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

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 $0<\alpha \le 1$ we define $$D_{\xi }^{\alpha }f\cdot g:=(D_{\xi }^{\alpha }f)g-f(D_{\xi }^{\alpha }g),$$ equivalently through the two-variable extension $D_{{\xi }_1}^{\alpha }-D_{{\xi }_2}^{\alpha }$. In Fourier variables this is a bilinear multiplier with symbol $(i{k}_1{)}^{\alpha }-(i{k}_2{)}^{\alpha }$. For $0<\alpha <1$ we prove a Marchaud-type singular integral representation, and we use it to establish basic algebraic identities (bilinearity, skew-symmetry and $D_{\xi }^{\alpha }f\cdot f=0$), a Sobolev estimate $H^s\times H^s\to H^{s-\alpha }$ for $s>\frac{1}{2}$, and convergence to the classical Hirota derivative as $\alpha \to 1^{-}$. As an application we derive a Hirota bilinear form for a spectral time-fractional KdV equation and construct explicit one- and two-soliton $\tau$-functions. The fractional order changes the dispersion relation to ${\omega }^{\alpha }=-k^3$, while the two-soliton interaction coefficient agrees with the classical KdV value.