A Spectral Fractional Hirota Bilinear Operator: Analysis and Application to a Time-Fractional KdV Equation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866909999881519104 |
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| author | Ray, S. |
| author_facet | Ray, S. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_17347 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Spectral Fractional Hirota Bilinear Operator: Analysis and Application to a Time-Fractional KdV Equation Ray, S. Analysis of PDEs 35Q53, 35R11, 35S05 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. |
| title | A Spectral Fractional Hirota Bilinear Operator: Analysis and Application to a Time-Fractional KdV Equation |
| topic | Analysis of PDEs 35Q53, 35R11, 35S05 |
| url | https://arxiv.org/abs/2601.17347 |