By I. S. Novikov

This booklet considers the speculation of 'integrable' non-linear partial differential equations. the idea was once constructed initially via mathematical physicists yet later mathematicians, fairly from the Soviet Union, have been interested in the sector. during this quantity are reprinted a few primary contributions, initially released in Russian Mathematical Surveys, from many of the best Soviet staff. Dr George Wilson has written an advent meant to tender the reader's direction via many of the articles.

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**Example text**

E~ y £ { | * - * i l + | x i - * 2 | + . . + |xjv-*l> w / a ,) N+l J J w(a: A ')da: 1 . . V iv/ 1 dxN. iv To find an asymptotic expansion of this expression, we represent u(x;) as a series 32 /. M. GeVfand and L A. Dikii We obtain the asymptotic form as y y ( — if — Zi - ZJ v l > N=0 (fc) Z N. where (4) j ... J The coefficients are non-zero only if kx + . . iM=2 2 N=lhi+ ... uik»). " This is, in fact, a polynomial in u, u' .. , and is homogeneous in the grading kx + . . + kN + 2N, that is, in the sum of twice the degree TV in the variables u, u\ .

MR 15-720. [7] L. D. Fadeev, An expression for the trace of the difference between two singular differential operators of Sturm-Liouville type, Dokl. Akad. Nauk SSSR 115(1957), 878-880. MR 20-1029. [8] V. S. Buslaev and L. D. Faddeev, On trace formulae for a singular differential SturmLiouville operator, Dokl. Akad. Nauk SSSR 132 (1960), 13-16. = Soviet Math. Dokl. 1 (1960), 451-454. [9] R. T. Seeley, The index of elliptic systems of singular integral operators, J. Math. Anal. Appl. 7 (1963), 289-309.

Therefore, the quantities (11) must be invariants of the equation, because they are the coefficients of the asymptotic expansion of the trace of the resolvent and therefore expressible in terms of the spectrum. How can such an operator A be constructed? One of the possible ways is given in [15]. ), which is most In an infinite-dimensional space we can introduce a symplectic structure by means of the 2-form (see [13]): oo x o) (6a, 6i>) = f f [6M (*) 6v fa) - bu fa) 6v (*)] dxi dx. — oo — oo Then for any functional H = \ H [u] dx as Hamiltonian, we can construct the Hamiltonian equation J n+1 cfc a 6 ut = — —- H.