We show that universal rogue wave patterns exist in integrable systems. These rogue patterns comprise fundamental rogue waves arranged in shapes such as a triangle, pentagon and heptagon, with a possible lower-order rogue wave at the center. These patterns appear when one of the internal parameters in bilinear expressions of rogue waves gets large. Analytically, these patterns are determined by the root structures of the Yablonskii–Vorob’ev polynomial hierarchy through a linear transformation. Thus, the induced rogue patterns in the space–time plane are simply the root structures of Yablonskii–Vorob’ev hierarchy polynomials under actions such as dilation, rotation, stretch, shear and translation. Which level of the Yablonskii–Vorob’ev hierarchy is determined by which internal parameter is chosen to be large, and which polynomial at that level of the hierarchy is determined by the order of the underlying rogue wave. As examples, these universal rogue patterns are explicitly determined and graphically illustrated for the generalized derivative nonlinear Schrödinger equations, the Boussinesq equation, and the Manakov system. Similarities and differences between these rogue patterns and those reported earlier in the nonlinear Schrödinger equation are discussed.

我们表明可积系统中存在通用的流氓波浪模式。这些流氓模式包括以三角形、五边形和七边形等形状排列的基本流氓波，中心可能是低阶流氓波。当流氓波的双线性表达式中的一个内部参数变大时，就会出现这些模式。从分析上讲，这些模式是由 Yablonskii-Vorob'ev 多项式层次结构的根结构通过线性变换确定的。因此，时空平面中的诱导流氓模式只是 Yablonskii-Vorob'ev 层次多项式在膨胀、旋转、拉伸、剪切和平移等作用下的根结构。Yablonskii-Vorob'ev 层次结构的哪个级别取决于选择哪个内部参数较大，层次结构中那个级别的多项式由底层流氓波的顺序决定。例如，对于广义导数非线性薛定谔方程、Boussinesq 方程和 Manakov 系统，明确确定并以图形方式说明了这些通用流氓模式。讨论了这些流氓模式与之前在非线性薛定谔方程中报告的那些模式之间的异同。