发表机构
Escuela de Ingenierías Industriales, Universidad de Málaga（马拉加大学工业工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出双指数色散族，分离紧致化与谱结构机制，揭示两条转变曲线，边缘尖点紧致子为分布解，线性化谱可精确求解，数值实验验证理论预测。
AI 中文摘要
我们引入一个保守的双指数色散族，其中$p>1$为非线性通量指数，$r>0$为梯度色散指数，该色散族在保持显式行波理论的同时，将控制紧致化、自由边界正则性和线性化谱结构的机制分离开来。当$r=1$时，该族退化为绝对Rosenau-Hyman方程。该变形不改变振幅-速度定律，但除$r=1$外，使得宽度依赖于振幅。出现两条独立的转变曲线：$pr=1$将无限尾孤立波与紧致子分开，而$r(p-1)=2$将具有正则自由边界的紧致子与具有发散斜率的边缘尖点紧致子分开。后者仍为分布意义下的解，在自由边界或波峰处均无奇异测度。对于$r\ge1$，线性化通过一个奇异Sturm-Liouville算子分解，其自然的Friedrichs实现是可解的：经过紧致子适配的变量替换后，特征值方程为Jacobi-超几何类型，具有两个显式谱序列、恰好一个负特征值和一个一维平移核。数值实验将这些精确结构与逼近和动力学联系起来。测得的傅里叶尾部和投影误差与自由边界指数预测的速率一致，而填充傅里叶-伽辽金时间积分以较小的相位漂移相干地传播尖点波。对完整生成器进行结构适配的Jacobi-伽辽金诊断，在测试的截断点谱中未检测到可察觉的实部。因此，该族提供了一个显式测试平台，可在单个退化色散模型中研究紧致子的几何奇异性、弱容许性、谱结构和数值正则性。
英文摘要
We introduce a conservative two-exponent dispersive family, with $p>1$ the nonlinear-flux exponent and $r>0$ the gradient-dispersion exponent, that separates the mechanisms governing compactification, free-boundary regularity and linearised spectral structure while retaining an explicit travelling-wave theory. The family reduces to the absolute Rosenau-Hyman equation when $r=1$. The deformation leaves the amplitude-speed law unchanged, but makes the width amplitude dependent except at $r=1$. Two independent transition curves emerge: $pr=1$ separates infinite-tail solitary waves from compactons, whereas $r(p-1)=2$ separates compactons with a regular free boundary from edge-cusped compactons with divergent slope. The latter remain distributional solutions, with no singular measures at either the free boundary or the crest. For $r\ge1$, the linearisation factors through a singular Sturm-Liouville operator whose natural Friedrichs realisation is solvable: after a compacton-adapted change of variable, the eigenvalue equation is of Jacobi-hypergeometric type, with two explicit spectral sequences, exactly one negative eigenvalue and a one-dimensional translation kernel. Numerical experiments connect these exact structures to approximation and dynamics. Measured Fourier tails and projection errors agree with the rates predicted by the free-boundary exponent, while padded Fourier-Galerkin time integration propagates the cusped waves coherently with small phase drift. A structure-adapted Jacobi-Galerkin diagnostic of the full generator finds no detectable real part in the tested truncated point spectra. The family therefore provides an explicit testbed in which geometric singularity, weak admissibility, spectral structure and numerical regularity of compactons can be studied within a single degenerate dispersive model.
Comments19 pages, 6 figures. Reproducibility code and numerical data: https://doi.org/10.5281/zenodo.22671976
