https://doi.org/10.1140/epjc/s10052-026-15625-z
Regular Article - Theoretical Physics
Interior structure of black holes with nonlinear terms
1
Liaoning Key Laboratory of Cosmology and Astrophysics, College of Sciences, Northeastern University, 110819, Shenyang, China
2
Center for Gravitation and Astrophysics, Kunming University of Science and Technology, 650500, Kunming, China
3
Center for the Cross-disciplinary Research of Space Science and Quantum-technologies (CROSS-Q), College of Physics, Nanjing University of Aeronautics and Astronautics, 210016, Nanjing, China
4
MOE Key Laboratory of Data Analytics and Optimization for Smart Industry, Northeastern University, 110819, Shenyang, China
5
National Frontiers Science Center for Industrial Intelligence and Systems Optimization, Northeastern University, 110819, Shenyang, China
a
This email address is being protected from spambots. You need JavaScript enabled to view it.
b
This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
14
January
2026
Accepted:
26
March
2026
Published online:
2
May
2026
Abstract
We investigate the oscillation of the Kasner exponent 
near the critical point of hairy black holes dual to holographic superfluid and reveal a clear inverse periodicity 
in a large region below the critical temperature. We first introduce the fourth-power term with a coefficient 
to adjust the oscillatory behavior of the Kasner exponent near the critical point. Importantly, we show that the nonlinear coefficient provides accurate control of this periodicity: a positive stretches the region, while a negative compresses it. By contrast, the influence of another coefficient 
is more concentrated in regions away from the critical point. This work provides a new perspective for understanding the complex dynamical structure inside black holes and extends the active control from the fourth- and sixth-power terms into the black hole interior region.
© The Author(s) 2026
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Funded by SCOAP3.
