On soliton equations with \(\mathbb{Z}_h\) and \(\mathbb{D}_h\) reductions: conservation laws and generating operators. (English) Zbl 1293.35268

Gerdjikov, Vladimir S.; Yanovski, Alexandar B.

J. Geom. Symmetry Phys. 31, 57-92 (2013).

Summary: The Lax representations for the soliton equations with \(\mathbb Z_h\) and \(\mathbb D_h\) reductions are analyzed. Their recursion operators are shown to possess factorization properties due to the grading in the relevant Lie algebra. We show that with each simple Lie algebra one can relate \(r\) fundamental recursion operators \(\Lambda_{m_k}\) and a master recursion operator \(\Lambda\) generating NLEEs of MKdV type and their Hamiltonian hierarchies. The Wronskian relations are formulated and shown to provide the tools to understand the inverse scattering method as a generalized Fourier transform. They are also used to analyze the conservation laws of the above mentioned soliton equations.

MSC:

35Q51	Soliton equations
37K15	Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
17B80	Applications of Lie algebras and superalgebras to integrable systems
35Q15	Riemann-Hilbert problems in context of PDEs

Keywords:

Lie algebra; Wronskian relations; nonlinear evolution equations; inverse scattering method; generalized Zakharov-Shabat system; \(N\)-wave equations; Riemann-Hilbert problem; soliton solutions