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In this article, we discuss the findings of new developments in a class of new triangular functions that blend the quantity functions of the traditional triangular. Considering the significant role played by the triangular functions in applied mathematics, physics, and engineering, it is conceivable to predict that the theory of new triangular functions will provide us with additional interpretations and discoveries in mathematics and physics. The solutions which consider variable separation based on arbitrary functions are constructed to the (3+1)-dimensional Burgers model by presenting the Fibonacci Riccati technique and the linearly independent variable separation approach. This technique's fundamental concept is to describe the solution of the Burgers model as a polynomial in the Riccati Equation solution that satisfies the symmetrical hyperbolic and triangular Fibonacci functions. Depending on the choice of suitable functions for variable separation, an abundance of new localized solutions were obtained. Moreover, examples such as embedded solitons, rectangle-solitons, plateau-type ring solitons, taper-like solitons, and their interactions with each other, following the symmetrical hyperbolic and triangular Fibonacci functions, as well as the golden mean, could be explored. [ABSTRACT FROM AUTHOR]

In this paper, we analyze Riemann–Liouville (R-L) time-fractional (2 + 1) dimensional extended quantum Zakharov–Kuznetsov (EQZK) equation by using the Lie symmetry method which arises in hydrodynamic that describes the nonlinear propagation of the quantum ion-acoustic waves. By using its symmetry, we convert the equation under consideration to a fractional order non-linear ordinary differential equation (ODE). In this reduced ODE, we use a special type of derivative which is known as Erdélyi–Kober (EK) derivative. This enables us to obtain explicit solutions with convergence analysis of the considered problem. By using Ibragimov's conservation laws theorem, we compute the conservation laws of the problem under investigation. Moreover, by employing the two potent methods explicit power series and ( 1 G ′ )-expansion technique, we get the explicit solutions to the problem under discussion. This analysis leads to the derivation of various key findings, including the identification of symmetries, the establishment of similarity reductions involving the EK fractional differential operator, the determination of exact solutions, and the formulation of conservation laws for the considered equation. We have confidence that these remarkable findings can provide valuable insights and contribute to the exploration of additional evolutionary mechanisms associated with the studied equation. [ABSTRACT FROM AUTHOR]

In this paper, a rational difference equation with positive parameters and non-negative conditions is used to determine the presence and direction of the Neimark–Sacker bifurcation. The neimark–Sacker bifurcation of the system is first studied using the characteristic equation. In addition, we study bifurcation invariant curves from the perspective of normal form theory. A computer simulation is used to illustrate the analytical results. [ABSTRACT FROM AUTHOR]