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Journals / Proceedings of International Mathematical Sciences / 2026 / Volume: 8 Issue: 1

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

Subhasis Ray(Visva-Bharati)0000-0003-1100-4632
Pages
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Publication Date
2026-06-29
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Numerical residual plots illustrate the mismatch.

📌 3 Key Takeaways

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Objective & Hypothesis:We define a skew-adjoint spectral time-fractional operator $\mathcal{D}_t^\alpha$ ($0<\alpha\le1$) by the Fourier multiplier $\widehat{\mathcal{D}_t^\alpha\varphi}(\omega)=\mathrm{i}\,\mathrm{sgn}(\omega)|\omega|^\alpha\hat\varphi(\omega)$.
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Key Findings & Data: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 $\alpha=1$.
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Conclusion & Impact:Numerical residual plots illustrate the mismatch.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:time-fractional nonlinear Schr\"odinger equation · Hirota bilinear method · spectral fractional derivative

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.

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Cite This Article

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, Volume: 8 Issue: 1. https://doi.org/10.47086/pims.1866476

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