On Critical Behaviour in Systems of Hamiltonian Partial Differential Equations

@article{Dubrovin2013OnCB,
  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},
  year={2013},
  volume={25},
  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… 

Numerical study of fractional nonlinear Schrödinger equations

Using a Fourier spectral method, a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case is provided to study the possibility of finite time blow-up versus global existence, the nature of the blow- up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions.

Two-layer interfacial flows beyond the Boussinesq approximation: a Hamiltonian approach

The theory of integrable systems of Hamiltonian PDEs and their near-integrable deformations is used to study evolution equations resulting from vertical-averages of the Euler system for two-layer

Numerical study of the long wavelength limit of the Toda lattice

We present the first detailed numerical study of the Toda equations in 2   +   1 dimensions in the limit of long wavelengths, both for the hyperbolic and elliptic case. We first study the continuum

Two-Parameter Asymptotics in a Bisingular Cauchy Problem for a Parabolic Equation

  • S. Zakharov
  • Mathematics
    Proceedings of the Steklov Institute of Mathematics
  • 2018
The Cauchy problem for a quasilinear parabolic equation with a small parameter ε at the highest derivative is considered. The initial function, which has the form of a smoothed step, depends on a

Thermodynamic limit and dispersive regularization in matrix models.

This analysis explains the origin and mechanism leading to the emergence of chaotic behaviors observed in M^{6} matrix models and extends its validity to even nonlinearity of arbitrary order.

On the plane into plane mappings of hydrodynamic type. Parabolic case

Singularities of plane into plane mappings described by parabolic two-component systems of quasi-linear partial differential equations of the first order are studied. Impediments arising in the

Numerical study of the semiclassical limit of the Davey–Stewartson II equations

We present the first detailed numerical study of the semiclassical limit of the Davey–Stewartson II equations both for the focusing and the defocusing variant. We concentrate on rapidly decreasing

Universality Near the Gradient Catastrophe Point in the Semiclassical Sine‐Gordon Equation

We study the semiclassical limit of the sine‐Gordon (sG) equation with below threshold pure impulse initial data of Klaus‐Shaw type. The Whitham averaged approximation of this system exhibits a

References

SHOWING 1-10 OF 158 REFERENCES

On universality of critical behaviour in Hamiltonian PDEs

Our main goal is the comparative study of singularities of solutions to the systems of first order quasilinear PDEs and their perturbations containing higher derivatives. The study is focused on the

On Hamiltonian Perturbations of Hyperbolic Systems of Conservation Laws, II: Universality of Critical Behaviour

Hamiltonian perturbations of the simplest hyperbolic equation ut + a(u) ux = 0 are studied. We argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe

On the critical behavior in nonlinear evolutionary PDEs with small viscosity

The problem of general dissipative regularization of the quasilinear transport equation is studied. We argue that the local behavior of solutions to the regularized equation near the point of

On Properties of Hamiltonian Structures for a Class of Evolutionary PDEs

In a recent paper we proved that for certain class of perturbations of the hyperbolic equation ut = f (u)ux, there exist changes of coordinate, called quasi-Miura transformations, that reduce the

Numerical Study of Breakup in Generalized Korteweg-de Vries and Kawahara Equations

It is shown numerically for a large class of equations that the local behaviour of their solution near the point of gradient catastrophe for the transport equation is described locally by a special solution of a Painlev\'e-type equation.

Obstacles to Asymptotic Integrability

We study nonintegrable effects appearing in the higher order corrections of an asymptotic perturbation expansion for a given nonlinear wave equation, and show that the analysis of the higher order

On a class of nonlinear Schrödinger equations

AbstractThis paper concerns the existence of standing wave solutions of nonlinear Schrödinger equations. Making a standing wave ansatz reduces the problem to that of studying the semilinear elliptic

Semiclassical Limit for Generalized KdV Equations Before the Gradient Catastrophe

We study the semiclassical limit of the (generalized) KdV equation, for initial data with Sobolev regularity, before the time of the gradient catastrophe of the limit conservation law. In particular,

Resolution of a shock in hyperbolic systems modified by weak dispersion.

  • G. El
  • Mathematics
    Chaos
  • 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
...