Enlarged spectral problems and nonintegrability

TL;DR

The paper investigates Ma's method of enlarging spectral problems for integrable equations and shows this approach can produce nonintegrable coupled systems, demonstrated via the enlarged KdV spectral problem whose ZCR couples the original equation to extra variables, constructed as $X = U A00$ and $T = V B00$. With a fixed X and a specially chosen T, the ZCR yields a one-parameter class of triangular KdV-type systems, e.g. $u_t= u_{xxx}-6u u_x$, $p_t=4p_{xxx}-(k+2)u p_x$, $q_t=4q_{xxx}-6u q_x-3q u_x+(k-4)u p_{xx}+(k-1)u_x p_x$, and a Painlevé test shows many integer values of $k$. Using the cyclic basis method they obtain the complete class of local evolution systems with fixed X and any traceless block-form T, derive a recursion operator $R=4 M N^{-1}$ (the Gurses-Karasu operator) for the integrable GK-type system $u_t = u_{xxx}-6u u_x$, $v_t = 4v_{xxx}-6u v_x-3v u_x$, and show the continual class depends on an arbitrary function g and is not a discrete hierarchy, with reductions like $u=0$ not guaranteeing integrability. A gauge transformation with $\Psi' = G \Psi$, $X' = G X G^{-1} + (D_x G) G^{-1}$, $T' = G T G^{-1} + (D_t G) G^{-1}$ explains the anomaly: selecting G with p-dependent entries merges two extra variables into $v = p_x + q$, making $X'$ effectively two-variable, and yielding $p_t - g$ vanishing on solutions, i.e., the enlarged Lax pair is gauge-equivalent to a weaker one.

Abstract

The method of obtaining new integrable coupled equations through enlarging spectral problems of known integrable equations, which was recently proposed by W.-X. Ma, can produce nonintegrable systems as well. This phenomenon is demonstrated and explained by the example of the enlarged spectral problem of the Korteweg - de Vries equation.