On continual classes of evolution equations

TL;DR

The paper demonstrates that the original Miura transformation, $u = v_1 - \tfrac{1}{2} v^2$, extends beyond the KdV hierarchy to a continual class of evolution equations, including nonintegrable cases. It presents two complementary characterizations: (i) a direct derivation of all local evolution equations admitting the transformation independent of Lie-Bäcklund algebras, and (ii) derivation from a gauge-invariant zero-curvature representation with a fixed $x$-part. The main result is that any local evolution equation $u_t = f(x,t,u,u_1,\dots,u_n)$ admitting the transformation must satisfy $f = (D_x^3 + 2 u D_x + u_1) p$, with $p = p(x,t,u,u_1,\dots,u_{n-3})$ arbitrary; hence the continual class is $u_t = (D_x^3 + 2 u D_x + u_1) p(x,t,u,u_1,\dots,u_{n-3})$ of arbitrary order $n$. The same class is recovered from a gauge-invariant ZCR with fixed $x$-part using the cyclic basis method, yielding $f = D_x^3 p + 2 u D_x p + u_1 p$ with $p$ independent of $v$, so the order remains undetermined; the work also notes a potential link to pseudosymmetries and that not every Miura-type substitution exists.

Abstract

The original Miura transformation, considered as a nonlinear potential transformation, is applicable to a continual class of evolution equations, not only to discrete integrable equations and their hierarchies. The same continual class of evolution equations appears from a different problem, namely, from the gauge-invariant description of a zero-curvature representation with a definite x-part containing no essential parameter.