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

A. Sergyeyev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 10 (2004) no. 1, pp. 243-253.

Résumé

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 $N\le n+1$. For this, we develop a refinement of the technique from our earlier work, where we completely characterized all $(1+1)$-dimensional evolution systems ${\mathit{u}}_t=\mathit{F}(x,t,\mathit{u},\partial \mathit{u}/\partial x,\dots ,{\partial }^n\mathit{u}/\partial x^n)$ that are conditionally invariant under a given generalized (Lie–Bäcklund) vector field $\mathit{Q}(x,t,\mathit{u},\partial \mathit{u}/\partial x,\dots ,{\partial }^k\mathit{u}/\partial x^k)\partial /\partial \mathit{u}$ under the assumption that the system of ODEs $\mathit{Q}=0$ 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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