Exercise 2. Suppose that the l-th prolongation of an equation E is a smooth submanifold in J^k+l p. Show that in this case one has (E^(l))^(t) = E^(l+t) for all t ³ 0.

Exercise 3. Construct example(s) of an equation E such that E⁽¹⁾ is not a smooth submanifold in J^k+1 p.

Exercise 4. Construct example(s) of differential equation(s) for which the mapping p_k+1,k :   E⁽¹⁾ ®E is not surjective.

Exercise 5. Let submanifolds N, N₁ Ì P be tangent to each other at a point a Î N ÇN₁ with order k, m = {f Î C^¥(N) | f(a) = 0}, m^k+1 be the (k+1)-st degree of the ideal m. Show that if g Î C^¥(P) and g |_N1 = 0 then g |_N Î m^k+1.

Problem 1. Consider a differential equation with delay of the form

Hint: First describe all prolongations of this equation.

Exercise 6. Construct example(s) of map(s) G :   J^¥ p®J^¥x such that G^*(F(x)) Ì F(p) and for any l and k₀ there exists an integer k ³ k₀ such that G^*(F_k(x)) Ë F_k+l(p).

Exercise 7. The Cartan distribution can be also defined on J^¥ p. What is the dimension of this distribution?

Exercise 8. Construct example(s) of derivation(s) X of the algebra F(p) such that for any l and k₀ there exists an integer k ³ k₀ such that X(F_k(p)) Ë F_k+l(p).

Exercise 9. Show that if X, Y Î D(p) and f Î F, then X+Y, fX, and [X,Y] = X °Y - Y °X are also vector fields on J^¥p. In addition, show that D (p) is a Lie algebra over IR and D^v(p) is its Lie subalgebra.

Problem 2. Let X Î D(p) and K(X) Ì F(p) be the subalgebra generated by functions f, X(f), ¼, X^k(f) = X(X^k-1(f)), ¼, where f Î F₀(p), k > 1. Prove that if the vector field X is integrable, then there exists an integer l ³ 0 such that K(X) Ì F_l(p).

Exercise 10. Using Problem 2, construct examples of

(i) integrable vector field(s) that raise(s) the filtration of the algebra F(p),

(ii) nonintegrable vector field(s).

Exercise 11. Show that the module D(p) is dual to L¹(p), i.e.,

Exercise 12. Construct the map Y_D,x^¥ starting from the smooth map Y_D^¥ :  J^¥(p)®J^¥(h).

Exercise 13. Show that Spec  injlim_k®¥ F_k(p) = projlim_k®¥ Spec  F_k(p).

Exercise 14. Let X be a derivation of the algebra F(p), l ³ 0, and there exists an integer k₀ such that X(F_k(p)) Ì F_k+l(p) for any k ³ k₀. Show that X Î D(p).

Exercise 15. Prove that a distribution P is integrable iff d(PL^*) Ì PL^*.

Exercise 16. Prove that X Î D_P(p) iff [X,PD(p)] Ì PD(p).

Exercise 17. Show that D_P(p) is a Lie algebra and PD(p) is its ideal.

Exercise 18. Prove that the equality C_q(p) = T_q(j_¥(s)(M)) holds for any point q = [s]_x^¥ Î J^¥(p).

Hint: Use the equalities T_q(j_¥(s)(M)) = j_¥(s)_*(T_x(M)) and C_qk^k = (p_k,k-1)_*^-1(L_qk).

Exercise 19. Show that j_¥(s)^* °[^X] = X °j_¥(s)^* for any s Î G(p) and X Î D(M).

Exercise 20. Using Exercise 19, show that the Cartan connection is flat.

Exercise 21. Let X Î D(M), D_j :  G(p)®C^¥(M) be the scalar differential operator associated to a function j Î F_k(p). Show that the operator X °D_j is associated to the function [^X](j): D_[^X](j) = X °D_j.

Exercise 22. Show that [^X] Î D(p), with deg ([^X]) = 1, for any X Î D(M).

Exercise 23. Prove that the Cartan distributions on J^¥ p and E^¥ are integrable.

Exercise 24. Show that X Î CD(E) iff X  û  U_C(f) = 0 for any f Î F(E).

Exercise 25. Show that

Exercise 26. Prove that U_C(f g) = f U_C(g) + g U_C(f) for any f,g Î F(E).

Exercise 27. Write down the formula for U_C in the case of the KdV equation u_t = u_xxx + u u_x.

Exercise 28. Let p :  IR×IR®IR be the trivial bundle. Show that any infinitesimal automorphism of the Cartan distribution on J^¥ p has the form

Exercise 29. Prove that the module F(p,p) is a Lie IR-algebra with respect to the higher Jacobi bracket.

Exercise 30. Show that if D :  F(p,x)®F(p,h) is a C-differential operator and D(p_¥^*(j)) = 0 for any j Î G(x), then D = 0.

Exercise 31. Show that the two definitions of linearization are equivalent.

Exercise 32. Let E be an evolution equation, j be a symmetry of E, ' _j^E and l_j^E be the restrictions of the corresponding operators to E^¥. Prove that [ ' _j^E -l_j^E, l_E] = 0 (the commutator relation).