Spectral time-fractional nonlinear Schrödinger equation: Formal Hirota framework, multi-soliton templates, and breather diagnostics

Alvaro H. Salas; Weaam Alhejaili; Samir A. El-Tantawy; Alvaro H. Salas; Weaam Alhejaili; Samir A. El-Tantawy

doi:10.3934/math.20261249

AIMS Mathematics

2026, Volume 11, Issue 9: 31710-31738. doi: 10.3934/math.20261249

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Spectral time-fractional nonlinear Schrödinger equation: Formal Hirota framework, multi-soliton templates, and breather diagnostics

1.
Department of Mathematics and Statistics, Universidad Nacional de Colombia, FIZMAKO Research Group, Colombia
2.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
3.
Department of Physics, Faculty of Science, Al-Baha University, P.O. Box 1988, Al-Baha 65779, Saudi Arabia
4.
Department of Physics, Faculty of Science, Port Said University, Port Said 42521, Egypt

Received: 10 February 2026 Revised: 03 July 2026 Accepted: 22 July 2026 Published: 24 September 2026
MSC : 35C08, 35Q55, 35R11, 65M70

In this work, we investigated a spectral time-fractional deformation of the focusing nonlinear Schrödinger equation in which the first-order time derivative was replaced by a translation-invariant Fourier-multiplier operator of order $ 0 < \alpha\leq1 $. The analysis was deliberately formulated as a formal Hirota-type construction: For $ \alpha < 1 $, the fractional operator did not obey the classical Leibniz and quotient rules required for an exact Hirota bilinearization, and the tau functions obtained in this work were therefore not asserted to be exact closed-form solutions of the fractional equation. Instead, the spectral action of $ D_t^\alpha $ on exponential modes was used to preserve the classical mode-by-mode tau-function algebra while deforming the dispersion relation. This led to explicit one-, two-, and $ N $-phase tau-function templates and to fractional Akhmediev-, Kuznetsov-Ma-, and Peregrine-type breather ansatz. The breather deformation was tied to the modulational-instability growth law through an explicitly stated reference wavenumber $ K $, with the natural Akhmediev choice $ K_a = 2\sqrt{1-2a} $ and the long-wave reference value $ K_{\rm ref} = 0.1 $ used in the limiting numerical tests. The accuracy of the resulting profiles was assessed a posteriori by Fourier-spectral residuals and by scaled residual criteria that separate quantitative approximation from qualitative morphology preservation. New central time-section diagnostics showed how the fractional order modified the focusing and recurrence scales, even when two-dimensional density plots looked similar. The results indicate that the proposed profiles are useful analytical benchmarks for weak-to-moderate spectral time-fractional deformation, while the Kuznetsov-Ma case should be regarded primarily as a qualitative recurrent breather template because of its larger relative residuals.
- nonlinear Schrödinger equation,
- spectral fractional derivative,
- formal Hirota framework,
- tau-function templates,
- modulational instability,
- breathers,
- rogue waves,
- residual diagnostics
Citation: Alvaro H. Salas, Weaam Alhejaili, Samir A. El-Tantawy. Spectral time-fractional nonlinear Schrödinger equation: Formal Hirota framework, multi-soliton templates, and breather diagnostics[J]. AIMS Mathematics, 2026, 11(9): 31710-31738. doi: 10.3934/math.20261249

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Abstract

In this work, we investigated a spectral time-fractional deformation of the focusing nonlinear Schrödinger equation in which the first-order time derivative was replaced by a translation-invariant Fourier-multiplier operator of order $ 0 < \alpha\leq1 $. The analysis was deliberately formulated as a formal Hirota-type construction: For $ \alpha < 1 $, the fractional operator did not obey the classical Leibniz and quotient rules required for an exact Hirota bilinearization, and the tau functions obtained in this work were therefore not asserted to be exact closed-form solutions of the fractional equation. Instead, the spectral action of $ D_t^\alpha $ on exponential modes was used to preserve the classical mode-by-mode tau-function algebra while deforming the dispersion relation. This led to explicit one-, two-, and $ N $-phase tau-function templates and to fractional Akhmediev-, Kuznetsov-Ma-, and Peregrine-type breather ansatz. The breather deformation was tied to the modulational-instability growth law through an explicitly stated reference wavenumber $ K $, with the natural Akhmediev choice $ K_a = 2\sqrt{1-2a} $ and the long-wave reference value $ K_{\rm ref} = 0.1 $ used in the limiting numerical tests. The accuracy of the resulting profiles was assessed a posteriori by Fourier-spectral residuals and by scaled residual criteria that separate quantitative approximation from qualitative morphology preservation. New central time-section diagnostics showed how the fractional order modified the focusing and recurrence scales, even when two-dimensional density plots looked similar. The results indicate that the proposed profiles are useful analytical benchmarks for weak-to-moderate spectral time-fractional deformation, while the Kuznetsov-Ma case should be regarded primarily as a qualitative recurrent breather template because of its larger relative residuals.

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