Edge-cusped compactons from gradient-power nonlinear dispersion: exact transitions and singular spectral structure
Abstract
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≥1, 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.
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BibTeX
@misc{villatoro2026edgecusped,
title = {{Edge-cusped compactons from gradient-power nonlinear dispersion: exact transitions and singular spectral structure}},
author = {Francisco R. Villatoro},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00552},
eprint = {2610.00552},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00552},
}
RIS
TY - UNPB
TI - Edge-cusped compactons from gradient-power nonlinear dispersion: exact transitions and singular spectral structure
AU - Francisco R. Villatoro
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00552
UR - https://arxiv.org/abs/2610.00552
AB - 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≥1, 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.
ER -