On Critical Behaviour in Systems of Hamiltonian Partial Differential Equations

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• 作者：Boris Dubrovin ; Tamara Grava ; Christian Klein ; Antonio Moro
• 关键词：Hamiltonian PDEs ; Hyperbolic and Elliptic systems ; Gradient catastrophe and elliptic umbilic catastrophe ; Quasi ; integrable systems ; Painlevé equations ; 35Q55 ; 37K05 ; 34M55
• 刊名：Journal of Nonlinear Science
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：25
• 期：3
• 页码：631-707
• 全文大小：4,728 KB
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• 作者单位：Boris Dubrovin (1) (2) (3)
  Tamara Grava (1) (4)
  Christian Klein (5)
  Antonio Moro (6)
  
  1. SISSA, Via Bonomea 265, 34136, Trieste, Italy
  2. Steklov Mathematics Institute, Moscow, Russia
  3. N. N. Bogolyubov Laboratory of Geometric Methods in Mathematical Physics, Moscow State University, 119899, Moscow, Russia
  4. School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK
  5. Institut de Mathématiques de Bourgogne, Université de Bourgogne, 9 Avenue Alain Savary, 21078, Dijon Cedex, France
  6. Department of Mathematics and Information Sciences, University of Northumbria at Newcastle upon Tyne, Pandon Building, Camden Street, Newcastle upon Tyne, NE2 1XE, UK
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Mathematical and Computational Physics
  Mechanics
  Applied Mathematics and Computational Methods of Engineering
  Economic Theory
  
• 出版者：Springer New York
• ISSN：1432-1467

文摘

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 Schr?dinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.