Hydrodynamic type systems in Riemann invariants arise in a whole range of applications in fluid dynamics, Whitham averaging procedure, differential geometry and the theory of Frobenius manifolds. In this paper we discuss parabolic (Jordan block) analogues of diagonalisable systems. Our main observation is that integrable quasilinear systems of Jordan block type are parametrised by solutions of the modified Kadomtsev–Petviashvili hierarchy. Such systems appear naturally as degenerations of quasilinear systems associated with multi-dimensional hypergeometric functions, in the context of parabolic regularisation of the Riemann equation, as finite-component reductions of hydrodynamic chains, and as hydrodynamic reductions of linearly degenerate dispersionless integrable PDEs in multi-dimensions.

中文翻译：

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Jordan块类型和mKP层次的拟线性系统

黎曼不变量的流体力学类型系统在流体动力学，惠特姆平均程序，微分几何和Frobenius流形理论中的各种应用中都有出现。在本文中，我们讨论对角化系统的抛物线型（乔丹块）类似物。我们的主要观察结果是，约旦块类型的可积拟线性系统由修改后的Kadomtsev–Petviashvili层次结构的解参数化。这样的系统自然而然地表现为与多维超几何函数相关的准线性系统的退化，在黎曼方程的抛物线正则化情况下，流体动力学链的有限分量减少，以及线性退化的无色可分散PDE的流体动力学减少。尺寸。