An algorithm to generate integrable systems is extended to the super case. Some new examples of superextensions of integrable systems are illustrated. We also generalize the trace identity due to Tu to the super case and use it to establish Hamiltonian structures of superextensions of integrable systems under consideration.

https://iopscience.iop.org/article/10.1088/0305-4470/30/2/023/pdf

An approach to generate superextensions of integrable systems

Nonlocal Fordy-Kulish equations on symmetric spaces

We present some nonlocal integrable systems by using the Ablowitz–Musslimani nonlocal reductions. We first present all possible nonlocal reductions of nonlinear Schrodinger (NLS) and modified Korteweg–de Vries (mKdV) systems. We give soliton solutions of these nonlocal equations by using the Hirota method. We extend the nonlocal NLS equation to nonlocal Fordy–Kulish equations by utilizing the nonlocal reduction to the Fordy–Kulish system on symmetric spaces. We also consider the super AKNS system and then show that Ablowitz–Musslimani nonlocal reduction can be extended to super integrable equations. We obtain new nonlocal equations namely nonlocal super NLS and nonlocal super mKdV equations.

Integrable Nonlocal Reductions

By using AKNS scheme and soliton connection taking values in N  = 1 superconformal algebra we obtain new coupled super nonlinear Schrodinger equations.

Nonlinear Schrödinger equations and N = 1 superconformal algebra

The finite-dimensional super integrable system of a super NLS-mKdV equation

The work of A.K.N.S. which is based on the sl(2, R) valued soliton connection is extended to obtain new integrable coupled nonlinear partial differential equations. This is achieved by assuming the soliton connection having values in a simple Lie, Kac-Moody, Lie superalgebras. Extensions of some of the integrable nonlinear partial differential equations are given explicitly. In particular the coupled NLS equations on various homogeneous spaces and the coupled modified KdV integro-differential equations are obtained on symmetric spaces.

Ö. Oǧuz

Papers

Nonlinear partial differential equations on homogeneous spaces