Symmetry Methods for Differential Equations

Dear Colleagues,

Symmetry plays a fundamental role in numerous aspects of the physical world and art, serving as a cornerstone in both disciplines. The concept of symmetry has been a pivotal element in many approaches to art, as it greatly contributes to the creation of esthetic appeal. In the late 19th century, Sophus Lie, in a series of books, published his groundbreaking work on the theory of infinitesimal transformations for the analysis of differential equations. Lie's work introduced the revolutionary idea of considering the infinitesimal representations of finite transformations of continuous groups. This approach allowed for the transition from group theory to a local algebraic representation, enabling a deeper exploration of the invariance properties under these transformations. The primary goal of determining the invariant transformations that leave a given differential equation unchanged is to simplify the solution of the equation under study. The presence of symmetries enables one to solve differential equations through repeated reduction of order, often using a reverse series of quadratures or by determining a sufficient number of first integrals. The latter can be used to systematically reduce the complexity of a given differential equation.

A few decades later, Emmy Noether presented her work which profoundly relates variational symmetries to conserved quantities. The novelty of Noether's work was her introduction of coefficient functions of the differential operator, which generated the infinitesimal transformation of the Action Integral depending on the derivatives of the dependent variable(s), leading to the so-called generalized, or dynamical, symmetries. Noether's work was crucial in the evolution of Analytic Mechanics, since it provides a systematic way for the construction of conservation laws and the derivation of new integrable systems. In physics, Noether's conservation laws are related to conserved quantities of the physical world. Applications of these laws are extended in gravitational theory and quantum mechanics.

This Special Issue aims to explore advanced topics in symmetry analysis, particularly in the context of nonlinear differential equations across diverse areas of applied mathematics. It highlights both classical and modern developments in the field, with an emphasis on innovative approaches to invariant functions, reduction techniques and the discovery of hidden or generalized symmetries. The following is a breakdown of the themes that will be covered:

• Invariant functions and the reduction of differential equations.
• Generalized, hidden and nonlocal symmetries.
• Variational symmetries for regular and singular dynamical systems.
• The effects of Dirac's constraint on variational symmetries.
• New techniques and innovative methods (computational tools, algebraic tools, etc.) for the treatment of symmetry analysis for nonlinear differential equations.
• The classification of differential equations and its relation to other methods for the construction of solutions.
• Applications of symmetry analysis in all areas of applied mathematics.

Dr. Andronikos Paliathanasis
Prof. Dr. Peter Gavin Lawrence Leach
Guest Editors

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• invariant functions and the reduction of differential equations
• generalized, hidden and nonlocal symmetries
• variational symmetries for regular and singular dynamical systems
• the effects of Dirac's constraint on variational symmetries
• new techniques and innovative methods (computational tools, algebraic tools, etc.) for the treatment of symmetry analysis for nonlinear differential equations
• the classification of differential equations and its relation to other methods for the construction of solutions
• applications of symmetry analysis in all areas of applied mathematics