Analytic-Bilinear Approach to Integrable Hierarchies

1 Introduction.- 1.1 Hirota Bilinear Identity.- 1.2 Meromorphic Loops, Determinant Formula and the ?-Function.- 1.3 Integrable Discrete Equations.- 1.4 From Discrete Equations to the Continuous Hierarchy.- 2 Hirota Bilinear Identity for the Cauchy Kernel.- 2.1 Boundary Problems for the ¯?-Operator in the Unit Disc.- 2.2 General Boundary Problems with Zero Index.- 2.3 Rational Deformations of the Boundary Problems.- 2.4 Hirota Bilinear Identity.- 2.5 Determinant Formula for Action of Meromorphic Loops on the Cauchy Kernel.- 2.6 ?-Function for the One-Component Case.- 3 Rational Loops and Integrable Discrete Equations. I: Zero Local Indices.- 3.1 One-Component Case.- 3.2 General Matrix Equations for the Multicomponent Case.- 4 Rational Loops and Integrable Discrete Equations. II: Two-Component Case.- 4.1 DS case.- 4.2 2DTL Case.- 5 Rational Loops and Integrable Discrete Equations. III: The General Case.- 5.1 General Multicomponent Case.- 5.2 ?-Function for the Multicomponent Case.- 5.3 Three-Component Case.- 5.4 Four-Component Case.- 5.5 Five-Component and Six-Component Cases.- 6 Generalized KP Hierarchy.- 6.1 Generalized Hirota Identity from the ¯?-Dressing Method.- 6.2 The Generalized KP Hierarchy.- 6.3 KP Hierarchy in the ‘Moving Frame’. Darboux Equations as the Horizontal Subhierarchy.- 6.4 Combescure Symmetry Transformations.- 6.5 ?-Function and Addition Formulae.- 6.6 ?-Function as a Functional.- 6.7 From the Discrete Case to the Continuous.- 7 Multicomponent Kp Hierarchy.- 7.1 Multicomponent Case with Zero Local Indices.- 7.2 ?-Function and Closed 1-Form for ?+N.- 7.3 Generalized DS Hierarchy.- 7.4 Loop Group ? and 2DTL Hierarchy.- 8 On The ¯?-Dressing Method.- 8.1 General Scheme.- 8.2 Matrix Lattice and q-Difference Darboux Equations.- 8.3 Special Cases of Nonlocal ¯?-Problem.- 8.4 On Some Equations, Integrable Via ¯?-Dressing Method.- 8.5 Solutions with Special Properties.- 8.6 Boussinesq Equation.- 8.7 Relativistically-Invariant Systems.- 8.8 Inverse Problems for the Differential Operator of Arbitrary Order on the Line.