The travelling wave solutions of the Aizhan-Gudekli-Nurshuak-Zhanbota equation and its numerical treatment

Abstract

The cornerstone of this paper is the developing of novel analytical travelling wave solutions for the Aizhan-Gudekli-Nurshuak-Zhanbota equation (AGNZE), which was recently introduced as an integrable model arising in the study of space curves and surfaces. Therefore, its analytical exploration is vital for understanding its physical applications. We will explore solitons and various solitary wave solutions with diverse physical structures of this model. Significantly, this model characterizes the generalized Heisenberg ferromagnetic model. The innovation of this work lies in the enriched and distinct soliton solutions obtained, and in performing a comparative analysis of the proposed method, which has not been previously addressed. These achieved solutions correspond to the spin waves in the context of the ferromagnetic model. The construction of these new travelling wave solutions is carried out within the framework of notable schemes, the modified exp [−$\xi$]-expansion method and the modified simple equation method which are governed by the balance rule. Furthermore, the numerical solutions for all obtained analytical travelling wave solutions have been implemented by using the differential transform method (DTM), which is considered one of the most effective semi-analytical and numerical methods. The extracted solutions display distinct physical structures, including forms like bright solitons, dark solitons, combination of bright solitons and dark solitons, peakon-type bright and dark waves, anti-kink wave structures, W-like solitons, M-like solitons, hyperbolic solitons, singular solitons, and other rational solitons forms. The obtained solutions may be applied to model ultra-short pulse propagation in some nonlinear fields such as optical fibers, photonic crystals, waveguides, and solitary waves in shallow water. Additionally, we will use 2D and 3D graphical simulations to illustrate the new dynamic properties of our obtained solutions by using a mathematical tool.