Generalized matrix exponential solutions to the AKNS hierarchy
type: Item
instance of: scholarly article
title: Generalized matrix exponential solutions to the AKNS hierarchy
zbMATH Open document ID: 1409.37072
DOI: 10.1155/2018/1375653
author: Y. Aharonov
published in: Advances in Mathematical Physics
publication date: 2018-12-11T00:00:00Z
review text: The authors consider the AKNS hierarchy and discuss its integrability in the framework of the modified inverse scattering transformation. The latter is essentially a consequence of the usual Gelfand-Levitan-Marchenko integral equation, and relies on a resulting pair of coupled Sylvester equations, which allow to write the explicit soliton solutions in matrix form. The key observation, that seems to simplify considerably the construction of the soliton solutions, is based on the freedom to choose a set of six matrices, subject to some week conditions, in order to produce the required pair of coupled Sylvester equations. This is clearly demonstrated by the authors by giving a number of explicit one- and multi-soliton solutions. The authors also generalize the AKNS hierarchy and their approach to integrability via a modified inverse scattering transformation. The paper is very straightforward, clearly written and is easy to follow. The authors also provide the complete details for their theorems and statements.
reviewed by: A. Melikyan
Mathematics Subject Classification ID: 37K15
Mathematics Subject Classification ID: 37K10
zbMATH DE Number: 6991921
zbMATH Keywords: integrable models
zbMATH Keywords: exact solutions
zbMATH Keywords: solitons
zbMATH Keywords: AKNS hierarchy
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2018/1375653
OpenAlex ID: W2790374740
Wikidata QID: Q115522020
cites work: Q3995225
cites work: Symmetries and Integrability
cites work: From the conservation laws to the Hamiltonian structures of discrete soliton systems
cites work: Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States
cites work: Method for Solving the Korteweg-deVries Equation
cites work: Q3925229
cites work: Integration of the soliton hierarchy with self-consistent sources
cites work: Exact solutions to a mixed Toda lattice hierarchy through the inverse scattering transform
cites work: Inverse scattering transform for the Toda hierarchy with steplike finite-gap backgrounds
cites work: Soliton solutions of the Toda hierarchy on quasi-periodic backgrounds revisited
cites work: Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation
cites work: Solutions to ABS Lattice Equations via Generalized Cauchy Matrix Approach
cites work: Soliton solutions for ABS lattice equations: I. Cauchy matrix approach
cites work: Q3719163
cites work: Direct Linearization of Extended Lattice BSQ Systems
cites work: Direct linearizing transform for three-dimensional discrete integrable systems: the lattice AKP, BKP and CKP equations
cites work: Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions
cites work: Generalized Casorati Determinant and Positon–Negaton-Type Solutions of the Toda Lattice Equation
cites work: New double Wronskian solutions of the AKNS equation
cites work: Solving the AKNS Hierarchy by Its Bilinear Form: Generalized Double Wronskian Solutions
cites work: Solutions of the nonlocal nonlinear Schrödinger hierarchy via reduction
cites work: Explicit solutions to the Korteweg–de Vries equation on the half line
cites work: High-energy analysis and Levinson's theorem for the selfadjoint matrix Schrödinger operator on the half line
cites work: Small-energy analysis for the self-adjoint matrix Schrödinger operator on the half line
cites work: Exact solutions to the sine-Gordon equation
cites work: Nonlinearization of the Lax system for AKNS hierarchy
description: scientific article; zbMATH DE number 6991921