与广义Hermite和Okamoto多项式相关的向量怪波模式

中文摘要 AI 辅助

我们在多分量非线性薛定谔方程和Hirota方程中建立了与第四Painlevé方程（$\mathrm{P}_{\mathrm{IV}}$）相关的怪波模式。广义Hermite多项式和广义Okamoto多项式出现在$\mathrm{P}_{\mathrm{IV}}$的有理解表示中，我们证明当怪波解的一个内部参数较大时，它们的根决定了两类怪波模式。具体而言，广义Hermite多项式源于由具有连续指标的Schur多项式行列式表示的怪波解，而广义Okamoto多项式则源于指标跳跃为三的类似行列式。两个方程的数值例子与预测一致。

英文摘要

We establish rogue wave patterns associated with the fourth Painlevé equation $(\mathrm{P}_{\mathrm{IV}})$ in the multi-component nonlinear Schrödinger and Hirota equations. The generalized Hermite and generalized Okamoto polynomials arise in representations of rational solutions of $\mathrm{P}_{\mathrm{IV}}$, and we show that their roots determine two classes of rogue wave patterns when one of the internal parameters of rogue wave solutions is large. Specifically, the generalized Hermite polynomials arise from rogue wave solutions represented by Schur-polynomial determinants with consecutive indices, whereas the generalized Okamoto polynomials arise from analogous determinants with index jumps of three. Numerical examples for both equations agree with the predictions.

发表机构

• Institute for Advanced Study, Shenzhen University（深圳大学高等研究院）
• School of Mathematical Sciences, Shenzhen University（深圳大学数学科学学院）
• Department of Mathematics, City University of Hong Kong（香港城市大学数学系）

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