Advanced Research in Mathematical Physics, 2nd Edition: Bäcklund Symmetries of Painlevé and Soliton Equations and Their Applications

Dear Colleagues,

Bäcklund transformations play a fundamental role in the study of nonlinear evolution equations. They originated in the late nineteenth century in differential geometry, in the work of Bäcklund, Darboux, and Bianchi, where they were introduced as transformations between surfaces of constant curvature and as a method for generating new surfaces from known ones. Although initially motivated by geometry, these transformations later acquired a much broader significance in the theory of nonlinear partial differential equations.

In soliton theory, beginning with the work of Gardner, Greene, Kruskal, Miura, and Zabusky on the KdV equation, Bäcklund transformations became a major constructive tool. They provide an algebraic or differential mechanism for generating new solutions from given ones while preserving integrability. In particular, they allow the systematic construction of multi–soliton solutions and are closely connected with nonlinear superposition principles.

In a general sense, a Bäcklund transformation relates solutions of two (possibly distinct) nonlinear PDE systems, or two different field configurations of the same equation. It is typically given by a set of first–order differential relations whose compatibility conditions reproduce the original equations. When applied iteratively, these relations generate infinite families of solutions. In this way, Bäcklund transformations act as solution–generating symmetries and provide a bridge between analytic and algebraic structures of integrable systems.

Within the Lax formalism and the zero-curvature representation of integrable models, Bäcklund transformations admit a natural interpretation as gauge transformations of the associated linear problem. When the system is formulated using loop or affine Lie algebras and their graded decompositions, these transformations appear as special gauge maps preserving the flatness condition. This viewpoint explains why they extend consistently to all higher flows of an integrable hierarchy (such as the KdV, AKNS, or Kaup–Newell hierarchies) and clarifies their compatibility with the Hamiltonian and bi-Hamiltonian structures of the theory.

When the transformation interpolates between two solutions of the same equation, it may be regarded as a discrete symmetry of the equation of motion. From this perspective, Bäcklund transformations connect continuous integrable PDEs with integrable lattice or discrete systems. Classical examples include the Toda lattice, the Volterra system, and other differential–difference equations, which can often be interpreted as discrete maps consistent with repeated Bäcklund steps. Thus, they provide an important link between continuous and discrete integrability.

For the Painlevé equations, which arise as similarity reductions of integrable hierarchies or as isomonodromic deformation equations, Bäcklund-type transformations are inherited from the symmetries of the parent hierarchies. These transformations leave the differential form of a given Painlevé equation invariant but act nontrivially on its parameters and solutions. A fundamental property is that they generate infinite families of solutions from a single seed solution. Moreover, the set of all such transformations forms an affine Weyl group, as clarified in the work of Okamoto and others, revealing a deep connection between Painlevé equations and the theory of root systems and Lie algebras.

Discrete Painlevé equations may likewise be understood as difference equations produced by Bäcklund transformations of the continuous Painlevé systems. In this sense, they represent the discrete counterpart of integrable dynamics, and their derivation and classification are closely tied to the algebraic structure underlying these transformations.

This volume is devoted to developments in the theory and applications of Bäcklund transformations in the realm of soliton and Painlevé equations, with particular emphasis on methods for constructing new solutions, elucidating symmetry structures, and uncovering new integrable models and their properties.

Prof. Dr. Henrik Aratyn
Guest Editor

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• mathematical physics
• integrable models
• Painlevé equations and discrete Painleve systems
• Backlund transformations
• applications of symmetry
• affine Weyl algebras and groups
• soliton solutions
• special functions and transcendental functions
• rational solutions
• polynomials

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