Decouple a Coupled KdV System of Nutku and Og̃uz

A coupled KdV system with a free parameter proposed by Nutku and Og̃uz is considered. It is shown that the system passes the WTC’s Painlevé test for arbitrary value of the parameter. A further analysis yields that the parameter can be scaled away and the system can be decoupled. Soliton or integrable equations are those systems which have rich structures, such as infinite number of conservation laws, multi-Hamiltonian structures, infinite symmetries, Bäcklund transformations. A important question is how integrable systems couple without loss of integrablity. The first coupled KdV system was proposed by Hirota and Satsuma in 1981 [5]. Since then, many such systems are constructed. We mention here Ito’s system [6], the coupled systems resulted from Drinfeld-Sokolov framework [3], those from the group theoretical approach of Kyoto school [7] and the ones constructed by means of energy dependent Schröedinger operator [1]. Nutku and Og̃uz [9] considered the following system qt = qxxx + 2aqqx + rrx + (qr)x, rt = rxxx + 2brrx + qqx + (qr)x, (1) where a and b are constants. They pointed out this system (1) decouples if a = ±b. It is further shown that subject to a+ b = 1, (2) the system (1) is a bi-Hamiltonian system with two local Hamiltonian structures. (We notice that the dispersionless version of this system with the condition (2) was studied recently by Matsuno [8].) Thus, one has a bi-Hamiltonian system which contains a free parameter. Our motivation is to consider the integrabilty of this system. In general, a bi-Hamiltonian system is supposed to be integrable since it has a infinite number of conserved quantities. However, it is peculiar that a system is integrable with arbitrary value of parameter. We believe that either the system is integrable for a certain value of the parameter or the parameter can be removed. For the system of Nutku and Og̃uz, we will show that it is the latter case. WTC’s Painlevé analysis will be used to explore the system. (see [2] and the references there for Painlevé property and applications)