Algebraic and Analytic Aspects of Soliton Type Equations

  title={Algebraic and Analytic Aspects of Soliton Type Equations},
  author={Vladimir S Gerdjikov},
  journal={arXiv: Exactly Solvable and Integrable Systems},
  • V. Gerdjikov
  • Published 11 June 2002
  • Mathematics
  • arXiv: Exactly Solvable and Integrable Systems
This is a review of two of the fundamental tools for analysis of soliton equations: i) the algebraic ones based on Kac-Moody algebras, their central extensions and their dual algebras which underlie the Hamiltonian structures of the NLEE; ii) the construction of the fundamental analytic solutions of the Lax operator and the Riemann-Hilbert problem (RHP) which they satisfy. The fact that the inverse scattering problem for the Lax operator can be viewed as a RHP gave rise to the dressing Zakharov… 

Figures from this paper

Basic Aspects of Soliton Theory
This is a review of the main ideas of the inverse scattering method (ISM) for solving nonlinear evolution equations (NLEE), known as soliton equations. As a basic tool we use the fundamental analytic
, We consider A-wave type equations related to symplectic and or­ thogonal algebras. We obtain their soliton solutions in the case when two different Z2 reductions (or equivalently one Z2 v Z2
Nonlocal Reductions of The Multicomponent Nonlinear Schrödinger Equation on Symmetric Spaces
Our aim is to develop the inverse scattering transform for multicomponent generalizations of nonlocal reductions of the nonlinear Schrödinger (NLS) equation with $$\mathcal{PT}$$PT symmetry related
On the stability of N-solitons in integrable systems
The dynamical stability of reflectionless N-solitons for a large class of integrable systems is considered. The underlying eigenvalue problem is the Zakharov–Shabat problem on for any r ≥ 1. Physical
Two-dimensional Toda field equations related to the exceptional algebra $\mathfrak{g}_2 $: Spectral properties of the lax operators
We analyze spectral properties of the Lax operator corresponding to the two-dimensional Toda field equations related to the algebra $\mathfrak{g}_2 $. We construct two minimal sets of scattering data
The Multicomponent Higher-Order Chen–Lee–Liu System: The Riemann–Hilbert Problem and Its N-Soliton Solution
It is well known that multicomponent integrable systems provide a method for analyzing phenomena with numerous interactions, due to the interactions between their different components. In this paper,
Algebro-geometric Solutions for the Degasperis-Procesi Hierarchy
A third order algebraic curve is introduced with characteristic polynomial of a Lax matrix for the DP hierarchy, from which the associated Baker--Akhiezer functions, meromorphic function and Dubrovin-type equations are established.


Generalised Fourier transforms for the soliton equations. Gauge-covariant formulation
The direct and the inverse scattering problems for the generalised Zakharov-Shabat spectral problem L related to a given semi-simple Lie algebra are solved. The fundamental analytic solutions and the
N-wave interactions related to simple Lie algebras. ℤ2-reductions and soliton solutions
The reductions of the integrable N-wave type equations solvable by the inverse scattering method with the generalized Zakharov-Shabat systems L and related to some simple Lie algebra are analysed.
The generating operator for the n × n linear system
Hamiltonian methods in the theory of solitons
The Nonlinear Schrodinger Equation (NS Model).- Zero Curvature Representation.- The Riemann Problem.- The Hamiltonian Formulation.- General Theory of Integrable Evolution Equations.- Basic Examples
Soliton Equations and Hamiltonian Systems
Integrable Systems Generated by Linear Differential nth Order Operators Hamiltonian Structures Hamiltonian Structure of the GD Hierarchies Modified KdV and GD. The Kupershmidt-Wilson Theorem The KP
Expansions over the 'squared' solutions and the inhomogeneous nonlinear Schrodinger equation
The inhomogeneous nonlinear Schrodinger equation (INLSE) with vanishing boundary conditions is studied using the expansions over the 'squared' solutions of the Zakharov-Shabat system L. The authors
Solitons and Infinite Dimensional Lie Algebras
Introduction §1. Fock Representation of gf(°°) §2. T Functions and the KP Hierarchy §3. Reduction to A[" §4. Fermions with 2 Components §5. Algebras B^ and Co §6. Spin Representation of J&TO §7.
Lie algebras and equations of Korteweg-de Vries type
The survey contains a description of the connection between the infinite-dimensional Lie algebras of Kats-Moody and systems of differential equations generalizing the Korteweg-de Vries and