Lax pair, cyclic basis, and a new integrable system Supriya Mukherjee * , A. Roy Chowdhury High Energy Physics Division, Department of Physics, Jadavpur University, Kolkata 700032, India Accepted 18 June 2003 Communicated by Prof. M. Wadati Abstract A direct approach to zero-curvature representation, as introduced by Marvan is applied to generate a new class of integrable equations by starting with the generic Lax operator (x-part) for a coupled set of nonlinear equations. The class of equation so obtained is more general than those usually obtained with the help of standard recursion operator. Finally some particular type of equations are identified by the special choice of the arbitrary functions occurring in the final solution of the time part of the Lax equation. The methodology is specially useful when the x-part of the Lax operator does not contain any spectral parameter. Ó 2003 Elsevier Ltd. All rights reserved. 1. Introduction Zero-curvature formalism was introduced in the theory of completely integrable systems by Zakharov and Shabat [1]. This is actually a different way of writing the Lax pair for a system of equation. But for a long time the practical approach of obtaining the Lax pair was that of prolongation structure introduced by Wahlquist and Estabrook [2]. The procedure always gives rise to an incomplete Lie algebra which is to be ÔclosedÕ in various algorithmic ways [3]. A generalised version was proposed by KrasilÕshchik and Vinogradov [4] using the concept of spectral sequences. Recently Marvan [5] showed how one can use the Guage invariance of Lax equation, to search for the new type of time part when the corresponding space part is known. At this point one should mention the technique of AKNS [6], who took the advantage of the dependence of the space part on an arbitrary spectral parameter ÔkÕ. But their procedure of expanding the time part in k fails if the space part of the Lax equation does not depend on k. Here, in this communication we have considered a set of coupled dispersive equation, whose symmetry property showed a rich structure of infinite dimen- sional Lie algebra [7], and have applied the technique developed by Marvan and Sakovich [8]. Their procedure of using the cyclic basis for the search of new time part of Lax equation is seen to generate a new class of integrable equations in five dependent variables. The final solution for the time part involves five arbitrary functions which can be chosen in a consistent fashion to reproduce some particular nonlinear problem. 2. Formulation Two important approach for the generation of nonlinear integrable equations are––the theory of prolongation structure, which starts from a given nonlinear partial differential equation and the other one is that of AKNS which starts from the x-part of a Lax operator and generate the time part by assuming the analyticity of the matrix elements in the spectral parameter k. The second approach leads naturally to a recursion operator leading to a heiarchy of * Corresponding author. 0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0960-0779(03)00312-6 Chaos, Solitons and Fractals 19 (2004) 1225–1229 www.elsevier.com/locate/chaos