发表机构
Ochanomizu University（御茶水女子大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过统一重整化群框架分析行波解，发现其标度维度必须为零，导致所有扰动尺度不变，从而形成无普适类的重整化群不动点，解释了行波对初始条件的强记忆性。
AI 中文摘要
行波是非线性物理系统中的基本渐近结构。虽然它们与自相似性的联系已被认识到，但其重整化群（RG）地位仍然难以捉摸。我们将最近为非线性偏微分方程开发的统一重整化群框架扩展到行波解，以Burgers方程和KdV方程作为典型例子。通过采用对数变换，我们将行波映射到渐近自相似解，从而允许进行系统的重整化群处理。我们的分析揭示了与标准重整化群范式的显著偏离：对于行波，尺度不变性唯一地迫使场的标度维度消失（A=0）。这个消失的维度意味着重整化群变换重新标度空间和时间，而保持场的幅度不变。因此，所有解析扰动都变得尺度不变，消除了将扰动分类为相关和无关结构的传统分类。虽然经典冲击波解和孤子解作为静态重整化群不动点出现，但普适类形成的机制——即逐步消除无关结构——从根本上缺失。因此，行波代表了一类独特的无普适类的重整化群不动点。这一发现确立了这两个概念之间的关键区别，并为行波为何对初始条件和系统参数表现出强烈记忆提供了严格的理论基础。
英文摘要
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.
Comments6 pages, no figures
