微分算子与移动边界的热-艾里传输
Heat--Airy Transport of Differential Operators and Moving Boundaries
AI总结：
本文研究微分算子在两参数可交换演化下的传输，结合移动吸收边界导出相容性方程，证明经典热理论是其\\(s=0\\)切片的特殊情况，还给出二阶、三阶算子的具体结果。
AI中文摘要：
我们研究多项式微分算子在两参数可交换演化\\( P_{t,s} = \exp\left( \frac{t}{2}D^2-\frac{s}{3}D^3 \right) \\)（其中\\( D=\frac{d}{dx} \\)）下的传输问题。位置算子的共轭运算给出\\( P_{t,s}xP_{t,s}^{-1} = x+tD-sD^2 \\)，这一结果导出了递归的正规序展开，并产生了一类无导数系数对应的热-艾里多项式。本文的主要目的是研究该传输与移动吸收边界的相互作用。对于满足\\( v_t=\frac12v_{xx} \\)、\\( v_s=-\frac13v_{xxx} \\)、\\( v(t,s,f(t,s))=0 \\)的函数，边界条件在\\( v \\)的空间喷流间生成了一系列关系。结合传输微分方程及其空间导数的约束，这些恒等式为\\( f \\)导出了相容性方程。我们证明了一个通用限制原理：在每个空间喷流层级，经典热问题的相容性方程可通过将对应的热-艾里层级限制到\\( s=0 \\)的切片得到。对于二阶算子，该框架给出了有限特征系统、非线性边界方程及显式可解示例；对于三阶线性势算子，额外的艾里流恒等式允许更强的顺序恢复：第一个热-艾里相容性方程确定\\( f_s(t,0) \\)，下一个空间喷流方程则重现了纯热理论中由\\( 4\times4 \\)特征行列式得到的非线性相容性条件。因此，额外的可交换流将移动边界的相容性问题重组为层级结构，同时保留经典热理论作为特殊限制。
英文摘要：
We study the transport of polynomial differential operators under the two-parameter commuting evolution \[ P_{t,s} = \exp\left( \frac{t}{2}D^2-\frac{s}{3}D^3 \right), \qquad D=\frac{d}{dx}. \] Conjugation of the position operator gives \[ P_{t,s}xP_{t,s}^{-1} = x+tD-sD^2, \] which leads to a recursive normal-ordering expansion and to a family of Heat--Airy polynomials arising as its derivative-free coefficients. Our main purpose is to study the interaction of this transport with moving absorbing boundaries. For functions satisfying \[ v_t=\frac12v_{xx}, \qquad v_s=-\frac13v_{xxx}, \qquad v(t,s,f(t,s))=0, \] the boundary condition generates a hierarchy of relations among the spatial jets of \(v\). Combined with the restrictions of transported differential equations and their spatial derivatives, these identities produce compatibility equations for \(f\). We prove a general restriction principle showing that, at every spatial-jet level, the compatibility equations of the classical Heat problem are obtained by restricting the corresponding Heat--Airy hierarchy to the slice \(s=0\). For second-order operators, this framework yields a finite characteristic system and nonlinear boundary equations, together with explicit solvable examples. For a third-order linear-potential operator, the additional Airy-flow identity permits a stronger sequential recovery: the first Heat--Airy compatibility equation determines \(f_s(t,0)\), and the next spatial-jet equation then reproduces the nonlinear compatibility condition obtained in the pure Heat theory from a \(4\times4\) characteristic determinant. Thus the additional commuting flow reorganizes the moving-boundary compatibility problem into a hierarchy while retaining the classical Heat theory as a distinguished restriction.
