平稳分级与热输运移动边界：从三次情形到两个四阶分支
Stationary Grading and Heat-Transported Moving Boundaries: From the Cubic Case to Two Fourth-Order Branches
- Centro de Estudios Monetarios Latinoamericanos（拉丁美洲货币研究中心）
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AI总结：
本文研究热半群输运平稳解零点形成的移动边界，从三次精确边界推广到四阶算子，通过去奇异化得到两个解析分支，分别对应留数1和3的平稳模式。
AI中文摘要：
我们研究由稀疏算子 $L_n=c_{n,0}\D^n+c_{0,n-1}x^{n-1}$ 的平稳解经热半群输运的零点。起点是 Hernández-del-Valle 和 Guerra-Polania~\cite{HerGuerra} 获得的精确三次边界。我们证明其中使用的超几何函数是自然分级的平稳解空间中的留数-$1$ 分量，并且热输运在缩放 $x=t^ny$ 后保持该分级。这确定了三次边界是特定平稳模式的输运零点。随后我们研究 $L_4=c_{4,0}\D^4+c_{0,3}x^3$。其 $6\times6$ 边界射流行列式导致普适缩放 \\[ F(t)=ρ\left(\frac t2\right)^4 Φ\\!\left(ρ^2\left(\frac t2\right)^7\right), \qquad ρ=\frac{c_{0,3}}{c_{4,0}}. \\] 由此得到的 $Φ$ 的非线性方程在原点处奇异。经过显式去奇异化后，其初始相容条件分解为 \\[ (5Φ'(0)-1)(6Φ'(0)-1)=0. \\] 因此恰好存在两个归一化的分级解析边界芽。我们将去奇异化方程置于 Briot--Bouquet 形式，证明两个分支的局部存在性和唯一性，并给出其所有 Taylor 系数的三角递推。最后，我们证明这两个分支分别由留数-$1$ 和留数-$3$ 的平稳解生成。二次和三次情形提供了既定基线；四阶问题是不同平稳模式产生不同解析分支的第一个点。我们不声称一般的分支计数定理。
英文摘要：
We study zeros transported by the heat semigroup from stationary solutions of the sparse operators $L_n=c_{n,0}\D^n+c_{0,n-1}x^{n-1}$. 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 $L_4=c_{4,0}\D^4+c_{0,3}x^3$. Its $6\times6$ boundary-jet determinant leads to the universal scaling \[ F(t)=ρ\left(\frac t2\right)^4 Φ\!\left(ρ^2\left(\frac t2\right)^7\right), \qquad ρ=\frac{c_{0,3}}{c_{4,0}}. \] 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.
