N-soliton solutions to a -dimensional integrable equation

We give explicitly N -soliton solutions of a (2C 1)-dimensional equation, xtC xxxz=4CxxzCxxz=2C@ 1 x zzz=4 D 0. This equation is obtained by unifying two directional generalizations of the potential KdV equation: the closed ring with the potential KP equation, and the Calogero-Bogoyavlenskij-Schiff equation. This equation is also a reduction of the KP hierarchy. We also find the Miura transformation which yields the same ring of the corresponding modified equations. The study of higher-dimensional integrable systems is one of the central themes in integrable systems. A typical example of a higher-dimensional integrable system is obtained by modifying the Lax operators of a basic equation, the potential KdV (p-KdV) equation in this paper. The Lax pair of the p-KdV equation have the form L.x;t/D@ 2 xCx.x;t/ (1) T.x;t/D.L.x;t/ 3 2/