Huygens' Principle in Minkowski Spaces and Soliton Solutions of the Korteweg–de Vries Equation

@article{Berest1997HuygensPI,
title={Huygens' Principle in Minkowski Spaces and Soliton
Solutions of the Korteweg–de Vries Equation
},
author={Yuri Yu. Berest and Igor Loutsenko},
journal={Communications in Mathematical Physics},
year={1997},
volume={190},
pages={113-132}
}
• Published 18 April 1997
• Mathematics, Physics
• Communications in Mathematical Physics
Abstract:A new class of linear second order hyperbolic partial differential operators satisfying Huygens' principle in Minkowski spaces is presented. The construction reveals a direct connection between Huygens' principle and the theory of solitary wave solutions of the Korteweg–de Vries equation.
On Huygens’ Principle for Dirac Operators and Nonlinear Evolution Equations
Abstract We exhibit a class of Dirac operators that possess Huygens’ property, i.e., the support of their fundamental solutions is precisely the light cone. This class is obtained by considering theExpand
On Huygens' Principle for Dirac Operators and Nonlinear Evolution Equations
• Physics
• 2001
We exhibit a class of Dirac operators that possess Huygens’ property, i.e., the support of their fundamental solutions is precisely the light cone. This class is obtained by considering the rationalExpand
HUYGENS' PRINCIPLE AND SEPARATION OF VARIABLES
• Mathematics
• 2000
We demonstrate a close relation between the algebraic structure of the (local) group of conformal transformations on a smooth Lorentzian manifold and the existence of nontrivial hierarchies ofExpand
Hierarchies of Huygens' Operators and Hadamard's Conjecture
We develop a new unified approach to the problem of constructing linear hyperbolic partial differential operators that satisfy Huygens' principle in the sense of J. Hadamard. The underlying method isExpand
Iso-Huygens Deformations of Homogeneous Differential Operators Related to a Special Cone of Rank $${\text{3}}$$
AbstractWe consider iso-Huygens deformations of homogeneous hyperbolic Gindikin operators related to a special cone of rank $${\text{3}}$$ . The deformations are carried out with the use ofExpand
Heat Kernel Coefficients for Two-Dimensional Schrödinger Operators
• Mathematics, Physics
• 2007
In this note, we compute the Hadamard coefficients of algebraically integrable Schrödinger operators in two dimensions. These operators first appeared in [BL] and [B] in connection with Huygens’Expand
Multidimensional Baker–Akhiezer Functions and Huygens' Principle
• Mathematics
• 1999
Abstract:A notion of the rational Baker–Akhiezer (BA) function related to a configuration of hyperplanes in Cn is introduced. It is proved that the BA function exists only for very specialExpand
Equilibrium of charges and differential equations solved by polynomials
We study limits of particular importance of the bilinear hypergeometric equation introduced in [8]. As part of this study, we examine connections between the rationality of certain indefiniteExpand
Algebro-geometric Schrödinger operators in many dimensions
• O. Chalykh
• Medicine, Mathematics
• Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
• 2007
These algebro-geometric operators L=−d2/dx2+u(x) with rational, trigonometric and elliptic potential which appear in the finite-gap theory are generalized to higher dimension. Expand
Harmonic oscillator and Darboux transformations in many dimensions
• Physics
• 2000
We consider a class of multidimensional Schrodinger operators L sy D q ux which can be intertwined with the ¨ 22 . quantum harmonic oscillator L sy D q v x : LDs DL q l for some partial differentialExpand

References

SHOWING 1-10 OF 63 REFERENCES
Hierarchies of Huygens' Operators and Hadamard's Conjecture
We develop a new unified approach to the problem of constructing linear hyperbolic partial differential operators that satisfy Huygens' principle in the sense of J. Hadamard. The underlying method isExpand
Integrability in the theory of Schrödinger operator and harmonic analysis
• Mathematics
• 1993
The algebraic integrability for the Schrödinger equation in ℝn and the role of the quantum Calogero-Sutherland problem and root systems in this context are discussed. For the special values of theExpand
On a class of polynomials connected with the Korteweg-deVries equation
• Mathematics
• 1978
A new and simpler construction of the family of rational solutions of the Korteweg-deVries equation is given. This construction is related to a factorization of the Sturm-Liouville operators intoExpand
Fundamental solutions for partial differential equations with reflection group invariance
• Mathematics
• 1995
The Dunkl differential‐difference operator associated with a finite reflection group is used to extend the Weyl–Heisenberg algebra. The subalgebra of mastersymmetries graded by the characters of theExpand
Solitons, Nonlinear Evolution Equations and Inverse Scattering
• Physics, Mathematics
• 1991
1. Introduction 2. Inverse scattering for the Korteweg-de Vries equation 3. General inverse scattering in one dimension 4. Inverse scattering for integro-differential equations 5. Inverse scatteringExpand
Darboux Transformations and Solitons
• Mathematics
• 1992
In 1882 Darboux proposed a systematic algebraic approach to the solution of the linear Sturm-Liouville problem. In this book, the authors develop Darboux's idea to solve linear and nonlinear partialExpand
Lacunae of Hyperbolic Riesz Kernels and Commutative Rings of Partial Differential Operators
The purpose of this Letter is to demonstrate a close connection between the problem of describing supercomplete commutative rings of partial differential operators (in the sense of Krichever–Veselov)Expand
Algebraic integrability for the Schrödinger equation and finite reflection groups
• Mathematics
• 1993
Algebraic integrability of ann-dimensional Schrödinger equation means that it has more thann independent quantum integrals. Forn=1, the problem of describing such equations arose in the theory ofExpand
Huygens' principle and integrability
• Mathematics
• 1994
The physical notion of Huygens’Principle goes back to the classical “Traite de la Lumiere” by Christian Huygens, published in 1690. Various aspects of this fundamental principle in the theory of waveExpand
The Wave Equation on a Curved Space–Time
F G Friedlander London: Cambridge University Press 1976 pp ix + 282 price £15 The continuing interest in general relativity has created the need for a book on the rigorous mathematical theory of waveExpand