An analogue of the KP theory in dimension 2

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1 An analogue of the KP theory in dimension 2 A.Zheglov 1 1 Moscow State University, Russia XVII Geometrical Seminar, Zlatibor, Serbia, September 3-8, 2012

2 Outline 1 History: 1-dimensional KP theory Isospectral deformations of ordinary differential operators Sato Grassmanian Geometric solutions Examples 2 An analogue of the KP theory in dimension 2 Isospectral deformations of partial differential operators Quasi elliptic rings Schur pairs Geometric data Examples

3 Isospectral deformations of ordinary differential operators The Kadomtsev-Petviashvili (KP) equation The Kadomtsev-Petviashvili (KP) equation is a non-linear partial differential equation: (4u t u 12uu ) = 3u yy for u(t, x, y). It was found by Kadomtsev and Petviashvili in 1970, and the motivation for introducing it was to study transversal stability of the soliton solutions of the famous Korteveg and de Vries (KdV) equation 4u t u 12uu = 0 for u(t, x). The KdV equation was derived in 1895 from the Navier-Stokes equation of fluid dynamics as a special limit to give a model of nonlinear wave motions of shallow water observed in a canal.

4 Isospectral deformations of ordinary differential operators The Kadomtsev-Petviashvili (KP) equation It is one of the most interesting non-linear PDE, because 1 There is a rich theory connecting this equation with many other famous equations as well as with different problems in various branches of mathematics 2 There are many exact solutions of this equation Below is a brief explanation of the KP theory.

5 Isospectral deformations of ordinary differential operators A general ring of pseudo-differential operators A classical method to deal with a non-linear PDE is to find its Lax pair. In this case it is easier to find solutions. Let me remind that a Lax pair is a pair of operators L(t), P(t) dependent on time and acting on a fixed (Hilbert) space, and verifying the Lax s equation: L t = [P, L]. Usually P depends on L in a prescribed way, so this is a nonlinear equation for L as a function of t. It can then be shown that the eigenvalues and more generally the spectrum of L are independent of t. The operators L are said to be isospectral as t varies.

6 Isospectral deformations of ordinary differential operators Isospectral deformations of ordinary differential operators In the case of the KP equation one can proceed as follows. Definition We say {P(t) t M}, where M is an open domain of C N, is a family of isospectral deformations if there exist ordinary differential operators Q 1 (t), Q 2 (t),..., Q N (t) depending on the parameter t M analytically such that the following system of equations has a nontrivial solution ψ(x, t; λ) for every eigenvalue λ of P(t): P(t)ψ(x, t; λ) = λψ(x, t; λ) t 1 ψ(x, t; λ) = Q 1 (t)ψ(x, t; λ)... t N ψ(x, t; λ) = Q N (t)ψ(x, t; λ) (1)

7 Isospectral deformations of ordinary differential operators A general ring of pseudo-differential operators One can show then that the defining equation for a universal family of isospectral deformations is the KP system, which can be defined using the formal language of pseudo-differential operators as follows. We consider an algebra R with a derivation : R R (ab) = (a)b + a (b), a, b R. Here R can be thought of as a ring of functions (analytical complex or real and with prescribed properties, etc.). We construct a ring R(( 1 )) : a i i, a i R 1 a = a 1 + ( 1 1 i + [, a] = (a) ) (a) 2 + ( 1 2 ) 2 (a) ,

8 Isospectral deformations of ordinary differential operators A ring of pseudo-differential operators in one variable: decomposition where ( ) i = k i(i 1)... (i k + 1), if k > 0. k(k 1)... 1 Let s consider for simplicity the ring R = C[[x]] with usual derivation (x) = 1. We add infinite number of formal variables: t 1, t 2,.... There is a unique decomposition in the ring E = R[[t 1, t 2,...]](( 1 )): if A E, then A = A + + A, where A + R[[t 1, t 2,...]][ ], A R[[t 1, t 2,...]][[ 1 ]] 1.

9 Isospectral deformations of ordinary differential operators KP-hierarchy Let L R[[t 1, t 2,...]](( 1 )) be of the following type: L = + u u , u i R[[t 1, t 2,...]]. Definition The classical KP-hierarchy is the following infinite system of equations (a generalized Lax equation): L t n = [(L n ) +, L], n N.

10 Isospectral deformations of ordinary differential operators Partial equations of the KP-hierarchy: KP and KdV equations From this system it follows the KP equation (4u t u 12uu ) = 3u yy where u(t, x, y) = u 1, x = t 1, y = t 2, t = t 3, and the KdV equation 4u t u 12uu = 0 if L 2 = 2 +u. The KP hierarchy can be considered as a dynamical system on the infinite-dimensional manifold: the Sato Grassmanian.

11 Sato Grassmanian Sato Grassmanian Definition A C-subspace W in C((u)) is called a Fredholm subspace if dim C W C[[u]] < and dim C C((u)) W + C[[u]] <. Remark One the set of Fredholm subspaces of C((u)) one can define the structure of infinite-dimensional projective algebraic variety, the Sato Grassmanian Gr(C((u))). Since W C((u)), we can consider the stabilizer subring A C((u)): A W W. We define its rank ranka = GCD{ν u (a) a A}, where ν u is a discrete valuation on the field C((u)).

12 Geometric solutions Geometric solutions of the KP-hierarchy Some solutions of the KP-hierarchy can be obtained geometrically from flows on Picard varieties of algebraic curves (for example, so called finite zone solutions of Krichever). Exactly these solutions give also isospectral deformations of an ordinary differential operator (or of a ring of commuting ordinary differential operators). Namely, there is the following one-to-one correspondence: where (C, P, F, π, φ)/ (A, W )/ B/,

13 Geometric solutions Definitions Definition B D is a commutative C-subalgebra, which is elliptic, i.e. B has a monic element P of order greater than zero. For an elliptic commutative subalgebra B D we define rankb = GCD{ord Q Q B}. B 1 B 2 if there is an invertible element f R such that B 1 = fb 2 f 1.

14 Geometric solutions Schur pairs and geometric data Definition We call (C, P, F, π, φ) a geometric data of rank r if it consists of the following data: C is a reduced irreducible projective curve over C; P C is a regular on C closed point; π : ÔP k[[u]] is a ring homomorphism such that ν(π(f )) = r, where f ÔP is a local equation of the point P. F is a torsion free coherent sheaf on X of rank r satisfying H 0 (C, F) = H 1 (C, F) = 0. φ : F P k[[u]] (a trivialization of F at P)

15 Geometric solutions Schur pairs Definition (A, W ) is a pair such that W Gr + (0) and A k, where Gr + (0) = {W C((u)) W u C[[u]] = C((u))} We say that (A, W ) and (A, W ) are isomorphic if there is an invertible zeroth order operator T E such that (TAT 1, TW ) = (A, W ) and T T 1 k(( 1 )). The rank of a Schur pair is the number rk(a, W ) = rank(a) (the rank of the module W at a generic point of Spec A).

16 Examples Examples of KdV solutions Example If we take the pair W = 1 + t, t i, i 1 A = k[t 2, t 3 ] : then the corresponding ring B is the old known example of Burchnall and Chaundy P = 2 x 2x 2, Q = 3 x 3x 2 x 3x 3 and the corresponding curve is the rational cuspidal curve. The solution of the KdV equation with the initial condition corresponding to this data is the famous rational solution u(x) = (x 1 ) x.

17 Isospectral deformations of partial differential operators We can try to repeat the definition of isospectral deformations also in the case of partial differential operators. But now, if we will try to proceed the arguments from one dimensional case and derive a defining equation for a universal family of isospectral deformations, we will come to a necessity 1) to normalize in certain sense the operators 2) to take a completion of the ring of partial differential operators. After that we really can proceed and get the defining equation a modified Parshin s KP system. Below are the exact definitions.

18 Isospectral deformations of partial differential operators Definitions Definition Let R = C[[x 1, x 2 ]], denote by M = (x 1, x 2 ) the maximal ideal. Then define the ring ˆD 1 = {a = q 0 a q q 1 a q C[[x 1, x 2 ]] and for any N N there exists n N such that ord M (a m ) > N for any m n}. (2) Define ˆD = ˆD 1 [ 2 ] D = R[ 1, 2 ].

19 Isospectral deformations of partial differential operators Definitions Definition Define Ê = ˆD 1 (( 1 2 )) ˆD D. This is an associative ring of formal pseudo-differential operators. Denote by Ê ( 1) the algebra ˆD 1 [[ 1 2 ]] 1 2. We have a natural direct sum decomposition as a module. Ê = ˆD Ê ( 1)

20 Isospectral deformations of partial differential operators Generalization of the KP-hierarchy in dimension 2 Definition We consider L, M Ê[[{t k}]] such that L = 1 + u , M = 2 + v , where u i, v i ˆD 1 [[{t k }]]. Let N = (L, M) and [L, M] = 0, then hierarchy is N t k = V k N, where V k N = ([(Li M j ) +, L], [(L i M j ) +, M]), k = (i, j) Z + Z +.

21 Quasi elliptic rings Definitions Definition The ring B ˆD of commuting operators is called quasi elliptic if it contains two monic operators P, Q such that ord Γ (P) = (0, k) and ord Γ (Q) = (1, l) for some k, l Z, where we say that an operator P Ê+ has order ord Γ (P) = (k, l) if P = l s= p s s 2, where p s ˆD 1, p l k[[x 1, x 2 ]][ 1 ] = D 1, and ord(p l ) = k.

22 Schur pairs Schur pairs Definition Denote by ν t the discrete valuation on the field C((u))((t)) with respect to t. Denote by ν u the discrete valuation on the field C((u)). They form a rank two valuation ν = ord Γ on the field C((u))((t)). For the ring A k[[u]]((t)) define N A = GCD{ν t (a), a A, ν(a) = (0, )}, Ñ A = GCD{ν t (a), a A}. We ll say that the ring A is strongly admissible if ÑA = N A and if there is an element a A with ν(a) = (1, ). The same notion can be defined also for rings of commuting operators B ˆD.

23 Schur pairs Schur pairs Definition A pair (A, W ), where A, W k[[u]]((t)), is said to be a Schur pair of rank r if the following conditions are satisfied: 1 A is a ring with unity, A W W, Supp(W ) = u i t j i, j 0, i j 0, where Supp(W ) = HT (w) w W and HT (w) is the highest term of the series w. 2 A is a strongly admissible ring and N A = r. We say that (A, W ) and (A, W ) are isomorphic if there is an operator T 1 + Ê such that (TAT 1, TW ) = (A, W ) and T 1 T 1, T 2 T 1 C(( 1 1 ))(( 1 2 )).

24 Geometric data Geometric data Definition We call (X, C, P, F, π, φ) a geometric data of rank r if it consists of the following data: X is a reduced irreducible projective algebraic surface defined over a field C; C is a reduced irreducible ample Q-Cartier divisor on X; P C is a regular on C and on X closed point; π : ÔP k[[u, t]] is a ring homomorphism such that ν(π(f )) = (0, r), ν(π(g)) = (1, 0), where f O P is a local equation of the curve C in a neighbourhood of P, and g O P restricted to C is a local equation of the point P on C.

25 Geometric data F is a torsion free quasi-coherent sheaf on X. φ : F P k[[u, t]] is a ÔP-module embedding such that the compositions of natural homomorphisms H 0 (X, F(nC )) F P φ k[[u, t]] k[[u, t]]/(u, t) ndr+1, where C = dc is a very ample divisor, are isomorphisms for any n 0. Theorem or rank r. (X, C, P, F, π, φ)/ (A, W )/ B/

26 Examples Example Example Let s give one example of a pair of commuting operators in the ring ˆD connected with one of the simplest points from the analogue of the Sato Grassmanian. Let s take a subspace W = 1 + t, t i u j, i 1, 0 j i C[[u]]((t)). One can easily check that its ring of stabilizers contains elements t 2, t 3, ut 2. The maximal ring of stabilizers will be infinitely generated over C. The Schur pair (W, A) with a finitely generated ring A containing the elements above corresponds to a geometric data with a surface being singular toric surface.

27 Examples Example P = (1 x 2 ) 2 (: exp( x 1 1 ) :), Q = x 2 (: exp( x 1 1 ) :) 1, P = (1 x 2 ) 2 (: exp( x ) :) 2 3 (1 x 2 ) 3 (: exp( x 1 1 ) :). where (: exp( x 1 1 ) :) = 1 x x /2! x /3! +....

28 Examples Equations Example If we derive equations of isospectral deformations of the operators above, we obtain the following equations: s 1 t 1 = 1 4 (s 1) x2 x 2 x (s 1) 2 x 2, s 1 t 2 = (s 1 ) x2 (s 1 ) x1 1 2 (s 1) x2 x 2 1, s 1 t 3 = (s 1 ) 2 x 1 (s 1 ) x1 x 2 1 (s 1 ) x2 2 1, where s 1 (x 1, x 2, t 1, t 2, t 3 ) = s 1 (t) is the first coefficient of the operator S(t) = 1 + s 1 (t) , and S(0) = S is the conjugating operator: P = S 2 2S 1. (3)

29 Examples Notably s 1 (0) = 1 1 x 2 (: exp( x 1 1 ) :) is a solution of the equations above. This corresponds to the following fact from one-dimensional KP theory: the function u(x) = (x 1 ) x is the rational solution of the KdV equation (and this function is the halved coefficient of the operator P in example below).

30 Examples Final remark A simple analysis of equations above show that even if we start with a commutative ring of partial differential operators (what means that s 1 (0) k[[x 1, x 2 ]][ 1 ] = D 1 ), the isospectral deformations will not be partial differential operators, but operators in ˆD, since s 1 (t) / D 1 for general t. This situation is similar to the problem of describing commutative rings of ordinary differential operators with polynomial coefficients in dimension one. In one dimensional KP theory, if we start with a commutative ring of ordinary differential operators with polynomial coefficients, its isospectral deformations (which are connected with solutions of the KP equation) will consist of operators with not polynomial coefficients though they will still be ordinary differential operators.

NORMALIZATION OF THE KRICHEVER DATA. Motohico Mulase

NORMALIZATION OF THE KRICHEVER DATA Motohico Mulase Institute of Theoretical Dynamics University of California Davis, CA 95616, U. S. A. and Max-Planck-Institut für Mathematik Gottfried-Claren-Strasse