Exact nonlinear wave structures in the conformable time-fractional Vakhnenko equation via bilinear neural networks

This study develops a conformable bilinear neural network framework for constructing exact nonlinear wave solutions of the conformable time-fractional Vakhnenko equation (CTFVE), a model of high-frequency wave motion in a relaxing medium in which the fractional order acts as a tunable time-scale parameter. By introducing a logarithmic dependent-variable transformation and a generalized Hirota bilinear operator compatible with the conformable fractional derivative, the governing fractional equation is recast in bilinear form. Several feedforward neural network structures with one and two hidden layers are then employed as structured ansätze for the bilinear auxiliary function, and the resulting algebraic systems are solved in closed form. This yields the integrable solution families of the Vakhnenko equation—the one-soliton, the two-soliton with its exact phase shift, the breather formed from a complex-conjugate soliton pair, and the cnoidal periodic wave—each verified symbolically against both the bilinear form and the original nonlinear equation. We show that the fractional order is physically equivalent to an algebraically time-varying relaxation rate of the medium, which makes the wave structures propagate with a power-law (decelerating for \(\epsilon <1\) ) speed while preserving their amplitude and width, and a numerical perturbation analysis confirms that the soliton is orbitally stable. Because the solutions are obtained in closed algebraic form, the method avoids the discretization and convergence errors of iterative schemes. The framework offers a flexible analytical route to coherent structures in fractional evolution equations and extends naturally to related nonlinear wave models.