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关于哈密顿偏微分方程系统中的临界行为

On Critical Behaviour in Systems of Hamiltonian Partial Differential Equations.

作者信息

Dubrovin Boris, Grava Tamara, Klein Christian, Moro Antonio

机构信息

SISSA, Via Bonomea 265, 34136 Trieste, Italy.

Steklov Mathematics Institute, Moscow, Russia.

出版信息

J Nonlinear Sci. 2015;25(3):631-707. doi: 10.1007/s00332-015-9236-y. Epub 2015 Feb 11.

DOI:10.1007/s00332-015-9236-y
PMID:28408786
原文链接:https://pmc.ncbi.nlm.nih.gov/articles/PMC5367859/
Abstract

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[Formula: see text]) equation or its fourth-order analogue P[Formula: see text]. 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.

摘要

我们研究了弱色散哈密顿系统解的临界行为，该系统被视为具有两个分量的椭圆型和双曲型流体动力学系统的扰动。我们认为，在无弥散系统梯度灾难的临界点附近，扰动方程合适初值问题的解可以由Painlevé-I（P[公式：见正文]）方程或其四阶类似物P[公式：见正文]的特解近似描述。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。对这些情况的数值研究为该猜想提供了有力证据。

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3
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