1303.3575 (Sergey Igonin)

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

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.

View original: http://arxiv.org/abs/1303.3575