We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlevé-I (P\(_I\)) equation or its fourth-order analogue P\(_I^2\). As concrete examples, we discuss nonlinear Schrödinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.

中文翻译：

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关于哈密顿偏微分方程系统中的临界行为。

我们研究弱色散哈密顿系统解的临界行为，该系统被认为是具有两个分量的流体动力学类型的椭圆和双曲线系统的扰动。我们认为，在无色散系统的梯度突变临界点附近，扰动方程的合适初值问题的解近似地由 Painlevé-I (P \(_I\) ) 方程或其第四个方程的特定解来描述-阶模拟 P \(_I^2\)。作为具体例子，我们讨论了半经典极限下的非线性薛定谔方程。对这些案例的数值研究为支持这一猜想提供了强有力的证据。