Multiple Solitons, Bifurcations, Chaotic Patterns And Fission/Fusion, Rogue Waves Solutions Of Two-Component Extended (2+1)-D Itã” Calculus System

The exploration of nonlinear phenomena entails the representation of intricate systems with space-time variables, and across this line, Itô calculus, as the stochastic calculus version of the change pertaining to the variables formula and chain rules, involves the second derivative of f, coming from the property that Brownian motion has non-zero quadratic variation. To this end, the extended (2+1)-dimensional two-component Itô equation, as an applicable mathematical tool employed in this study for enhancing our understanding of the complex dynamics inherent in multidimensional physical systems, serves the purpose of modeling and understanding dynamic phenomena pervading various disciplines. For modeling complex phenomena, fractional differential equations (FDEs), ordinary differential equations (ODEs), partial differential equations (PDEs) as well as the other ones provide benefits in terms of accuracy and tractability. Accordingly, our study provides the analysis of the two-component nonlinear extended (2+1)-dimensional Itô equation using the Hirota bilinear method to derive multiple soliton solutions, including novel variations along with their dispersion coefficients, which shed light into the intriguing attributes of the Itô equation. The investigation further encompasses diverse soliton types, such as the general first-order soliton, second-order soliton with fission and bifurcation, third-order soliton, and fourth-order soliton with fission and bifurcation. Besides these, the study also explores the rogue wave and lump solutions by varying parameters across distinct planes. Consequently, these results validate the characteristics and utility of the two-component nonlinear extended (2+1)-dimensional Itô equation and its relevance to related systems. The novel findings based on the Itô calculus systems revealed, through the analyses, theoretical and experimental aspects in combination with the graphical presentation of the parameter effects on solitons in line with the analyses obtained. These have enhanced the understanding of the dynamics of intricate attributes governed by the two-component nonlinear extended (2+1)-dimensional Itô equation, particularly concerning chaotic patterns, fission and bifurcation soliton nonlinear complexities.