A Deformation of the Method of Characteristics and the Cauchy Problem for Hamiltonian PDEs in the Small Dispersion Limit

@article{Masoero2012ADO,
title={A Deformation of the Method of Characteristics and the Cauchy Problem for Hamiltonian PDEs in the Small Dispersion Limit},
author={Davide Masoero and Andrea Raimondo},
journal={International Mathematics Research Notices},
year={2012},
volume={2015},
pages={1200-1238}
}
• Published 12 November 2012
• Mathematics
• International Mathematics Research Notices
We introduce a deformation of the method of characteristics valid for Hamiltonian perturbations of a scalar conservation law in the small dispersion limit. Our method of analysis is based on the 'variational string equation', a functional-differential relation originally introduced by Dubrovin in a particular case, of which we lay the mathematical foundation. Starting from first principles, we construct the string equation explicitly up to the fourth order in perturbation theory, and we show…
4 Citations
• Mathematics
• 2013
We propose an extension of the Dubrovin–Zhang perturbative approach to the study of normal forms for non-Hamiltonian integrable scalar conservation laws. The explicit computation of the first few
• Mathematics, Physics
Letters in Mathematical Physics
• 2013
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,
We give an overview of a program of Stochastic Deformation of Classical Mechanics and the Calculus of Variations, strongly inspired by the quantization method.

References

SHOWING 1-10 OF 31 REFERENCES

• Mathematics
• 2008
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
• Mathematics
• 2013
We propose an extension of the Dubrovin–Zhang perturbative approach to the study of normal forms for non-Hamiltonian integrable scalar conservation laws. The explicit computation of the first few
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
We begin with presentation of classification results in the theory of Hamiltonian PDEs with one spatial dimension depending on a small parameter. Special attention is paid to the deformation theory
• Mathematics
• 2008
AbstractWe obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation $$u_t+6uu_x+\epsilon^{2}u_{xxx}=0,\quad u(x,t=0,\epsilon)=u_0(x),$$for
• Mathematics
• 1982
In Part I the scattering transform method is used to study the weak limit of solutions to the initial value problem for the Korteweg-deVries (KdV) equation as the dispersion tends to zero. In that
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
The string equations of hermitian and unitary matrix models of 2D gravity are flatness conditions. These flatness conditions may be interpreted as the consistency conditions for isomonodromic
In this paper we construct a large family of special solutions of the KdV equation which are periodic in x and almost periodic in t. These solutions lie on N-dimensional tori; very likely they are
• Mathematics, Physics
• 2011
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,