Soliton Equations and Hamiltonian Systems.

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, "Preface to the Second Edition; Contents; Introduction to the First Edition; Chapter 1 Integrable Systems Generated by Linear Differential nth Order Operators; 1.1 Differential Algebra A; 1.2 Space of Functionals A; 1.3 Ring of Pseudodifferential Operators; 1.4 Lax Pairs. GD Hierarchies of Equations; 1.5 First Integrals (Constants of Motion); 1.6 Compatibility of the Equations of a Hierarchy; 1.7 Soliton Solutions; 1.8 Resolvent. Adler Mapping; Chapter 2 Hamiltonian Structures; 2.1 Finite-Dimensional Case; 2.2 Hamilton Mapping; 2.3 Variational Principles.

2.4 Symplectic Form on an Orbit of the Coadjoint Representation of a Lie Group2.5 Purely Algebraic Treatment of the Hamiltonian Structure; 2.6 Examples; Chapter 3 Hamiltonian Structure of the GD Hierarchies; 3.1 Lie Algebra V Dual Space Q1 and Module Q0; 3.2 Proof of Theorem 3.1.2; 3.3 Poisson Bracket; 3.4 Reduction to the Submanifold Un-1 = 0; 3.5 Variational Derivative of the Resolvent; 3.6 Hamiltonians of the GD Hierarchies; 3.7 Theory of the KdV-Hierarchy (n = 2) Independent of the General Case; Chapter 4 Modified KdV and GD. The Kupershmidt-Wilson Theorem.

4 1 Miura Transformation. The Kupershmidt-Wilson Theorem4.2 Modified KdV Equation. Backlund Transformations; 4.3 More on Modified GD Equations; Chapter 5 The KP Hierarchy; 5.1 Definition of the KP Hierarchy; 5.2 Reduction of the KP Hierarchy to GD; 5.3 First Integrals and Soliton Solutions; 5.4 Hamiltonian Structure; 5.5 Resolvent; 5.6 Hamiltonians of the KP Hierarchy; Chapter 6 Baker Function T-Function; 6.1 Dressing; 6.2 Baker Function; 6.3 Shift Operator and T-Function; 6.4 Resolvent and Baker Function. Fay Identities; 6.5 Vertex Operators; 6.6 T-Function and Fock Representation.

6A Appendix. List of Useful Formulas for the Faa di Bruno PolynomialsChapter 7 Additional Symmetries String Equation; 7.1 Additional Symmetries; 7.2 Generating Function for Additional Symmetries; 7.3 String Equation; Chapter 8 Grassmannian. Algebraic-Geometrical Krichever Solutions; 8.1 Infinite-Dimensional Grassmannian; 8.2 Modified Definition of the Grassmannian T-Function; 8.3 Algebraic-Geometrical Solutions of Krichever; 8A Appendix. Abel Mapping and the 0-Function; Chapter 9 Matrix First-Order Operator AKNS-D Hierarchy.

9.1 Hierarchy of Equations Generated by a First-Order Matrix Differential Operator9.2 Hamiltonian Structure; 9.3 Hamiltonians of the AKNS-D Hierarchy; 9.4 GD Hierarchies as Reductions of the Matrix Hierarchies (Drinfeld-Sokolov Reduction); 9A Appendix. Extension of the Algebra A to an Algebra Closed with Respect to the Indefinite Integration; Chapter 10 Generalization of the AKNS-D Hierarchy: Single-Pole and Multi-Pole Matrix Hierarchies; 10.1 Single-Pole Matrix Hierarchy; 10.2 Single-Pole Hierarchy. Presentation not Depending on a Distinguished Operator 1.

10.3 Multi-Pole (General Zakharov-Shabat) Hierarchy.

The theory of soliton equations and integrable systems has developed rapidly during the last 30 years with numerous applications in mechanics and physics. For a long time,

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