Traveling Waves as Renormalization-Group Fixed Points without Universality Classes

Abstract

Traveling waves are fundamental asymptotic structures in nonlinear physical systems. While their connection to self-similarity is recognized, their renormalization-group (RG) status remains elusive. We extend a recently developed unified RG framework for nonlinear PDEs to traveling-wave solutions, using Burgers' and KdV equations as paradigmatic examples. By employing a logarithmic transformation, we map traveling waves onto asymptotically self-similar solutions, allowing for a systematic RG treatment. Our analysis reveals a striking departure from the standard RG paradigm: for traveling waves, scale invariance uniquely forces the field's scaling dimension to vanish (A=0). This vanishing dimension implies that the RG transformation rescales space and time while leaving the field magnitude unchanged. Consequently, all analytic perturbations become scale-invariant, eliminating the conventional classification into relevant and irrelevant structures. While classical shock-wave and soliton solutions emerge as stationary RG fixed points, the mechanism for universality class formation - the progressive elimination of irrelevant structures - is fundamentally absent. Traveling waves thus represent a unique class of RG fixed points without universality classes. This finding establishes a crucial distinction between these two concepts and provides a rigorous theoretical basis for why traveling waves exhibit strong memory of initial conditions and system parameters.