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

TL;DR

This paper introduces a spectral fractional Hirota bilinear operator, analyzes its properties, and applies it to derive a time-fractional KdV equation with explicit soliton solutions, extending classical integrable systems to fractional orders.

Contribution

It develops a spectral fractional Hirota bilinear calculus, establishes its fundamental properties, and applies it to formulate and solve a time-fractional KdV equation with explicit soliton solutions.

Findings

Defined a spectral fractional Hirota derivative with Fourier multiplier representation.

Proved algebraic identities and Sobolev estimates for the fractional operator.

Derived explicit one- and two-soliton solutions for the 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 $0<\alpha\le 1$ 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 $(ik_1)^{\alpha}-(ik_2)^{\alpha}$. 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 $s>\tfrac12$, 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.