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 (PI) equation or its fourth-order analogue P. 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.
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机译：我们研究了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水动力型椭圆和双曲系统的摄动。我们认为，在无色散系统的梯度突变的临界点附近，对于Painlevé-I（PI）方程或其四阶模拟量P <数学xmlns：mml =“ http://www.w3.org/1998/Math/MathML” id =“ M4” overflow =“ scroll”> I 2 。作为具体示例，我们讨论半经典极限中的非线性Schrödinger方程。对这些案例的数值研究提供了有力的证据支持这一推测。
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