On Critical Behaviour in Systems of Hamiltonian Partial Differential Equations

  title={On Critical Behaviour in Systems of Hamiltonian Partial Differential Equations},
  author={Boris Dubrovin and Tamara Grava and Christian Klein and Antonio Moro},
  journal={Journal of Nonlinear Science},
  pages={631 - 707}
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$$_I$$I) equation or its fourth-order analogue P$$_I^2$$I2. As… 

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    Proceedings of the Steklov Institute of Mathematics
  • 2018
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  • 2005
We present a way to deal with dispersion-dominated "shock-type" transition in the absence of completely integrable structure for the systems that one may characterize as strictly hyperbolic