A Spectral Fractional Hirota Bilinear Operator: Analysis and Application to a Time-Fractional KdV Equation
S. Ray
TL;DR
The paper addresses defining a Hirota-type bilinear calculus compatible with a spectral (Fourier-multiplier) fractional derivative on $\mathbb{R}$. It introduces the operator $D_{\xi}^{\alpha}f\cdot g = (D_{\xi}^{\alpha}f)\,g - f\,(D_{\xi}^{\alpha}g)$, derives a Marchaud-type integral representation, proves key algebraic and Sobolev properties, and shows convergence to the classical Hirota derivative as $\alpha\to1^-$. The framework is then applied to a spectral time-fractional KdV equation, yielding a Hirota bilinear form and explicit one- and two-soliton $\tau$-functions with dispersion $\omega^{\alpha}=-k^{3}$; the two-soliton interaction coefficient coincides with the classical KdV value. A remark extends the approach to a time-fractional KP equation, illustrating the method's potential for broad fractional integrable models. Overall, the work provides a rigorous, Fourier-compatible bilinear calculus for fractional derivatives and demonstrates its utility in constructing fractional soliton solutions.
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 $D_{ξ_1}^α-D_{ξ_2}^α$. 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 $D_ξ^αf\cdot f=0$), 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.