This study aims to explore the bifurcating soliton solutions of the time-fractional complex Maccari system (FCMS) with M-fractional derivatives. The focus is on applying the Generalized Kudryashov auxiliary method (GKAM) to solve the nonlinear fractional partial differential equations that describe hydraulic systems widely used in mechanics, physics and engineering. The goal is to derive soliton solutions that provide insights into the dynamic behavior of FCMS, specifically in terms of bifurcated cnoidal wave phenomena. The results aim to deepen the understanding of complex dynamics in systems described by fractional derivatives.
The GKAM is applied to the FCMS, a set of nonlinear fractional partial differential equations, to derive bifurcating soliton solutions. The process involves transforming these fractional differential equations into nonlinear ordinary differential equations (NODEs) via a variable transformation. The resulting NODEs are solved using series form solutions, which lead to systems of nonlinear algebraic equations. These systems are solved to obtain soliton solutions expressed in Jacobian elliptic functions. The investigation employs contour and 3D graphs to illustrate the dynamic behavior and bifurcation phenomena of the identified solitons.
The application of GKAM reveals soliton solutions of the time-FCMS, expressed in Jacobian elliptic functions. These solitons exhibit bifurcated cnoidal wave phenomena, demonstrating a rich dynamic behavior. The results are validated through visual representations, including contour and 3D graphs, which clearly show the intricate nature of these soliton solutions. The study finds that the GKAM technique effectively generates a variety of soliton solutions, providing new insights into the dynamic complexity of FCMS. These findings contribute to a deeper understanding of hydraulic systems and their nonlinear fractional dynamics.
This research provides a novel application of the GKAM to the time-FCMS, offering a fresh perspective on bifurcating soliton solutions in hydraulic systems. By using a variable transformation to convert nonlinear fractional partial differential equations into NODEs, the study presents a systematic approach for deriving soliton solutions. The use of Jacobian elliptic functions to express these solutions is an original contribution that highlights bifurcated cnoidal wave phenomena. The study’s results significantly expand the understanding of complex fractional dynamics, providing valuable insights into various engineering and physics applications.
