Friday, March 15, 2013

1303.3575 (Sergey Igonin)

Lie algebras responsible for zero-curvature representations of scalar
evolution equations

Sergey Igonin
Zero-curvature representations (ZCRs) are well known to be one of the main tools in soliton theory. In particular, Lax pairs in the (1+1)-dimensional case can be interpreted as ZCRs. For any (1+1)-dimensional scalar evolution equation, we define a sequence of Lie algebras F^n, n=0,1,2,3,..., which classify all ZCRs of this equation up to gauge transformations. The algebra F^n classifies ZCRs whose x-part depends on jets of order not greater than n. We prove some results on generators of F^n. This allows us to compute the explicit structure of F^n for some examples. In particular, we study the structure of F^n for equations of the form u_t=u_{2q+1}+f(x,t,u,u_1,...,u_{2q-1}) for all q>0, which include KdV, Kaup-Kupershmidt, Sawada-Kotera type equations. Here u_k is the k-th derivative of u=u(x,t) with respect to x. For such equations, it is shown that the algebra F^n is isomorphic to a central extension of the algebra F^{n-1} for all n>2q-2. This result allows one to obtain necessary conditions for integrability of such equations. Some applications to classification of scalar evolution equations with respect to Backlund transformations are also discussed.
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