Applications of Riccati–Bernoulli and Bäcklund Methods to the Kuralay-II System in Nonlinear Sciences

The Kuralay-II system (K-IIS) plays a pivotal role in modeling sophisticated nonlinear wave processes, particularly in the field of optics. This study introduces novel soliton solutions for the K-IIS, derived using the Riccati–Bernoulli sub-ODE method combined with Bäcklund transformation and conformable fractional derivatives. The obtained solutions are expressed in trigonometric, hyperbolic, and rational forms, highlighting the adaptability and efficacy of the proposed approach. To enhance the understanding of the results, the solutions are visualized using 2D representations for fractional-order variations and 3D plots for integer-type solutions, supported by detailed contour plots. The findings contribute to a deeper understanding of nonlinear wave–wave interactions and the underlying dynamics governed by fractional-order derivatives. This work underscores the significance of fractional calculus in analyzing complex wave phenomena and provides a robust framework for further exploration in nonlinear sciences and optical wave modeling.