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Pure mathematics, Applied mathematics
A certain class of integrable nonlinear differential equations related to A.III-type symmetric spaces and having additional reductions is analysed via the Inverse Scattering Method (ISM). The class contains systems of nonlinear evolution equations (NLEEs) associated with a Lax operator (for the time-evolution) polynomial in the spectral parameter. Using the Wronskian relations, the mapping between the potential and the minimal sets of scattering data is constructed. Furthermore, completeness relations for the `squared solutions' (generalised exponentials) are derived. Next, expansions of Q and its variation are obtained. This demonstrates that the interpretation of the inverse scattering method as a generalized Fourier transform holds true. Next, by using the (generating) recursion operator (derived here via Gurses-Karasu-Sokolov method and via "squared solutions") all fundamental properties of the class of NLEE are described: the class of NLEE related to a given Lax operator, the infinite set of integrals of motion (conserved quantities). Finally, the Hamiltonian structures of these generalised multi-component Heisenberg Ferromagnetic (MHF) type integrable models on A.III-type symmetric spaces are briefly analysed.
Gerdjikov, Vladimir; Grahovski, Georgi; Mikhailov, Alexander V.; and Valchev, Tihomir, "Polynomial Bundles and Generalised Fourier Transforms for Integrable Equations on A.III-type Symmetric Spaces" (2011). Articles. Paper 116.