A simple-looking relative of the Novikov, Hirota-Satsuma and Sawada-Kotera equations

Abstract

We study the simple-looking scalar integrable equation $f_{xxt} - 3(f_x f_t - 1) = 0$, which is related (in different ways) to the Novikov, Hirota-Satsuma and Sawada-Kotera equations. For this equation we present a Lax pair, a B\"acklund transformation, soliton and merging soliton solutions (some exhibiting instabilities), two infinite hierarchies of conservation laws, an infinite hierarchy of continuous symmetries, a Painlev\'e series, a scaling reduction to a third order ODE and its Painlev\'e series, and the Hirota form (giving further multisoliton solutions).