DIPS 4/2004 [tex source, PostScript, PDF file, dvi]

Title: Towards classification of conditionally integrable evolution systems in (1+1) dimensions.

Author: A. Sergyeyev

We generalize earlier results of Fokas and Liu and find all locally analytic (1+1)-dimensional evolution equations of order n that admit an N-shock type solution with N< n+2. To this end, we develop a refinement of the technique from our earlier work (A. Sergyeyev, J. Phys. A: Math. Gen, 35 (2002), 7653-7660), where we completely characterized all (1+1)-dimensional evolution systems u_t=F(x,t,u,pu\p x,...p^nup x^n) that are conditionally invariant under a given generalized (Lie-Baecklund) vector field Q(x,t,u,pup x,...,p^kup x^k)p/pu under the assumption that the system of ODEs Q=0 is totally nondegenerate. Every such conditionally invariant evolution system admits a reduction to a system of ODEs in t, thus being a nonlinear counterpart to quasi-exactly solvable models in quantum mechanics.

11 pages, AmS-LaTeX (LaTeX-2e).

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