Solitary wave structures, conservation laws and dynamical analysis of the Heisenberg ferromagnet-type equation

In this paper, we study the integrable Zhaidary equation, which is a kind of Heisenberg ferromagnet-type equation in (1+1)-dimensions. By using a class of ordinary differential equations, we derive several types of solitary wave solutions to the Zhaidary equation, which include bell-shaped, W-shaped, and sinusoidal profile solitary wave solutions, as well as singular solutions and periodic wave solutions. Further, the multiplier technique is employed as a systematic framework to derive the conservation laws corresponding to the integrable Zhaidary equation. This method enables the identification of conserved quantities by determining appropriate multipliers that satisfy the necessary compatibility conditions. In addition, by employing the Galilean transformation, we present a comprehensive analysis of the system’s dynamics, including bifurcation structures, chaotic behavior, and sensitivity to initial conditions. The stability of the equation is investigated by using the linear-stability analysis and the modulation instability gain spectrum is acquired. Finally, the physical explanations of the obtained solutions are demonstrated graphically.