Abstract
The Hirota bilinear method is employed for searching the localized waves, lump–solitons, and solutions between lumps and rogue waves for the fractional generalized Calogero–Bogoyavlensky–Schiff–Bogoyavlensky–Konopelchenko (CBS-BK) equation. We probe three cases including lump (combination of two positive functions as polynomial), lump–kink (combination of two positive functions as polynomial and exponential function) called the interaction between a lump and one line soliton, and lump–soliton (combination of two positive functions as polynomial and hyperbolic cos function) called the interaction between a lump and two-line solitons. At the critical point, the second-order derivative and the Hessian matrix for only one point will be investigated and the lump solution has one maximum value. The moving path of the lump solution and also the moving velocity and the maximum amplitude will be obtained. The graphs for various fractional orders α are plotted to obtain 3D plot, contour plot, density plot, and 2D plot. The physical phenomena of this obtained lump and its interaction soliton solutions are analyzed and presented in figures by selecting the suitable values. That will be extensively used to report many attractive physical phenomena in the fields of fluid dynamics, classical mechanics, physics, and so on.
Highlights
1 Introduction The nonlinear partial differential equation is a physical and natural model which can be used for model constructs by scientists and researchers
The utilized methods which employed by powerful researchers are such methods as the Exp-function method, the multiple Exp-function method, (G’-G)expansion method—but we should not forget to mention that these methods continue to attract a wave of criticism
We address solving the fractional generalized CBS-BK equation in the sense of the modified Riemann–Liouville derivative which has been derived by [37]
Summary
The nonlinear partial differential equation is a physical and natural model which can be used for model constructs by scientists and researchers. The aim of this study is to construct the invariant solutions of the (2 + 1)-dimensional fractional generalized CBS-BK equation in the following form: Dαt + xxy + 3 x y + δ1 y + δ2 yy + δ3 x. Some important work related with recent development in fractional calculus and its applications can be pointed out referring to the valuable papers containing studies of general fractional derivatives: theory, methods and applications by Yang [50]; anomalous diffusion equations with the decay exponential kernel by the Laplace transform [51]; new fractal nonlinear Burgers’ equation arising in the acoustic signals propagation by Yang and Machado [52]; time fractional nonlinear diffusion equation from diffusion process by fractional Lie group approach [53]; the generalized time fractional diffusion equation by symmetry analysis [54]; investigating a time fractional nonlinear heat conduction equation with applications in mathematics physics, integrable system, fluid mechanics and nonlinear areas, by means of applying the fractional symmetry group method [55]; and determining the time fractional extended (2 + 1)-dimensional Zakharov–Kuznetsov equation in quantum magneto-plasmas by using a group analysis approach [56].
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