Application of Kolmogorov-Arnold network (KAN) for solitary-Peakon investigation of Lax model

This study utilizes novel Kolmogorov-Arnold Networks to solve the fifth-order KdV-Lax problem, employing both periodic and Peakon solutions. Several soliton solutions, including solitary wave, Peakon forms, are presented using the KANs technique for the Lax problem of fifth order. The novelty of this investigation lies in its combination of a practical numerical approach for KANs with a powerful analytical strategy (the exp-function method) to verify reliability and authenticity, utilizing illustrations and tables to identify various types of soliton solutions. Although KANs provide an approximation for solving the KdV-Lax equation, their ability to tackle complex problems makes them a viable choice when an analytical solution is not possible. The use of KANs over the exponential function method's approximate solution is exceptional; it offers a thorough analysis of the solution's uniqueness and convergence using loss plots, error histograms, mean square logarithmic error, mean poisson deviation, and regression plots, among others. The numerical results of thorough simulations, with minimal error ( ), regression metric value ( ), and histogram with the majority of instances ( ) nearly close to zero, unquestionably supported or confirmed the accuracy of KANs against the analytical (exp-function) method. Furthermore, the robustness of the KANs model is demonstrated by consistently low absolute error metrics.