发表机构
Innopolis University（因诺波利斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过构造二次表达式和快速衰减环，将线性化第四Painlevé方程的解表示为积分，并证明其线性无关性及推导变分积分公式。
AI 中文摘要
我们将第四Painlevé方程的线性化表述为一个线性化Hamilton系统，并将标量变分方程变换为正规形式，其中一阶导数项被消除。从Jimbo--Miwa Lax对的典型基本解出发，我们构造满足恒等式$\partial_x^2Q_{IV}-U_{IV}Q_{IV}=\partial_\lambda R_{IV}$的二次表达式$Q_{IV}$和$R_{IV}$。这里$U_{IV}$是正规形式方程中的系数。一个具有消失边界项的快速衰减环给出线性化方程的一个积分解。每个非平凡的此类环在至少两个不同的Stokes扇形中具有渐近分支；否则积分由Cauchy定理消失。第二个快速衰减同调类给出另一个积分解。两个积分解的Wronskian行列式对monodromy数据全纯依赖，因此单个非零值意味着一般的线性无关性。对于一组复数数据，直接求积给出了非零Wronskian行列式的数值证据。我们还推导了Stokes乘子、连接矩阵和局部monodromy的变分的积分公式，并通过代入线性化方程验证了围道表示。
英文摘要
We formulate the linearization of the fourth Painlevé equation as a linearized Hamiltonian system and transform the scalar variational equation to normal form, with the first-derivative term eliminated. From the canonical fundamental solutions of the Jimbo--Miwa Lax pair, we construct quadratic expressions $Q_{IV}$ and $R_{IV}$ that satisfy the identity $\partial_x^2Q_{IV}-U_{IV}Q_{IV}=\partial_λR_{IV}$. Here $U_{IV}$ is the coefficient in the normal-form equation. A rapid-decay cycle with a vanishing boundary term yields an integral solution of the linearized equation. Every nontrivial such cycle has asymptotic branches in at least two distinct Stokes sectors; otherwise the integral vanishes by Cauchy's theorem. A second rapid-decay homology class gives another integral solution. The Wronskian of the two integral solutions depends holomorphically on the monodromy data, so a single nonzero value implies generic linear independence. For one set of complex data, direct quadrature gives numerical evidence of a nonzero Wronskian. We also derive integral formulas for variations of the Stokes multipliers, the connection matrix, and the local monodromy, and verify the contour representation by substitution into the linearized equation.
Comments43pages, 6 figures
