Conservation laws for a complexly coupled KdV system, coupled Burgers’ system and Drinfeld–Sokolov–Wilson system via multiplier approach R. Naz Department of Mathematics, Faculty of Basic and Applied Sciences, International Islamic University, Islamabad, Pakistan article info Article history: Received 1 March 2009 Received in revised form 15 April 2009 Accepted 31 May 2009 Available online 2 July 2009 PACS: 11.30.j Keywords: Multiplier Complexly coupled KdV system Coupled Burgers’ system Drinfeld–Sokolov–Wilson system abstract The multiplier approach (variational derivative method) is used to derive the conservation laws for some nonlinear systems of partial differential equations. Firstly, the multipliers (characteristics) are computed and then conserved vectors are obtained for the each mul- tiplier. Examples of the third-order complexly coupled KdV system, second-order coupled Burgers’ system and third-order Drinfeld–Sokolov–Wilson system are considered. For all three systems the local conservation laws are established by utilizing the multiplier approach. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction The conservation laws are important in the solution and reductions of partial differential equations. In the last few dec- ades, active research efforts were focused on the derivation of conservation laws for partial differential equations. Many powerful methods have been developed for the construction of conservation laws, such as the Noether’s theorem [1] for var- iational problems, Laplace’s Direct method [2], characteristic form introduced by Stuedel [3], multiplier approach [4,5], Kara and Mahomed [6] symmetry condition, partial Noether approach [7]. The computer packages for the direct method and char- acteristic methods were developed by Wolf [8] and Wolf et al. [9]. Göktas and Hereman [10] and Hereman et al. [11–13] developed powerful software packages to compute conservation laws for partial differential equations. Cheviakov [14] dis- covered Maple code to compute conservation laws based on the multiplier approach. All different approaches to construct conservation laws were discussed by Naz et al. [15]. The multiplier approach (also known as variational derivative method) was proposed by Stuedel [3] who wrote the con- servation law in characteristic form as D i T i ¼ K a E a . Later, Olver [4] modified the method of determining the characteristics (multipliers) by taking the variational derivative of D i T i ¼ Q a E a for the arbitrary functions not only for solutions of system of partial differential equations. It was successfully applied to the construction of conservation laws for partial differential equations, for the two-dimensional and radial jets [16], for the nonlinear field equation describing the relaxation to a Max- wellian distribution and the nonlinear diffusion equation for the spreading of an axisymmetric thin liquid drop [15]. 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.05.071 E-mail address: rehananaz_qau@yahoo.com Commun Nonlinear Sci Numer Simulat 15 (2010) 1177–1182 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns