Deep learning of Hirota τ-function for soliton solutions of a (3+1)-dimensional nonlocal evolution equation
Year: 2026
Type: article
Abstract: We study a ( 3 + 1 ) -dimensional nonlinear evolution equation carrying a nonlocal term through the operator ∂ x − 1 , and we build a single machine-learning framework that unifies forward solution learning, inverse coefficient discovery, and automatic solution certification. The central step is to let a neural network carry the Hirota bilinear τ -function rather than the physical field, to train it with the residual of the bilinear equation, and to recover the field through the exact logarithmic map ψ = − 3 ( ln G ) x x . The bilinear equation obtained in this way is, up to sign, the bilinear form of the Jimbo–Miwa equation, so the model inherits the conditional integrability of that equation. This choice removes the nonlocal operator by construction, represents singular and complex waves directly, because singularities of the field are zeros of a smooth τ -function, and yields a differentiable functional that certifies whether any candidate closed form is a true solution. Using the certification analysis we revisit the exact solutions recently reported for this equation and prove, with symbolic derivations and high-precision numerical certificates, that the non-singular and singular complexiton families do not satisfy the equation and that the equation admits Hirota one- and two-soliton solutions but fails the three-soliton condition for generic wave numbers, the hierarchy surviving only on restricted parameter families. The corrected phase shift is supplied in closed form. The framework is validated on the true solutions, attains a relative error of about one percent on the forward problem, recovers the governing coefficients from noisy data, and continues to operate on a non-integrable perturbation for which no closed form exists.
Source: Nuclear Physics B
Authors: Mohammad Akram, Umar Ishtiaq, Ioan-Lucian Popa
Language: English
Topic: Nonlinear Waves and Solitons
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: diamond
Funder: Islamic University of Madinah
OpenAlex ID: W7223448423