A spectral fractional Hirota bilinear formulation]{A spectral fractional Hirota bilinear formulation for a time-fractional nonlinear Schr\"odinger flow: one-soliton solution and obstruction to finite two-soliton truncation
Abstract
We define a skew-adjoint spectral time-fractional operator $\mathcal{D}_t^\alpha$ ($0<\alpha\le1$) by the Fourier multiplier $\widehat{\mathcal{D}_t^\alpha\varphi}(\omega)=\mathrm{i}\,\mathrm{sgn}(\omega)|\omega|^\alpha\hat\varphi(\omega)$. This provides a single-valued exponential Hirota calculus. Using the induced fractional Hirota $D_t^\alpha$-operator we derive a bilinear system for a time-fractional focusing NLS flow, recovering the classical Hirota form when $\alpha=1$. A stationary bright one-soliton $\tau$-function is obtained with dispersion relation $\omega=k^{2/\alpha}$. For $0<\alpha<1$ we prove that the classical finite two-soliton Hirota truncation is generically inconsistent: interaction-mode matching forces identities such as $(2\omega_2-\omega_1)^\alpha=2\omega_2^\alpha-\omega_1^\alpha$, contradicting strict concavity. Numerical residual plots illustrate the mismatch.
Keywords
- time-fractional nonlinear Schr\"odinger equation;
- Hirota bilinear method
- spectral fractional derivative
- soliton
- $\tau$-function
Supporting Institution
None
Ethical Statement
This study is purely theoretical and does not involve human participants, animals, or any personal/identifiable data. Therefore, ethical approval and informed consent are not applicable.
Thanks
None
References
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- I. Podlubny, Fractional Differential Equations, Academic Press, 1999.
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- N. Laskin, Fractional Schr¨odinger equation, Phys. Rev. E 66 (2002), 056108. DOI: 10.1103/PhysRevE.66.056108.
- R. Grande, Space–time fractional nonlinear Schr¨odinger equation, SIAM J. Math. Anal. 51 (2019), 4172–4212. DOI: 10.1137/19M1247140.
- S. Ray, A spectral fractional Hirota bilinear operator: analysis and application to a timefractional KdV equation, arXiv:2601.17347 (2026). DOI: 10.48550/arXiv.2601.17347.
Details
Primary Language
English
Subjects
Applied Mathematics (Other)
Journal Section
Research Article
Authors
Publication Date
June 29, 2026
Submission Date
January 18, 2026
Acceptance Date
June 28, 2026
Published in Issue
Year 2026 Volume: 8 Number: 1
APA
Ray, S. (2026). A spectral fractional Hirota bilinear formulation]{A spectral fractional Hirota bilinear formulation for a time-fractional nonlinear Schr\"odinger flow: one-soliton solution and obstruction to finite two-soliton truncation. Proceedings of International Mathematical Sciences, 8(1), 9-15. https://doi.org/10.47086/pims.1866476
AMA
1.Ray S. A spectral fractional Hirota bilinear formulation]{A spectral fractional Hirota bilinear formulation for a time-fractional nonlinear Schr\"odinger flow: one-soliton solution and obstruction to finite two-soliton truncation. PIMS. 2026;8(1):9-15. doi:10.47086/pims.1866476
Chicago
Ray, Subhasis. 2026. “A Spectral Fractional Hirota Bilinear Formulation]{A Spectral Fractional Hirota Bilinear Formulation for a Time-Fractional Nonlinear Schr\"odinger Flow: One-Soliton Solution and Obstruction to Finite Two-Soliton Truncation”. Proceedings of International Mathematical Sciences 8 (1): 9-15. https://doi.org/10.47086/pims.1866476.
EndNote
Ray S (June 1, 2026) A spectral fractional Hirota bilinear formulation]{A spectral fractional Hirota bilinear formulation for a time-fractional nonlinear Schr\"odinger flow: one-soliton solution and obstruction to finite two-soliton truncation. Proceedings of International Mathematical Sciences 8 1 9–15.
IEEE
[1]S. Ray, “A spectral fractional Hirota bilinear formulation]{A spectral fractional Hirota bilinear formulation for a time-fractional nonlinear Schr\"odinger flow: one-soliton solution and obstruction to finite two-soliton truncation”, PIMS, vol. 8, no. 1, pp. 9–15, June 2026, doi: 10.47086/pims.1866476.
ISNAD
Ray, Subhasis. “A Spectral Fractional Hirota Bilinear Formulation]{A Spectral Fractional Hirota Bilinear Formulation for a Time-Fractional Nonlinear Schr\"odinger Flow: One-Soliton Solution and Obstruction to Finite Two-Soliton Truncation”. Proceedings of International Mathematical Sciences 8/1 (June 1, 2026): 9-15. https://doi.org/10.47086/pims.1866476.
JAMA
1.Ray S. A spectral fractional Hirota bilinear formulation]{A spectral fractional Hirota bilinear formulation for a time-fractional nonlinear Schr\"odinger flow: one-soliton solution and obstruction to finite two-soliton truncation. PIMS. 2026;8:9–15.
MLA
Ray, Subhasis. “A Spectral Fractional Hirota Bilinear Formulation]{A Spectral Fractional Hirota Bilinear Formulation for a Time-Fractional Nonlinear Schr\"odinger Flow: One-Soliton Solution and Obstruction to Finite Two-Soliton Truncation”. Proceedings of International Mathematical Sciences, vol. 8, no. 1, June 2026, pp. 9-15, doi:10.47086/pims.1866476.
Vancouver
1.Subhasis Ray. A spectral fractional Hirota bilinear formulation]{A spectral fractional Hirota bilinear formulation for a time-fractional nonlinear Schr\"odinger flow: one-soliton solution and obstruction to finite two-soliton truncation. PIMS. 2026 Jun. 1;8(1):9-15. doi:10.47086/pims.1866476
