Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

The Guest Editors for this special issue are

Niky Kamran (McGill University, Canada)
Gloria Mari Beffa (University of Wisconsin, USA)
Willard Miller (University of Minnesota, USA)
Guillermo R. Sapiro (Duke University, USA)

For over 35 years, Peter Olver has been immensely influential through his research contributions in shaping the development of a vast area of the mathematical sciences covering the symmetry and integrability properties of differential equations, Hamiltonian structures and conservation laws for partial differential equations, differential invariants of groups actions, moving frames, Lie pseudogroups, invariant variational problems, classical invariant theory, computer vision, invariant numerical algorithms and much more. In each of these areas, Peter has made fundamental discoveries which have led to new, rich and rapidly growing areas of research. To cite but two instances within this long list, we mention his epoch-making discovery of recursion operators as structures underlying the integrability of PDEs such as the KdV equation and his introduction with Mark Fels of the extraordinarily powerful method of equivariant moving coframes. All of these contributions have been presented through papers written with exemplary clarity, elegance and reader-friendliness, and books which have rightly become classics thanks to their breadth, clarity and scholarly nature. Peter has also been a trusted colleague and model mentor to a large number of us within the mathematical community, always fair and open in his judgment, and always wise in giving the best advice.

We are very happy to dedicate this special issue of SIGMA to him, with great admiration and friendship.

Niky Kamran, Gloria Mari-Beffa, Willard Miller, Jr., and Guillermo Sapiro

Papers in this Issue:

Invariant Discretization Schemes Using Evolution-Projection Techniques
Alexander Bihlo and Jean-Christophe Nave
SIGMA 9 (2013), 052, 23 pages   [ abs   pdf ]
Relative Critical Points
Debra Lewis
SIGMA 9 (2013), 038, 28 pages   [ abs   pdf ]
On Local Congruence of Immersions in Homogeneous or Nonhomogeneous Spaces
Jeongoo Cheh
SIGMA 9 (2013), 036, 21 pages   [ abs   pdf ]
The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces
João Paulo dos Santos and Keti Tenenblat
SIGMA 9 (2013), 033, 27 pages   [ abs   pdf ]
The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape Analysis
Mireille Boutin and Shanshan Huang
SIGMA 9 (2013), 031, 25 pages   [ abs   pdf ]
Solving Local Equivalence Problems with the Equivariant Moving Frame Method
Francis Valiquette
SIGMA 9 (2013), 029, 43 pages   [ abs   pdf ]
G-Strands and Peakon Collisions on Diff(R)
Darryl D. Holm and Rossen I. Ivanov
SIGMA 9 (2013), 027, 14 pages   [ abs   pdf ]
A Quasi-Lie Schemes Approach to Second-Order Gambier Equations
José F. Cariñena, Partha Guha and Javier de Lucas
SIGMA 9 (2013), 026, 23 pages   [ abs   pdf ]
Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type
Peter J. Vassiliou
SIGMA 9 (2013), 024, 21 pages   [ abs   pdf ]
Object-Image Correspondence for Algebraic Curves under Projections
Joseph M. Burdis, Irina A. Kogan and Hoon Hong
SIGMA 9 (2013), 023, 31 pages   [ abs   pdf ]
Integrable Flows for Starlike Curves in Centroaffine Space
Annalisa Calini, Thomas Ivey and Gloria Marí Beffa
SIGMA 9 (2013), 022, 21 pages   [ abs   pdf ]
The Cauchy Problem for Darboux Integrable Systems and Non-Linear d'Alembert Formulas
Ian M. Anderson and Mark E. Fels
SIGMA 9 (2013), 017, 22 pages   [ abs   pdf ]
Courant Algebroids. A Short History
Yvette Kosmann-Schwarzbach
SIGMA 9 (2013), 014,  8 pages   [ abs   pdf ]
Specialized Orthonormal Frames and Embedding
Frank B. Estabrook
SIGMA 9 (2013), 012,  5 pages   [ abs   pdf ]
Invertible Darboux Transformations
Ekaterina Shemyakova
SIGMA 9 (2013), 002, 10 pages   [ abs   pdf ]
Multi-Component Integrable Systems and Invariant Curve Flows in Certain Geometries
Changzheng Qu, Junfeng Song and Ruoxia Yao
SIGMA 9 (2013), 001, 19 pages   [ abs   pdf ]
Nonlocal Symmetries, Telescopic Vector Fields and λ-Symmetries of Ordinary Differential Equations
Concepción Muriel and Juan Luis Romero
SIGMA 8 (2012), 106, 21 pages   [ abs   pdf ]
The Klein-Gordon Equation and Differential Substitutions of the Form v=φ(u,ux,uy)
Mariya N. Kuznetsova, Aslı Pekcan and Anatoliy V. Zhiber
SIGMA 8 (2012), 090, 37 pages   [ abs   pdf ]
Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables
Philip Broadbridge, Claudia M. Chanu and Willard Miller Jr.
SIGMA 8 (2012), 089, 31 pages   [ abs   pdf ]
Sylvester versus Gundelfinger
Andries E. Brouwer and Mihaela Popoviciu
SIGMA 8 (2012), 075,  7 pages   [ abs   pdf ]
Motions of Curves in the Projective Plane Inducing the Kaup-Kupershmidt Hierarchy
Emilio Musso
SIGMA 8 (2012), 030, 20 pages   [ abs   pdf ]
Lagrange Anchor and Characteristic Symmetries of Free Massless Fields
Dmitry S. Kaparulin, Simon L. Lyakhovich and Alexey A. Sharapov
SIGMA 8 (2012), 021, 18 pages   [ abs   pdf ]

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