Algebraic aspects of nonlinear differential equations by Manin Yu.I. PDF

By Manin Yu.I.

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Example text

Z°*= < U >~ = [ < U ) ~ L°*\-[Zo < 2/ >~ S i n c e t h e r e s i d u e s of t h e commutators l i e i n Imd L°i-% by Lemma 3 . $. According t o t h e d e f i n i t i o n s of S e c . 7, Chap. I , t h i s means t h a t XLi(r) and *Li(s) commute. We shall now clarify the algebraic significance of commutativity. ) let B be some Hamiltonian operator, and let Q, PG^=fc[ay ]. Let H,=(a0, .. ,tt/v-2)> Returning to the conventions •. of Chap. 13. Proposition. If Q, P commute in the Hamiltonian structure with operator B , then XQ takes the minimal d -closed ideal JP, generated by the components of B-= into itself.

Therefore, the a-th element of the column (B+£<) ( l ) X'-^LY («+ * + c ~ ! ) " i ' V * - ^ ' ' »,c>0 In analogy with the computation at the start of the proof, the a-th element of the column X'^-(B + B')WY i s *£-K da' 2 Xbb(c+d)tt^d_xYe= »,cX> d—a+b-l 2b(a+b + c-l)u«J>b+c_2XbYc. »,cX> c) As above, the a - t h element of the column BWX'^M-Y is S abUaU+c-2XbYe . t>,c>0 On the other hand, the matrix y^ald,a+c-iYc 2 , and (B*)lJl = bui%d-i • Kjafrtta+j+e-z^Tj . Ycr~{Bacy= i s equal to The proof of the lemma i s complete.

3 where in place of the equations we will solve jointly the system ut=T(1=0 , where wt is an integral. ut=T(m)u It is just this procedure which distinguishes in an invariant way the class of multisolution and finite-zone solutions of the Korteweg—de Vries equation as already mentioned in the introduction. 18. Examples, a) Let n = 2r , and B = \ E ~ Q \ * where E is the identity matrix of —• go order r . The system of evolution at = B—=- is traditionally called a Hamiltonian system 6u with Hamiltonian P in "canonical coordinates".

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Algebraic aspects of nonlinear differential equations by Manin Yu.I.


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