双重色散方程行波解Hessian算子谱分析
Spectral Analysis of the Hessian Operator for Traveling Wave Solutions of the Double Dispersion Equation
DOI: 10.12677/AAM.2026.155213, PDF,   
作者: 李 涛：长沙理工大学数学与统计学院, 湖南 长沙；长沙理工大学工程数学建模与分析湖南省重点实验室, 湖南 长沙
关键词: 双重色散方程；行波解；临界速度；谱分析；Double Dispersion Equation； Traveling Wave Solutions； Critical Speed； Spectral Analysis
摘要: 谱分析的结果对于研究双重色散方程临界速度下行波解的稳定性情况具有重要意义。 本文研究临 界情况时双重色散方程行波解 Hessian 算子的谱分析。 首先，基于方程的晗密顿结构，梳理动量 与能量守恒律，构造由守恒律线性组合的泛函，分析该泛函在临界速度下的相关性质。 其次，对 该泛函在临界行波处的 Hessian 算子进行系统谱分析，明确其核空间，证明算子存在唯一负特征 值，且本质谱位于正半轴。
Abstract: The results of spectral analysis are of great significance for studying the stability of traveling wave solutions of the double dispersion equation at the critical speed. This paper investigates the spectral analysis of the Hessian operator for traveling wave solutions of the double dispersion equation in the critical case. Firstly, based on the Hamiltonian structure of the equation, we sort out the momentum and energy conservation laws, construct a functional formed by the linear combination of these conservation laws, and analyze the relevant properties of this functional at the critical speed. Secondly, we conduct a systematic spectral analysis of the Hessian operator of this functional at the critical traveling wave, clarify its kernel space, and prove that the operator has a unique negative eigenvalue with its essential spectrum lying on the positive semi-axis.
文章引用：李涛. 双重色散方程行波解Hessian算子谱分析[J]. 应用数学进展, 2026, 15(5): 121-132. https://doi.org/10.12677/AAM.2026.155213

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