Exact Soliton Solutions and Dynamical Behaviors of Hausdorff Fractal Fokas–Lenells Equation

• Published in: International journal of applied and computational mathematics Vol. 11; no. 2
• Main Authors: Abdelkhalek, A., El-Kalla, I. L., Elsaid, A., Kader, A. H. Abdel
• Format: Journal Article
• Language: English
• Published: New Delhi Springer India 01.04.2025; Springer Nature B.V
• Subjects: Applications of Mathematics; Bifurcations; Computational Science and Engineering; Differential equations; Dynamical systems; Elliptic functions; Fields (mathematics); Fractals; Mathematical and Computational Physics; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Nuclear Energy; Operations Research/Decision Theory; Original Paper; Solitary waves; Theoretical; Traveling waves; Bifurcation; Fokas–Lenells; Nonlinear dynamics; Hausdorff fractal; Doubly periodic; Solitons
• ISSN: 2349-5103; 2199-5796
• DOI: 10.1007/s40819-025-01874-1

In this paper, the Hausdorff Fractal Fokas-Lenells (HFFL) equation with full nonlinearity is investigated. The travelling wave reduction technique is utilized to transform the HFFL equation into an ordinary differential equation that can be converted to a two-dimensional planar dynamical system. By applying the direct integration method, some doubly periodic solutions are obtained for the HFFL equation in the form of Jacobi elliptic functions and Weierstrass function. New peakon, kink, bright, dark, W-shaped, and M-shaped solitons solutions for the HFFL equation are found under some constraints. The two-dimensional planar dynamical system is investigated to calculate the equilibrium points, which are classified as hyperbolic and nonhyperbolic equilibrium points. The theories of dynamical systems are employed to analyze the stability of individual equilibrium points as well as the bifurcation of the system's dynamics. Vector fields are plotted in the two-dimensional phase portraits to explore the dynamics near each equilibrium point.