Traveling, periodic and localized solitary waves solutions of the (4+1)-dimensional nonlinear Fokas equation
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Traveling, periodic and localized solitary waves solutions of the (4+1)‑dimensional nonlinear Fokas equation Hitender Khatri1 · Anand Malik2,3 · Manjeet Singh Gautam4 Received: 20 January 2019 / Accepted: 1 October 2020 © Springer Nature Switzerland AG 2020
Abstract By utilizing the three distinctive approaches specifically, the extended Fan sub-equation method, the exp[−G(𝜉)]-function expansion method, and the fractional transformation method, the traveling waves and soliton solutions of the (4+1)-dimensional nonlinear Fokas equation are extracted. Meanwhile, some parametric constraint conditions are described. The acquired solutions are singular and nonsingular soliton solutions, periodic solutions, breather solution, rational solutions, trigonometric periodic wave solutions, hyperbolic solutions, Weierstrass, and Jacobi elliptic doubly periodic wave solutions. The dynamics of some of the obtained solutions are investigated and described in 2-dimensional figures by choosing appropriate parameter values. The comparison of our obtained results with the other solutions in literature shows that the obtained solutions of this paper are new and have not been formulated before by other techniques. We believe that all these results are useful to enrich the knowledge of the important physical phenomenon characterized by the Fokas equation. The reported solutions illustrate the straightforwardness, reliability, and effectiveness of the used techniques that can be further employed to higher-dimensional nonlinear evolution equations. Keywords Soliton · Traveling wave solution · Jacobi elliptic function solution · The extended sub-equation method · The exp[−G(𝜉)]-function expansion method · The fractional transformation method
1 Introduction The nonlinear evolution equations (NLEEs) transpire in a wide spectrum of physical problems such as plasma physics, fluid dynamics, solid mechanics, nonlinear optics, oceanography, engineering, chemistry, biology, quantum field theory, and several others. It is requisite to extract traveling wave solutions of NLEEs to identify the mechanisms of the nonlinear physical phenomena characterized by these equations. Over the years much care has been paid by the research scholars for this purpose and evolved many powerful, reliable, and compact techniques for handling NLEEs. These are the inverse scattering transform method, auto Bäcklund
transformation method, Hirota’s bilinear transformation method, the tanh-coth function method, the projective Riccati equation method, the ansatz method, the F-expansion method, the simplest equation method, the modified and extended simplest equation method, the extended sinh-Gordon equation expansion method, � the auxiliary equation method, the (G ∕G) -expansion method and it’s extended versions, the fractional transformation, the extended and modified auxiliary equation method, the trial function method, the reductive perturbation method, the extended modified rational expansion method, the modified extended mapping method, the extended sub-equation me
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