Nonlinear Systems and Their Remarkable Mathematical Structures

Part A: Integrability, Lax Pairs and Symmetry. A1. Reciprocal transformations and their role in the integrability and classification of PDEs. A2. Contact Lax pairs and associated (3+1)-dimensional integrable dispersionless systems. A3. Lax Pairs for Edge-constrained Boussinesq Systems of Partial Difference Equations. A4. Lie point symmetries of delay ordinary differential equations. A5. The symmetry approach to integrability: recent advances. A6. Evolution of the concept of λ--symmetry and main applications. A7. Heir-equations for partial dfferential equations: a 25-year review. Part B: Algebraic and Geometric Methods. B1. Coupled nonlinear Schrodinger equations: spectra and instabilities of plane waves. B2. Rational solutions of Painleve systems. B3. Cluster algebras and discrete integrability. B4. A review of elliptic difference Painleve equations. B5. Linkage mechanisms governed by integrable deformations of discrete space curves. B6. The Cauchy problem of the Kadomtsev-Petviashvili hierarchy and infinite-dimensional groups. B7. Wronskian solutions of integrable systems. Part C: Applications. C1. Global gradient catastrophe in a shallow water model: evolution unfolding by stretched coordiates. C2. Vibrations of an elastic bar, isospectral deformations, and modified Camassa-Holm equations. C3. Exactly solvable (discrete) quantum mechanics and new orthogonal polynomials.