R-manifolds and multi-valued solutions of PDE. Part 2
Vladimir Chetverikov
A program of the course at the 4-th Italian Diffiety School,
(Forino, July 17-29, 2000)
Singularities of solutions of PDE
  1. Caustics and wave front sets.
  2. Definitions. R-manifolds, multi-valued solutions of PDE, singularities of solutions of PDE, labels of singular points.
  3. Examples. Second order equations with two independent variables and one dependent variable, branched Riemann surfaces.
Characteristic 1-forms and characteristic vector fields.
  1. Associated 1-singularity equations. Characteristic covectors.
  2. Method of characteristics. Characteristic symmetries of distributions. Example of the eikonal equation.
Multi-valued solutions and Schwartz generalized solutions.
  1. Schwartz distribution. Definitions and examples.
  2. Schwartz generalized solutions of PDE. Its relation with multi-valued solutions of PDE.

Exercises and examination problems
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To pass the exam one should solve 4 problems.

    Exercise 11. Prove that the surface S introduced in lecture determines a multi-valued solution of the eikonal equation.

    Problem 12. Let q Î singV, W = Tq (V), P = (pk,k-1)* (W), P0 = ker(pk,k-1|V)*,q. Prove that P0 Ì Tq (l(P)), where l(P) is the s-ray, s = dimP.

    Problem 13. Let an equation E Ì J2(2,1) be determined by the equality
    F(x1,x2,u,u1,u2,u11,u12,u22) = 0.
    Find 1) elliptic points; 2) hyperbolic points; 3) parabolic points of this equation.

    Problem 14. Find the multi-valued solution of the Cauchy-Riemann system corresponding to the analytic function w = 2/3 z3/2. Show that 1) this solution has one singular point; 2) the type of this singular point is 2.

    Problem 15. Find a finite intrinsic symmetry of the Cauchy-Riemann system E that maps vertical fibers of E to horizontal submanifolds. Also, find the image under this symmetry of the solution from the previous problem.

    Exercise 12 Prove the theorem about the relation characteristic covectors of an equation E with points of its associated 1-singularity equation E[1].

    Exercise 13. Let E = {F = 0} Ì J1(n,1). Show that the vector field
    YF = n
    å
     i = 1 
    		 æ
    ç
    è     		- F
    pi
    		  
    xi
    		 + æ
    è     		F
    xi
    		 +pi F
    u
    		 ö
    ø     		 
    pi
    		 ö
    ÷
    ø     		- æ
    ç
    è     		n
    å
     i = 1 
    		 pi F
    pi
    		 ö
    ÷
    ø     		 
    u
    is a characteristic symmetry of E.

    Questions and suggestions should go to Jet NESTRUEV, jet @ diffiety.ac.ru.