Stationary Grading and Heat-Transported Moving Boundaries: From the Cubic Case to Two Fourth-Order Branches
Abstract
We study zeros transported by the heat semigroup from stationary solutions of the sparse operators Ln=cn,0Dn+c0,n−1xn−1. The starting point is the exact cubic boundary obtained by Hernández-del-Valle and Guerra-Polania~\cite{HerGuerra}. We show that the hypergeometric function used there is the residue-$1$ component of a naturally graded stationary solution space, and that heat transport preserves this grading after the scaling x=tny. This identifies the cubic boundary as the transported zero of a specific stationary mode. We then study L4=c4,0D4+c0,3x3. Its 6×6 boundary-jet determinant leads to the universal scaling F(t)=ρ(t/2)4 Φ (ρ2(t/2)7), ρ=c0,3/c4,0. The resulting nonlinear equation for Φ is singular at the origin. After an explicit desingularization, its initial compatibility condition factors as (5Φ'(0)−1)(6Φ'(0)−1)=0. Consequently there are exactly two normalized graded analytic boundary germs. We place the desingularized equation in Briot--Bouquet form, prove local existence and uniqueness of both branches, and give a triangular recursion for all their Taylor coefficients. Finally, we prove that the two branches are generated respectively by the residue-$1$ and residue-$3$ stationary solutions. The quadratic and cubic cases provide the established baseline; the fourth-order problem is the first point at which distinct stationary modes produce distinct analytic branches. We do not claim a general branch-count theorem.
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BibTeX
@misc{hernandezdelvalle2026stationary,
title = {{Stationary Grading and Heat-Transported Moving Boundaries: From the Cubic Case to Two Fourth-Order Branches}},
author = {Gerardo Hernández-del-Valle},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27914},
eprint = {2609.27914},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27914},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Stationary Grading and Heat-Transported Moving Boundaries: From the Cubic Case to Two Fourth-Order Branches
AU - Gerardo Hernández-del-Valle
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27914
UR - https://arxiv.org/abs/2609.27914
AB - We study zeros transported by the heat semigroup from stationary solutions of the sparse operators L_n=c_n,0D^n+c_0,n−1x^n−1. The starting point is the exact cubic boundary obtained by Hernández-del-Valle and Guerra-Polania~\cite{HerGuerra}. We show that the hypergeometric function used there is the residue-$1$ component of a naturally graded stationary solution space, and that heat transport preserves this grading after the scaling x=t^ny. This identifies the cubic boundary as the transported zero of a specific stationary mode. We then study L_4=c_4,0D^4+c_0,3x^3. Its 6×6 boundary-jet determinant leads to the universal scaling F(t)=ρ(t/2)^4 Φ (ρ^2(t/2)^7), ρ=c_0,3/c_4,0. The resulting nonlinear equation for Φ is singular at the origin. After an explicit desingularization, its initial compatibility condition factors as (5Φ'(0)−1)(6Φ'(0)−1)=0. Consequently there are exactly two normalized graded analytic boundary germs. We place the desingularized equation in Briot--Bouquet form, prove local existence and uniqueness of both branches, and give a triangular recursion for all their Taylor coefficients. Finally, we prove that the two branches are generated respectively by the residue-$1$ and residue-$3$ stationary solutions. The quadratic and cubic cases provide the established baseline; the fourth-order problem is the first point at which distinct stationary modes produce distinct analytic branches. We do not claim a general branch-count theorem.
ER -