A Spectral Fractional Hirota Bilinear Operator: Analysis and Application to a Time-Fractional KdV Equation

Abstract: We develop a fractional version of Hirota's bilinear calculus that is built directly from the spectral (Fourier-multiplier) fractional derivative on $\mathbb{R}$. For $0<α\le 1$ we define [ D_ξ^αf\cdot g := (D_ξ^αf)\,g - f\,(D_ξ^αg), ] equivalently through the two-variable extension $D_{ξ1}^α-D{ξ2}^α$. In Fourier variables this is a bilinear multiplier with symbol $(ik_1)^α-(ik_2)^α$. For $0<α<1$ we prove a Marchaud-type singular integral representation, and we use it to establish basic algebraic identities (bilinearity, skew-symmetry and $Dξ^αf\cdot f=0$), a Sobolev estimate $H^{s}\times H^{s}\to H^{s-α}$ for $s>\tfrac12$, and convergence to the classical Hirota derivative as $α\to 1^-$. As an application we derive a Hirota bilinear form for a spectral time-fractional KdV equation and construct explicit one- and two-soliton $τ$-functions. The fractional order changes the dispersion relation to $ω^α=-k^{3}$, while the two-soliton interaction coefficient agrees with the classical KdV value.