-2023-0006

Abstract

This paper deals with the nonlinear (1+1)-dimensional Phi-four equation in the sense of the Katugampola operator, which can be used to model a variety of real-world applications. To solve this equation, we propose a generalized double auxiliary equation method that yields several new exact solutions. We also use linear stability analysis to discuss the instability modulation analysis for stationary solutions. Other partial differential equations can have their exact solutions found using the proposed methodology.

References

Triki H., Wazwaz A.M., Envelope solitons for generalized forms of the phi-four equation, Journal of King Saud University Science, 25, 129-133, 2013.
Search in Google Scholar Back to article
Katugampola U.N., A new fractional derivative with classical properties, arXiv:1410.6535v2, 2014.
Search in Google Scholar Back to article
Khater M.M.A., De broglie waves and nuclear element interaction; abundant waves structures of the nonlinear fractional phi-four equation, Chaos Solitons and Fractals, 163, 112549, 2022.
Search in Google Scholar Back to article
Alam L.M.B., Jiang X., Mamun A.A., Exact and explicit traveling wave solution to the time-fractional phi-four and (2+ 1) dimensional CBS equations using the modified extended tanh-function method in mathematical physics, Partial Differential Equations in Applied Mathematics, 4, 100039, 2021.
Search in Google Scholar Back to article
Alam L.M.B., Xingfang J., Mamun A.A., Anna S.N., Investigation of lump, soliton, periodic, kink, and rogue waves to the time-fractional phi-four and (2+1) dimensional CBS equations in mathematical physics, Partial Differential Equations in Applied Mathematics, 4, 100122, 2021.
Search in Google Scholar Back to article
Wang M., Solitary wave solutions for variant Boussinesq equations, Physics Letters A, 199(3-4), 169-172, 1995.
Search in Google Scholar Back to article
Gao W., Veeresha P., Prakasha D.G., Baskonus H.M., Yel G., New numerical results for the time-fractional phi-four equation using a novel analytical approach, Symmetry, 12(3), 478, 2020.
Search in Google Scholar Back to article
Parkes E.J., Duffy B.R., Travelling solitary wave solutions to a compound KdV-Burgers equation, Physics Letters A, 229(4), 217-220, 1997.
Search in Google Scholar Back to article
Fu Z., Liu S., Liu S., Zhao Q., New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Physics Letters A, 290(1-2), 72-76, 2001.
Search in Google Scholar Back to article
Wang M., Zhou Y., The periodic wave solutions for the Klein-Kordon-Schrödinger equations, Physics Letters A, 318(1-2), 8492, 2003.
Search in Google Scholar Back to article
He J.H., Homotopy perturbation method for bifurcation of nonlinear problems, International Journal of Nonlinear Sciences and Numerical Simulation, 6(2), 207-208, 2005.
Search in Google Scholar Back to article
Li J., Liu Z., Traveling wave solutions for a class of nonlinear dispersive equations, Chinese Annals of Mathematics, 23(03), 397-418, 2002.
Search in Google Scholar Back to article
Deng X., Zhao M., Li X., Travelling wave solutions for a nonlinear variant of the PHI-four equation, Mathematical and Computer Modelling, 49(3-4), 617-622, 2009.
Search in Google Scholar Back to article
Shahen N.H.M., Foyjonnesa, Ali M.S., Mamun A.A., Rahman M.M., Interaction among lump, periodic, and kink solutions with dynamical analysis to the conformable time-fractional Phi-four equation, Partial Differential Equations in Applied Mathematics, 4, 100038, 2021.
Search in Google Scholar Back to article
Mahmud F., Samsuzzoha M., Akbar M.A., The generalized Kudryashov method to obtain exact traveling wave solutions of the PHI-four equation and the Fisher equation, Results in Physics, 7, 4296-4302, 2017.
Search in Google Scholar Back to article
Mahmud A.A., Tanriverdi T., Mahmud K.A., Baskonus H.M., Structure of the analytic solutions for the complex nonlinear (2+ 1)-dimensional conformable time-fractional Schrödinger equation, Thermal Science, 27(1), 211-225, 2023.
Search in Google Scholar Back to article
Muhamad K.A., Tanriverdi T., Mahmud A.A., Baskonus H.M., Interaction characteristics of the Riemann wave propagation in the (2+1)-dimensional generalized breaking soliton system, International Journal of Computer Mathematics, 100(6), 1340-1355, 2023.
Search in Google Scholar Back to article
Baskonus H.M., Gao W., Investigation of optical solitons to the nonlinear complex Kundu-Eckhaus and Zakharov-Kuznetsov-Benjamin-Bona-Mahony equations in conformable, Optical and Quantum Electronics, 54(388), 1-23, 2022.
Search in Google Scholar Back to article
Alhakim L.A., Moussa A.A., The double auxiliary equations method and its application to space-time fractional nonlinear equations, Journal of Ocean Engineering and Science, 4(1), 713, 2019.
Search in Google Scholar Back to article
Bashir M.A., Moussa A.A., New approach of ( G[prime]G ) expansion method: Applications to KDV equation, Journal of Mathematics Research, 6(1), 24-32, 2014.
Search in Google Scholar Back to article
Bashir M.A., Moussa A.A., Alhakim L.A., The new generalized of exp(−f (x )) expansion method and its application to some complex nonlinear partial differential equations, Journal of Advances in Mathematics, 9(8), 2905-2926, 2015.
Search in Google Scholar Back to article
Bashir M.A., Moussa A.A., The cotha(x ) expansion method and its application to the Davey-Stewartson equation, Applied Mathematical Sciences, 8(78), 3851-3868, 2014.
Search in Google Scholar Back to article
Bashir M.A., Alhakim L.A., New f expansion method and its applications to modified KdV equation, Journal of Mathematics Research, 5(4), 83-96, 2013.
Search in Google Scholar Back to article
Gasmi B., Kessi A., Hammouch Z., Various optical solutions to the (1+1)-Telegraph equation with space-time conformable derivatives, International Journal of Nonlinear Analysis and Applications, 12(Special Issue), 767-780, 2021.
Search in Google Scholar Back to article
Arshad M., Seadawy A.R., Lu D., Jun W., Modulation instability analysis of modify unstable nonlinear Schrödinger dynamical equation and its optical soliton solutions, Results in Physics, 7, 4153-4161, 2017.
Search in Google Scholar Back to article
Li Y.X., Celik E., Guirao J.L.G., Saeed T., Baskonus H.M., On the modulation instability analysis and deeper properties of the cubic nonlinear Schrödingers equation with repulsive d -potential, Results in Physics, 25, 104303, 2021.
Search in Google Scholar Back to article

New analytical solutions and modulation instability analysis for the nonlinear (1+1)-dimensional Phi-four model

Abstract

Paradigm

My account