Symmetries and Integrability of Difference Equations
A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".
Deadline for manuscript submissions: closed (31 July 2019) | Viewed by 4478
Special Issue Editors
Interests: mathematical physics; integrable nonlinear systems; theories of groups and symmetries
Special Issue Information
Dear Colleagues,
The concept of integrability was introduced in classical mechanics in the 19th century for finite dimensional continuous Hamiltonian systems. It was extended to certain classes of nonlinear differential equations in the second half of the 20th century with the discovery of the inverse scattering transform and the birth of soliton theory. Additionally at the end of the 19th century, Lie group theory was invented as a powerful tool for obtaining exact analytical solutions of large classes of differential equations. Together, Lie group theory and integrability theory in its most general sense provide the main tools for solving nonlinear differential equations. Like differential equations, difference equations play an important role in physics and other sciences. They occur very naturally in the description of phenomena that are genuinely discrete. Indeed, they may actually be more fundamental than differential equations if space-time is actually discrete at very short distances. On the other hand, even when treating continuous phenomena described by differential equations it is very often necessary to resort to numerical methods. This involves a discretization of the differential equation, i.e., a replacement of the differential equation by a difference one. Given the well-developed and understood techniques of symmetry and integrability for differential equations a natural question to ask is whether it is possible to develop similar techniques for difference equations. The aim is, on one hand, to obtain powerful methods for solving integrable difference equations and to establish practical integrability criteria, telling us when the methods are applicable. On the other hand, Lie group methods can be adapted to solve difference equations analytically. Finally, integrability and symmetry methods can be combined with numerical methods to obtain improved numerical solutions of differential equations.
Prof. Decio Levi
Dr. Giorgio Gubbiotti
Guest Editors
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Keywords
- Discrete, continuous and ultra-discrete Painlevé equations
- Orthogonal polynomials, special functions and their relation to discrete integrable systems
- Integrability criteria for single and multivariable difference equations and differential difference equations
- Discrete differential geometry
- Discrete integrable systems and isomonodromy transformations
- Yang-Baxter maps and quantum discrete integrable systems
- Continuous symmetries of discrete equations
- Structure preserving dis-cretization of differential equations and numerical methods
- Cluster algebras and discrete integrable systems
- Dynamics on graphs and combinatorics
- Difference Galois theory
- Lattices and Symmetries in Physical Applications
