Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

233Lie symmetriesof partial differential equationsJoël MerkerTable of contents1. Completely integrable systems of partial differential equations . . . . . . . . . . . . . . . . . . . .??.2. Submanifold of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??.3. Classification problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??.4. Punctual and infinitesimal Lie symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??.5. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .??.6. Transfer of Lie symmetries to the parameter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??.7. Equivalence problems and normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .??.8. Study of two specific examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??.9. Dual system of partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??.10. Fundamental pair of foliations and covering property . . . . . . . . . . . . . . . . . . . . . . . . . . . ??.11. Formal and smooth equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??.Journal of Mathematical Sciences (N. Y.), to appearThis memoir is divided in three parts 29 . Part I endeavours a general, new theory(inspired by modern CR geometry) of Lie symmetries of completely integrablePDE systems, viewed from their associated submanifold of solutions. Part II buildsgeneral combinatorial formulas for the prolongations of vector fields to jet spaces.Part III characterizes explicitly flatness of some systems of second order. The resultspresented here are original and did not appear in print elsewhere; most formulasof Parts II and III were checked by means of Maple Release 7.§1. COMPLETELY INTEGRABLE SYSTEMS OF PARTIAL DIFFERENTIALEQUATIONS1.1. General systems. Let K = R or C. Let n ∈ N with n 1 andlet x = (x 1 , . . .,x n ) ∈ K n . Also, let m ∈ N with m 1 and let y =(y 1 , . . .,y m ) ∈ K m . For α ∈ N n , we denote by a subscript y x α the partialderivative ∂ |α| y/∂x α of a local map K n ∋ x ↦→ y(x) ∈ K m .Let κ ∈ N with κ 1, let p ∈ N with p 1, choose a collection ofp multiindices β(1), . . ., β(p) ∈ N n with |β(q)| 1 for q = 1, . . .,p andmax 1qp |β(q)| = κ, and choose integers j(1), . . .,j(p) with 1 j(q) 29 Part II of [Me2005a] already appeared as [Me2005b].