Numerical solutions and analytical methods for the Kuralay equation: a path to understanding integrable systems

This study undertakes the challenging task of unraveling the intricacies embedded in the integrable Kuralay equation, a cornerstone in the field of mathematical physics. Specifically, our focus extends to the integrable dynamics governing space curves, encompassing both the Kuralay-I equation ( 𝒦-ℐℰ ) and the Kuralay-II equation ( 𝒦-ℐℐℰ ). Our methodological approach leverages innovative analytical techniques, namely the Bernoulli sub-equation function ( ℬ𝒮ℱ ) method and the Khater II ( 𝒦h II ) method. For the derivation of numerical solutions, we employ the exponential cubic-B-spline ( ℰ𝒞ℬ𝒮 ) scheme. The outcomes of our investigation signify a triumphant resolution of the Kuralay equation, thereby enriching our comprehension of its dynamic behavior. The profound impact of this study reverberates throughout the domain of mathematical physics, particularly in illuminating the intricate dynamics inherent in integrable systems. The distinctive aspect of our research lies in the application of the ℬ𝒮ℱ and 𝒦h II methods for analytical solutions, coupled with the utilization of the ℰ𝒞ℬ𝒮 scheme for numerical solutions. These methodologies, hitherto underexplored in the context of Kuralay equations, impart a unique quality to our investigation. In conclusion, this research constitutes a significant contribution to the progression of mathematical physics. It not only furnishes valuable insights, numerical solutions, and innovative methodologies for addressing the complex Kuralay equation but also advances our collective understanding of this foundational problem.