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Intro; Preface; Contents; Contributors; Acronyms; Introduction; Numerical Instability of the Akhmediev Breather and a Finite-Gap Model of It; 1 Introduction; 2 A Short Summary of Finite Gap Results; 3 The Numerical Instability of the Akhmediev Breather; 4 Numerical Experiments; 4.1 Effect of the Time Step; 4.2 Effect of Round-Off Errors; 4.3 Effect of the Cauchy Data; 4.4 Effect of the x-Grid Size; 5 Conclusions; References; Movable Poles of Painlevé I Transcendents and Singularities of Monodromy Data Manifolds; 1 Introduction; 2 Poles Parametrization in General Case
3 Singularities of the Monodromy Manifold P14 Truncated Solutions of PI Equation; 4.1 1-Truncated Solutions; 4.2 2-Truncated Solutions; 5 Formation of Truncated Solutions; References; Elliptic Calogero-Moser Hamiltonians and Compatible Poisson Brackets; 1 Introduction; 2 Polynomial Forms for Elliptic Quantum Calogero-Moser Hamiltonians; 3 Classical Case; 4 Conclusion; References; Bäcklund Transformations and New Integrable Systems on the Plane; 1 Introduction; 2 Divisor Arithmetic on Hyperelliptic Curves; 2.1 Some Explicit Formulae for Arithmetic on Genus 2 Hyperelliptic Curves
3 Group Law and New Integrable Systems on the Plane3.1 Hénon-Heiles System; 3.2 System with Quartic Potential; 4 Equivalence Relations and New Integrable Systems on the Plane; 4.1 Separation of Variables and Abel Differential Equation; 4.2 Construction of the New Integrable System on T*R; 4.3 Other New Integrable Systems on the Plane; 5 Conclusion; References; Quadro-Cubic Cremona Transformations and Feigin-Odesskii-Sklyanin Algebras with 5 Generators; 1 Introduction; 2 Poisson Algebras and Elliptic Poisson Algebras; 2.1 Poisson Algebras on Affine Varieties
2.2 Sklyanin Elliptic and Poisson Algebras2.3 Algebra Qn(Y,); 3 q5,k(Y)-Poisson Tensors; 3.1 Why n=5?; 3.2 H-Invariancy; 4 Cremona Transformation; 4.1 Generalities about Cremona Transformations; 4.2 Case n=4. Cremona transformation and Elliptic Curves.; 4.3 Quadro-Cubic Cremona Transformations; 4.4 A Remark on the Compatibility; 5 Klein and Moore Syzygies and Poisson Structures; 5.1 Klein Syzygy and Free Resolution Complex.; 5.2 Poisson Structure on P4 from Syzygies; 5.3 Moore Syzygies and Corresponding 3-Folds.; 5.4 Poisson Structure on P4 from Moore Syzygies
5.5 ``Quantum'' Cremona TransformationsReferences; From Reflection Equation Algebra to Braided Yangians; 1 Introduction; 2 Braidings: Definitions and Properties; 3 Braided Lie Algebras and their Affinization; 4 Characteristic Polynomials for Generating Matrices; 5 Braided Yangians; 6 Shifted Braided q-Yangian and its q=1 Limit; References; Survey of the Deformation Quantization of Commutative Families; 1 Introduction; 1.1 The Main Question; 1.2 Argument Shift Method and Enveloping Algebras; 1.3 Presentation and Organization of the Text; 2 The Argument Shift Method and Nijenhuis Operators; 2.1 Argument Shift on a Manifold
3 Singularities of the Monodromy Manifold P14 Truncated Solutions of PI Equation; 4.1 1-Truncated Solutions; 4.2 2-Truncated Solutions; 5 Formation of Truncated Solutions; References; Elliptic Calogero-Moser Hamiltonians and Compatible Poisson Brackets; 1 Introduction; 2 Polynomial Forms for Elliptic Quantum Calogero-Moser Hamiltonians; 3 Classical Case; 4 Conclusion; References; Bäcklund Transformations and New Integrable Systems on the Plane; 1 Introduction; 2 Divisor Arithmetic on Hyperelliptic Curves; 2.1 Some Explicit Formulae for Arithmetic on Genus 2 Hyperelliptic Curves
3 Group Law and New Integrable Systems on the Plane3.1 Hénon-Heiles System; 3.2 System with Quartic Potential; 4 Equivalence Relations and New Integrable Systems on the Plane; 4.1 Separation of Variables and Abel Differential Equation; 4.2 Construction of the New Integrable System on T*R; 4.3 Other New Integrable Systems on the Plane; 5 Conclusion; References; Quadro-Cubic Cremona Transformations and Feigin-Odesskii-Sklyanin Algebras with 5 Generators; 1 Introduction; 2 Poisson Algebras and Elliptic Poisson Algebras; 2.1 Poisson Algebras on Affine Varieties
2.2 Sklyanin Elliptic and Poisson Algebras2.3 Algebra Qn(Y,); 3 q5,k(Y)-Poisson Tensors; 3.1 Why n=5?; 3.2 H-Invariancy; 4 Cremona Transformation; 4.1 Generalities about Cremona Transformations; 4.2 Case n=4. Cremona transformation and Elliptic Curves.; 4.3 Quadro-Cubic Cremona Transformations; 4.4 A Remark on the Compatibility; 5 Klein and Moore Syzygies and Poisson Structures; 5.1 Klein Syzygy and Free Resolution Complex.; 5.2 Poisson Structure on P4 from Syzygies; 5.3 Moore Syzygies and Corresponding 3-Folds.; 5.4 Poisson Structure on P4 from Moore Syzygies
5.5 ``Quantum'' Cremona TransformationsReferences; From Reflection Equation Algebra to Braided Yangians; 1 Introduction; 2 Braidings: Definitions and Properties; 3 Braided Lie Algebras and their Affinization; 4 Characteristic Polynomials for Generating Matrices; 5 Braided Yangians; 6 Shifted Braided q-Yangian and its q=1 Limit; References; Survey of the Deformation Quantization of Commutative Families; 1 Introduction; 1.1 The Main Question; 1.2 Argument Shift Method and Enveloping Algebras; 1.3 Presentation and Organization of the Text; 2 The Argument Shift Method and Nijenhuis Operators; 2.1 Argument Shift on a Manifold