移位非局部NLS和MKdV方程精确解的环型几何折叠
Loop-type geometric folding of exact solutions of shifted nonlocal NLS and MKdV equations
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中文总结 AI 辅助
该研究基于折叠子概念，引入几何框架构造移位非局部NLS和MKdV方程精确解的折叠参数波表示，采用简化几何方法，通过空间坐标非单调参数化及折叠映射导数符号变化条件，分析参数影响，生成多种折叠轮廓。
中文摘要 AI 辅助
基于折叠子的概念，我们引入了一个几何框架，用于构造一些移位非局部非线性薛定谔方程和修正科特韦格 - 德弗里斯方程精确解的折叠参数波表示。与基于通用变量分离方法或速端曲线变换在(2 + 1)维可积模型中构造环的方法不同，我们考虑一种简化的几何方法，通过对与(1 + 1)维移位非局部方程精确解相关的空间坐标进行非单调参数化来构造环型折叠轮廓。以折叠映射导数的符号变化形式给出了发生折叠的充分条件。应用先前找到的各种移位非局部非线性薛定谔方程和修正科特韦格 - 德弗里斯方程的单孤子和双孤子解，我们展示了不同的折叠映射如何生成不同的环型折叠轮廓。特别地，我们分析了变形参数和解参数对折叠波几何形状的影响。我们表明，折叠的效果仅导致空间参数化的修改，并为某些参数值生成各种几何结构，如规则环型、振荡型和奇异型折叠轮廓。
英文摘要
Based on the notion of foldon, we introduce a geometric framework for constructing folded parametric wave representations of exact solutions of some shifted nonlocal nonlinear Schrödinger and modified Korteweg-de Vries equations. Unlike the method of constructing loops in $(2+1)$-dimensional integrable models based on universal variable separation approach or hodograph transformation, we consider a simplified geometric approach of constructing loop-type folded profiles via non-monotonic parametrization of the spatial coordinate associated with the exact solution of the $(1+1)$-dimensional shifted nonlocal equations. A sufficient condition under which folding takes place is provided in the form of sign change of the derivative of folding map. Applying one- and two-soliton solutions of various shifted nonlocal nonlinear Schrödinger and modified Korteweg-de Vries equations found earlier, we show how different folding maps generate different loop-type folded profiles. In particular, we analyze the influence of deformation parameters and solution parameters on the geometry of folded waves. We show that the effect of the folding leads only to the modification of the spatial parametrization and generates various geometric structures like regular loop-type, oscillating-type, and singular-type folded profiles for certain values of parameters.
