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

Partha Guha, Indranil Mukherjee

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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.

Original languageBritish English
Pages (from-to)1677-1695
Number of pages19
JournalDiscrete and Continuous Dynamical Systems - Series B
Volume24
Issue number4
DOIs
StatePublished - Apr 2019

Keywords

  • Adler-Kostant-Symes theorem
  • Bihamiltonian system
  • Gesztesy-Holden methods
  • Loop algebra
  • Nonlinear Schrödinger equation
  • Trace identity
  • Tu methodology

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