Advancing analytical solutions: Novel wave insights and methodologies for beta fractional Kuralay-II equations

Abstract This article investigates new analytical wave solutions within the beta ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>β</m:mi> </m:math> \beta ) fractional framework (F <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>κ</m:mi> </m:math> \kappa IIAE and F <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>κ</m:mi> </m:math> \kappa IIBE) of the Kuralay II equations, which are significant in the field of nonlinear optics. To achieve this, we employ the improved Kudryashov method, the R method, and the Sardar sub-equation method. The study successfully derives a variety of soliton solutions, including dark, bright, singular, and others, some of which are illustrated through 2D and 3D graphics. For the first time, this research graphically demonstrates the impact of the beta fractional derivative on these solutions. The findings provide valuable insights that may aid in the advancement of future models. The methodologies applied here are not only effective and straightforward to implement but also robust for addressing other fractional, nonlinear partial differential equations.