Introduction to Classical Integrable Systems
Cambridge University Press, Apr 17, 2003 - Science - 602 pages
This book provides a thorough introduction to the theory of classical integrable systems, discussing the various approaches to the subject and explaining their interrelations. The book begins by introducing the central ideas of the theory of integrable systems, based on Lax representations, loop groups and Riemann surfaces. These ideas are then illustrated with detailed studies of model systems. The connection between isomonodromic deformation and integrability is discussed, and integrable field theories are covered in detail. The KP, KdV and Toda hierarchies are explained using the notion of Grassmannian, vertex operators and pseudo-differential operators. A chapter is devoted to the inverse scattering method and three complementary chapters cover the necessary mathematical tools from symplectic geometry, Riemann surfaces and Lie algebras. The book contains many worked examples and is suitable for use as a textbook on graduate courses. It also provides a comprehensive reference for researchers already working in the field.
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10 The KP hierarchy
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action analytic asymptotic Baker–Akhiezer function branch points Cartan Chapter coadjoint action coadjoint orbit coefﬁcients components compute condition conserved quantities consider constant construction coordinates corresponding deﬁned deﬁnition denote diagonal matrix differential equation differential operator eigenvalues eigenvector equations of motion expression fermionic ﬁeld ﬁnd ﬁnite ﬁrst ﬁxed formula gauge transformation Hamiltonian Hence highest weight holomorphic integrable systems introduce invariant Jacobi identity KdV hierarchy Lax equation Lax matrix Lax pair Lie algebra Lie group line bundle linear Liouville loop algebra matrix elements meromorphic function monodromy matrix normalized Note obtained parameters phase space Poisson brackets polynomial Proof Proposition r-matrix relation representation residue Riemann surface root satisﬁes singular soliton solution spectral curve structure subalgebra symplectic form tau-function theorem theta-functions tion vanishes variables vector write zeroes