The Construction of Analytical Exact Soliton Waves of Kuralay Equation

The primary objective of this work is to examine the Kuralay equation, which is a complex integrable coupled system, in order to investigate the integrable motion of induced curves. The soliton solutions derived from the Kuralay equation are thought to be the supremacy study of numerous significant phenomena and extensive applications across a wide range of domains, including optical fibres, nonlinear optics and ferromagnetic materials. The inverse scattering transform is unable to resolve the Cauchy problem for this equation, so the analytical method is used to produce exact travelling wave solutions. The modified auxiliary equation and Sardar sub-equation approaches are used to find solitary wave solutions. As a result, singular, mixed singular, periodic, mixed trigonometric, complex combo, trigonometric, mixed hyperbolic, plane and combined bright–dark soliton solution can be obtained. The derived solutions are graphically displayed in 2-D and 3-D glances to demonstrate how the fitting values of the system parameters can be used to predict the behavioural responses to pulse propagation. This study also provides a rich platform for further investigation.