Renormalization-Group Fixed Points Not Associated with Universality Classes
Abstract
Universality classes have recently been identified for self-similar solutions of nonlinear partial differential equations (PDEs) as consequences of repeated renormalization-group (RG) transformations that eliminate irrelevant structures. Whether all RG fixed points are associated with universality classes, however, remains unclear. We address this question by extending a recently proposed predictive RG framework to traveling-wave solutions, using the Burgers and Korteweg--de Vries (KdV) equations as examples. Traveling waves can be mapped onto self-similar forms via a logarithmic transformation. Applying the predictive RG to the log-transformed governing equations yields an RG-flow analysis that reproduces shock and soliton solutions, thereby identifying traveling waves as RG fixed points and the logarithmic variable as RG-flow time. Although the predictive RG framework has shown that self-similar fixed points are associated with universality classes, we find that the present traveling-wave fixed points do not produce the irrelevant structures whose elimination normally gives rise to universality classes. This failure originates from the logarithmic representation, which renders analytic local functions of the field and its derivatives scale invariant, thereby preventing the emergence of irrelevant structures. Traveling waves thus provide explicit examples of RG fixed points without universality classes. These observations suggest that the absence of universality classes may be generic for traveling-wave RG fixed points formulated through logarithmic coordinates. The absence of universality classes suggests that information associated with initial conditions may persist asymptotically, promoting memory retention. The Burgers and KdV examples suggest that memory of initial conditions can survive RG transformations, although its form remains system dependent.
Related
- Predictive Renormalization-Group Theory of Universality Classes in Nonlinear Systems
- Real-Space renormalization for the drawups and drawdowns of stochastic trajectories: universality classes for diffusion processes and random walks via the scale function
- Fixed points of the CKM matrix renormalization group running to all orders in perturbation theory
- Chiral Doubling, Renormalization Group Fixed Points, and Dense Matter Equations of State
- Cluster Representation of Renormalization Group Transformations and a Rigorous Proof for Convergence of the RG-Flow of the Ising Model to Trivial Fixed Points away from Criticality
BibTeX
@misc{okumura2026renormalizationgroup,
title = {{Renormalization-Group Fixed Points Not Associated with Universality Classes}},
author = {Ko Okumura},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.18984},
eprint = {2609.18984},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18984},
}
RIS
TY - UNPB
TI - Renormalization-Group Fixed Points Not Associated with Universality Classes
AU - Ko Okumura
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2609.18984
UR - https://arxiv.org/abs/2609.18984
AB - Universality classes have recently been identified for self-similar solutions of nonlinear partial differential equations (PDEs) as consequences of repeated renormalization-group (RG) transformations that eliminate irrelevant structures. Whether all RG fixed points are associated with universality classes, however, remains unclear. We address this question by extending a recently proposed predictive RG framework to traveling-wave solutions, using the Burgers and Korteweg--de Vries (KdV) equations as examples. Traveling waves can be mapped onto self-similar forms via a logarithmic transformation. Applying the predictive RG to the log-transformed governing equations yields an RG-flow analysis that reproduces shock and soliton solutions, thereby identifying traveling waves as RG fixed points and the logarithmic variable as RG-flow time. Although the predictive RG framework has shown that self-similar fixed points are associated with universality classes, we find that the present traveling-wave fixed points do not produce the irrelevant structures whose elimination normally gives rise to universality classes. This failure originates from the logarithmic representation, which renders analytic local functions of the field and its derivatives scale invariant, thereby preventing the emergence of irrelevant structures. Traveling waves thus provide explicit examples of RG fixed points without universality classes. These observations suggest that the absence of universality classes may be generic for traveling-wave RG fixed points formulated through logarithmic coordinates. The absence of universality classes suggests that information associated with initial conditions may persist asymptotically, promoting memory retention. The Burgers and KdV examples suggest that memory of initial conditions can survive RG transformations, although its form remains system dependent.
ER -