Preface The Authors A1. Reciprocal transformations and their role in the integrability and classification of PDEs 1. Introduction 2. Fundamentals 3. Reciprocal transformations as a way to identify and classify PDEs 4. Reciprocal transformations to derive Lax pairs 5. A Miura-reciprocal transformation 6. Conclusions A2. Contact Lax pairs and associated (3+l)-dimensional integrable dispersionless systems 1. Introduction 2. Isospectral versus nonisospectral Lax pairs 3. Lax representations for dispersionless systems in (I+I)D and (2+1)D 4. Lax reprcsentations for dispersionless systems in (3+l)D 5. R-matrix approach for dispersionless systems with nonisospectral Lax representations A3. Lax pairs for edge-constrained Boussinesq systems of partial difference equations 1. Introduction 2. Gauge equivalence of Lax pairs for PDEs and PAEs 3. Derivation of Lax pairs for Boussinesq systems 4. Gauge and gauge-like equivalences of Lax pairs 5. Application to generalized Hietarinta systems 6. Summary of results 7. Software implementation and conclusions A4. Lie point symmetries of delay ordinary differential equations 1. Introduction 2. Illustrating example 3. Formulation of the problem for first-order DODEs 4. Construction of invariant first-order DODSs 5. First-order linearDODSs 6. Lie symmetry classification of first-order nonlinear DODSs 7. Exact solutions of the DODSs 8. Higher order DODSs 9. Traffic flow micro-model equation 10. Conclusions A5. The symmetry approach to integrability: recent advances 1. Introduction 2. The symmetry approach to integrability 3. Integrable non-abelian equations 4. Non-evolutionary systems A6. Evolution of the concept of h-symmetry and main applications 1. Introduction 2. Basic notions on Lie point symmetries and C∞- symmetries of ODEs 3. Analytical applications of C~-symmetries 4. Extensions and geometric interpretations of C∞-symmetries A7. Heir-equations for partial differential equations: a 25-year review 1. Introduction 2. Constructing the heir-equations 3. Symmetry solutions of heir-equations 4. Zhdanov's conditional Lie-B~cklund symmetries and heir-equations
5. Nonclassical symmetries as special solutions of heir-equations 6. Final remarks B1. Coupled nonlinear SchrSdinger equations: spectra and instabilities of plane waves 1. Introduction 2. Spectra 3. Dispersion relation and instability 4. Conclusions A. Case r = 0 B. Polynomials: a tool box B2. Rational solutions of Painlev systems 1. Introduction 2. Dressing chains and Painlevd systems 3. Hermite T-functions 4. Hermite-type rational solutions 5. Cyclic Maya diagrams 6. Examples of Hermite-type rational solutions B3. Cluster algebras and discrete integrability 1. Introduction 2. Cluster algebras: definition and examples 3. Cluster algebras with periodicity 4. Algebraic entropy and tropical dynamics 5. Poisson and symplectic structures 6. Discrete Painlev6 equations from coefficient mutation 7. Conclusions B4. A review of elliptic difference Painlev6 equations 1. Introduction 2. E-lattice 3. The initial-value space of the RCG equation 4. Cremona isometrics 5. Birational actions of the Cremona isometrics for the Jacobi setting 6. Special solutions of the RCG equation A. A-lattice B. General elliptic difference equations B5. Linkage mechanisms governed by integrable deformations of discrete space curves 1. Introduction 2. A mathematical model of linkage 3. Hinged network and discrete space curve 4. Deformation of discrete curves 5. Extreme Kaleidocycles B6. The Cauchy problem of the Kadomtsev=Petviashvili hierarchy and infinite-dimensional groups 1. Introduction 2. Diffeologies, Fr51icher spaces and the Ambrose-Singer theorem 3. Infinite-dimensional Lie groups and pseudodifferential operators 4. The Cauchy problem for the KP hierarchy 5. A non-formal KP hierarchy BT. Wronskian solutions of integrable systems 1. Introduction 2. Preliminary 3. The KdV equation 4. The mKdV equation
5. The AKNS and reductions 6. Discrete case: the lpKdV equation 7. Conclusions C1. Global gradient catastrophe in a shallow water model: evolution unfolding by stretched coordinates 1. Introduction 2. Exact solutions 3. Evolution beyond the gradient catastrophe 4. Discussion and conclusions C2. Vibrations of an elastic bar isospectral deformations and modified Camassa-Holm equations 1. Introduction 2. The Lax formalism: the boundary value problem 3. Liouville integrability 4. Forward map: spectrum and spectral data 5. Inverse problem 6. Multipeakons for N = 2K 7. Multipeakons for N = 2K + 1 8. Reductions of multipeakons A. Lax pair for the 2-mCH peakon ODEs B. Proof of Theorem 13 C. Proof of Theorem 17 C3. Exactly solvable (discrete) quantum mechanics and new orthogonal polynomials 1. Introduction 2. New orthogonal polynomials in ordinary QM 3. New orthogonal polynomials in discrete QM with real shifts 4. Summary and comments 編輯手記
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