Hierarchies and Hamiltonian structures of the Nonlinear Schrödinger family using geometric and spectral techniques

Partha Guha; Indranil Mukherjee

doi:10.3934/dcdsb.2018287

Article Contents

2019, Volume 24, Issue 4: 1677-1695. Doi: 10.3934/dcdsb.2018287

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Hierarchies and Hamiltonian structures of the Nonlinear Schrödinger family using geometric and spectral techniques

Partha Guha^1,2, and
Indranil Mukherjee^3,

1.
S.N. Bose National Centre for Basic Sciences, JD Block, Sector Ⅲ, Salt Lake, Kolkata - 700106, India
2.
Instituto de Física de São Carlos; IFSC/USP, Universidade de São Paulo Caixa Postal 369, CEP 13560-970, São Carlos-SP, Brazil
3.
School of Management and Sciences, Maulana Abul Kalam Azad University of Technology, West Bengal, BF 142, Sector I, Salt Lake, Kolkata-700064, India

Received: February 28, 2017

Revised: January 31, 2018

Early access: August 2018

Published: January 2019

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Abstract

This paper explores the class of equations of the Non-linear Schrödinger (NLS) type by employing both geometrical and spectral analysis methods. The work is developed in three stages. First, the geometrical method (AKS theorem) is used to derive different equations of the Non-linear Schrödinger (NLS) and Derivative Non-linear Schrödinger (DNLS) families. Second, the spectral technique (Tu method) is applied to obtain the hierarchies of equations belonging to these types. Third, the trace identity along with other techniques is used to obtain the corresponding Hamiltonian structures. It is found that the spectral method provides a simple algorithmic procedure to obtain the hierarchy as well as the Hamiltonian structure. Finally, the connection between the two formalisms is discussed and it is pointed out how application of these two techniques in unison can facilitate the understanding of integrable systems. In concurrence with Tu's method, Gesztesy and Holden also formulated a method of derivation of the trace formulas for integrable nonlinear evolution equations, this method is based on a contour-integration technique.

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Mathematics Subject Classification: 35Q55, 37K10, 37K30.

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