Whitham's Method and Dubrovin-Novikov Bracket in Single-Phase and Multiphase Cases

@article{Maltsev2012WhithamsMA,
  title={Whitham's Method and Dubrovin-Novikov Bracket in Single-Phase and Multiphase Cases},
  author={Andrei Ya. Maltsev},
  journal={Symmetry Integrability and Geometry-methods and Applications},
  year={2012},
  volume={8},
  pages={103}
}
  • A. Maltsev
  • Published 26 March 2012
  • Physics
  • Symmetry Integrability and Geometry-methods and Applications
In this paper we examine in detail the procedure of averaging of the local field- theoretic Poisson brackets proposed by B.A. Dubrovin and S.P. Novikov for the method of Whitham. The main attention is paid to the questions of justification and the conditions of applicability of the Dubrovin{Novikov procedure. Separate consideration is given to special features of single-phase and multiphase cases. In particular, one of the main results is the insensitivity of the procedure of bracket averaging… 
The multi-dimensional Hamiltonian structures in the Whitham method
We consider the averaging of local field-theoretic Poisson brackets in the multi-dimensional case. As a result, we construct a local Poisson bracket for the regular Whitham system in the
The averaging of multi-dimensional Poisson brackets for systems having pseudo-phases
We consider features of the Hamiltonian formulation of the Whitham method in the presence of pseudo-phases. As we show, an analog of the procedure of averaging of the Poisson bracket with the reduced
On the canonical forms of the multi-dimensional averaged Poisson brackets
We consider here special Poisson brackets given by the "averaging" of local multi-dimensional Poisson brackets in the Whitham method. For the brackets of this kind it is natural to ask about their
On the minimal set of conservation laws and the Hamiltonian structure of the Whitham equations
We consider the questions connected with the Hamiltonian properties of the Whitham equations in case of several spatial dimensions. An essential point of our approach here is a connection of the
Poisson Brackets of Hydrodynamic Type and Their Generalizations
Abstract In this paper, we consider Hamiltonian structures of hydrodynamic type and some of their generalizations. In particular, we discuss the questions concerning the structure and special forms

References

SHOWING 1-10 OF 69 REFERENCES
The conservation of the Hamiltonian structures in Whitham's method of averaging
The work is devoted to the proof of the conservation of local field-theoretical Hamiltonian structures in Whitham's method of averaging. The consideration is based on the procedure of averaging of
THE AVERAGING OF NONLOCAL HAMILTONIAN STRUCTURES IN WHITHAM'S METHOD
We consider the m-phase Whitham’s averaging method and propose the procedure of “averaging” nonlocal Hamiltonian structures. The procedure is based on the existence of a sufficient number of
PERIODIC AND CONDITIONALLY PERIODIC ANALOGS OF THE MANY-SOLITON SOLUTIONS OF THE KORTEWEG-DE VRIES EQUATION
A method of connecting the Korteweg-de Vries (KdV) equation, known from the theory of nonlinear waves, with the Schrodinger equation was discovered in 1967. (1) This method is applied in the present
NON-LINEAR EQUATIONS OF KORTEWEG-DE VRIES TYPE, FINITE-ZONE LINEAR OPERATORS, AND ABELIAN VARIETIES
The basic content of this survey is an exposition of a recently developed method of constructing a broad class of periodic and almost-periodic solutions of non-linear equations of mathematical
Whitham systems and deformations
We consider the deformations of Whitham systems including the “dispersion terms” and having the form of Dubrovin-Zhang deformations of Frobenius manifolds. The procedure is connected with the B. A.
Remark on the phase shift in the Kuzmak-Whitham ansatz
AbstractWe consider one-phase (formal) asymptotic solutions in the Kuzmak-Whitham form for the nonlinear Klein-Gordon equation and for the Korteweg-de Vries equation. In this case, the leading
Multiphase averaging and the inverse spectral solution of the Korteweg—de Vries equation
Inverse spectral theory is used to prescribe and study equations for the slow modulations of N-phase wave trains for the Korteweg-de Vries (KdV) equation. An invariant representation of the
Finite-zone, almost-periodic solutions in WKB approximations
It is shown that the recently discovered finite-zone, almost-periodic solutions may, on the one hand, serve as the foundation for the development of the multiphase WKB method in nonlinear equations
A PERIODICITY PROBLEM FOR THE KORTEWEG–DE VRIES AND STURM–LIOUVILLE EQUATIONS. THEIR CONNECTION WITH ALGEBRAIC GEOMETRY
1. As was shown in the remarkable communication [4] the Cauchy problem for the Korteweg–de Vries (KdV) equation ut = 6uux−uxxx, familiar in theory of nonlinear waves, is closely linked with a study
...
1
2
3
4
5
...