AI 中文总结
研究二维费米-帕斯塔-乌拉姆晶格中的局域模式和色散结构，通过降维推导修正KdV方程，利用其精确解建模，探索黎曼问题二维推广及KPII极限，比较数值动力学检验准连续长波渐近极限性能。
AI 中文摘要
本文研究了标量二维费米-帕斯塔-乌拉姆（FPU）晶格的类似物。在FPU晶格的数值模拟中，通过数值识别出各种色散波结构和局域模式，但据我们所知，所有这些特定波结构都没有解析封闭形式的表达式。为解决此问题，我们进行降维并推导修正的KdV方程。基于此，利用其精确局域解对FPU晶格中的相关波模式建模，探索FPU晶格黎曼问题的二维推广及相应修正KdV约化，提出并严格推导FPU晶格的KPII极限并研究相关楔形问题，最后比较所有这些数值动力学以检验这些准连续长波渐近极限的性能。
英文摘要
In this paper, we study an analog of the scalar two-dimensional Fermi-Pasta-Ulam (FPU) lattice. In particular, a variety of dispersive wave structures and localized patterns are numerically identified in the numerical simulations of the FPU lattice, but, to the best of our knowledge, all of these particular wave structures do not admit analytical closed-form expressions. In order to resolve this issue, we perform a dimensional reduction and accordingly derive a modified KdV equation. Based on this reduction, we first take advantage of some of its exact localized solutions to model the associated wave patterns in the FPU lattice. In addition, we explore the two-dimensional generalization of the Riemann problems for the FPU lattice and the corresponding modified KdV reduction, whose evolution dynamics lead to the formation of multiple composite dispersive structures. Moreover, we propose and rigorously derive the KPII limit of the FPU lattice and investigate their associated wedge problems. Finally, all these relevant numerical dynamics are compared to examine the performance of these quasi-continuum long-wave asymptotic limits.
Comments10 pages, 10 figures
