The closed-form soliton solutions of the time-fraction Phi-four and (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff model using the recent approach

Abstract

The Phi-four (PF) model is a variation of the familiar Klein–Fock–Gordon​ (KFG) model. Quantum effects, ultra-short pulse propagation in optical fibers, character of de-Broglie waves, wave-particle duality, composite particles of spineless relativistic, etc. are investigated through the PF model. The interface of Riemann waves in two spatial dimensions is explained by the Calogero–Bogoyavlenskii–Schiff (CBS) model. Many physical phenomena such as, the magneto-sound waves in plasmas, tsunami and tidal in rivers, and the internal waves in oceans can be describe through the Riemann wave. This article investigates the unique and wide-ranging analytic soliton solutions to the above-stated models in the sense of the fractional conformable derivative by using the auxiliary equation technique in terms of exponential, trigonometric, hyperbolic, and rational functions. To analyze the underlying wave structures and to examine the active behavior of the obtained solutions, three-dimensional (3D) and contour graphs are plotted using the Wolfram Mathematica program. The attained solutions demonstrate that the method is compatible, effective, and scientifically efficient.

Keywords

• Conformable derivative
• The time-fractional Phi-four equation
• The auxiliary equation method
• The time-fractional Calogero–Bogoyavlenskii–Schiff equation