Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 056, 26 pages      arXiv:2311.07195      https://doi.org/10.3842/SIGMA.2024.056
Contribution to the Special Issue on Symmetry, Invariants, and their Applications in honor of Peter J. Olver

Talbot Effect for the Manakov System on the Torus

Zihan Yin a, Jing Kang a, Xiaochuan Liu b and Changzheng Qu c
a) Center for Nonlinear Studies and School of Mathematics, Northwest University, Xi'an 710069, P.R. China
b) School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, P.R. China
c) Center for Nonlinear Studies and Department of Mathematics, Ningbo University, Ningbo 315211, P.R. China

Received November 13, 2023, in final form June 17, 2024; Published online June 25, 2024

Abstract
In this paper, the Talbot effect for the multi-component linear and nonlinear systems of the dispersive evolution equations on a bounded interval subject to periodic boundary conditions and discontinuous initial profiles is investigated. Firstly, for a class of two-component linear systems satisfying the dispersive quantization conditions, we discuss the fractal solutions at irrational times. Next, the investigation to nonlinear regime is extended, we prove that, for the concrete example of the Manakov system, the solutions of the corresponding periodic initial-boundary value problem subject to initial data of bounded variation are continuous but nowhere differentiable fractal-like curve with Minkowski dimension 3/2 at irrational times. Finally, numerical experiments for the periodic initial-boundary value problem of the Manakov system, are used to justify how such effects persist into the multi-component nonlinear regime. Furthermore, it is shown in the nonlinear multi-component regime that the interplay of different components may induce subtle different qualitative profile between the jump discontinuities, especially in the case that two nonlinearly coupled components start with different initial profile.

Key words: Talbot effect; dispersive fractalization; dispersive quantization; multi-component dispersive equation; Manakov system.

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