Mathematical problems of nonlinearity

A Study of Nonholonomic Deformations of Nonlocal Integrable Systems Belonging to the Nonlinear Schrödinger Family

I. Mukherjee^a, P. Guha<sup>b

a</sup> School of Natural and Applied Sciences, Maulana Abul Kalam Azad University of Technology, BF 142, BF Block, Sector 1, Bidhannagar, Kolkata, West Bengal, 700064 India
^b S.N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata, 700098 India

Abstract: The nonholonomic deformations of nonlocal integrable systems belonging to the nonlinear Schrödinger family are studied using the bi-Hamiltonian formalism as well as the Lax pair method. The nonlocal equations are first obtained by symmetry reductions of the variables in the corresponding local systems. The bi-Hamiltonian structures of these equations are explicitly derived. The bi-Hamiltonian structures are used to obtain the nonholonomic deformation following the Kupershmidt ansatz. Further, the same deformation is studied using the Lax pair approach and several properties of the deformation are discussed. The process is carried out for coupled nonlocal nonlinear Schrödinger and derivative nonlinear Schrödinger (Kaup Newell) equations. In the case of the former, an exact equivalence between the deformations obtained through the bi-Hamiltonian and Lax pair formalisms is indicated.

Keywords: nonlocal integrable systems, nonlinear Schröodinger equation, Kaup – Newell equation, bi-Hamiltonian system, Lax method, nonholonomic deformation.

MSC: 35Q55, 37K10, 37J60, 70F25, 32Gxx

Received: 01.05.2019
Accepted: 03.09.2019

DOI: 10.20537/nd190308

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