2025-06-05 (Czwartek)
Zapraszamy na spotkanie o godzinie 15:00 
Jakub Vašíček (Silesian University in Opava)Conservation laws and nonexistence of local Hamiltonian structures for generalized Infeld—Rowlands equation
We exhaustively characterize all cases when a certain natural generalization of the Infeld–Rowlands equation admits nontrivial local conservation laws of any order, and give explicit form of these conservation laws modulo trivial ones. The original Infeld–Rowlands equation arises inter alia in the study of solution stability for the Ginzburg–Landau equation. What is more, one of the special cases of our generalization can be seen as an extension of two-dimensional Kuramoto–Sivashinsky equation to dimension three.It turns out that even in the generic case the equation in question, to which we refer to as the generalized Infeld–Rowlands equation, admits an infinite family of local conservation laws parameterized by an arbitrary smooth function of one variable.Furthermore, we prove that the equation under study admits no nontrivial local Hamiltonian structures and no nontrivial local symplectic structures no matter the orders of the structures in question; the method of establishing the said nonexistence results can be readily applied to many other PDEs.For further details please see the article J. Vašíček, Conservation laws and nonexistence of local Hamiltonian structures for generalized Infeld—Rowlands equation, Rep. Math. Phys. 93 (2024), no. 3, 287-300, https://doi.org/10.1016/S0034-4877(24)00038-7.
2025-05-29 (Czwartek)
Zapraszamy na spotkanie o godzinie 15:15
Ian Anderson (Utah State University)Superposition Formulas for Differential Systems II
2025-05-22 (Czwartek)
Zapraszamy na spotkanie o godzinie 15:00
Piergiulio Tempesta (Universidad Complutense de Madrid)To be announced
2025-05-15 (Czwartek)
Zapraszamy na spotkanie o godzinie 15:00
Ian Anderson (Utah State University)Superposition Formulas for Differential Systems I
In these two seminar talks I will survey the Lie group theoretic approach to the classical integration method of Darboux -- based on the symmetry reduction of pairs of differential systems admitting a common symmetry group. Topics include:Differential Equations via Differential SystemsFundamental ConstructionsSymmetry Reduction of Differential Systems.Lie EquationsSymbolic ToolsExamplesDarboux Integrable (DI) SystemsThe Equivalence Problem for DI Systems ClassificationsFurther ApplicationsReferences The book by O. Stomark, Lie's Structural Approach to PDE Systems is a good introduction to the classical literature. Here are references to my work on superposition principles with Mark Fels, Peter Vassiliou, and Brandon Ashley. Exterior Differential Systems with Symmetry Acta. Appl. Math., 87 (2005) 3-31. Superposition Formulas for Darboux Integrable Exterior Differential Systems, Advances in Math., 221 (2009) 1910--1963Transformations of Darboux Integrable Equations,} Proceeding of the 2008 Abel Conference, Tromso, Norway. Springer.Backlund transformations for Darboux integrable differential systems} Selecta Math. New Series 21 (2014) 379-448.Backlund transformations for Darboux integrable differential systems: Examples and Applications Journal of Geometry and Physics 102 (2016) 1-31The Cauchy Problem for Darboux Integrable Systems and Non Linear d’Alembert Formulas, SIGMA 9 (2017) Darboux Integrable f-Gordon Equations and Rank 2 Distributions in Five Variables (in preparation)