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Almost everything you always wanted to know about the Toda equation. (English) Zbl 1039.37058
From the text: The present article reviews methods from spectral theory and algebraic geometry for finding explicit solutions of the Toda equation, namely for the \(N\)-soliton solution and quasi-periodic solutions. Along they way basic concepts like Lax pairs, associated hierarchies, and Bäcklund transformations for the Toda equation are introduced. The contents constitute an extended version of my talk given at the joint annual meeting of the Österreichische Mathematische Gesellschaft und der Deutsche Mathematiker-Vereinigung in September 2001, Vienna, Austria.

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37K60 Lattice dynamics; integrable lattice equations