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On Critical Behaviour in Systems of Hamiltonian Partial Differential Equations
Boris Dubrovin1,2,3, Tamara Grava1,4, Christian Klein5
1SISSA, Via Bonomea 265, 34136 Trieste, Italy.
Journal of Nonlinear Science
|April 15, 2017
Summary
Solutions to perturbed Hamiltonian systems near critical points are approximated by Painlevé equations. Numerical studies support this finding for nonlinear Schrödinger equations.
Area of Science:
- Mathematical physics
- Nonlinear dynamics
- Fluid mechanics
Background:
- Hamiltonian systems exhibit complex behavior, especially when weakly dispersive.
- Elliptic and hyperbolic systems of hydrodynamic type with two components are foundational in fluid dynamics.
- Gradient catastrophe signifies a critical point where solutions may break down.
Purpose of the Study:
- To investigate the critical behavior of solutions for weakly dispersive Hamiltonian systems.
- To establish an approximation for these solutions near the gradient catastrophe point.
- To connect the behavior to specific Painlevé equations.
Main Methods:
- Perturbation theory applied to Hamiltonian systems.
- Analysis of initial value problems for the perturbed equations.
- Numerical simulations of nonlinear Schrödinger equations in the semiclassical limit.
Main Results:
- Solutions near the critical point are approximated by specific solutions of the Painlevé-I (P I) and Painlevé-IV (P IV) equations.
- Nonlinear Schrödinger equations in the semiclassical limit serve as concrete examples.
- Numerical results strongly support the theoretical conjecture.
Conclusions:
- The Painlevé equations provide accurate approximations for critical phenomena in these systems.
- This work bridges the gap between dispersionless limits and dispersive effects.
- The findings have implications for understanding wave phenomena and singularities in physical systems.