FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2004, VOLUME 10, NUMBER 1, PAGES 243-253

On the classification of conditionally integrable evolution systems in $(1+1)$ dimensions

A. Sergyeyev

Abstract

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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+1. For this, we develop a refinement of the technique from our earlier work, where we completely characterized all $(1+1)$-dimensional evolution systems $$u_t = F(x,t,u,¶u/¶x,1,¶ⁿu/¶xⁿ) that are conditionally invariant under a given generalized (Lie--Bäcklund) vector field $$Q(x,t,u,¶u/¶x,1,¶^ku/¶x^k)¶/¶u 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.

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Last modified: October 25, 2004