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International Journal of Partial Differential Equations
Volume 2014 (2014), Article ID 125821, 3 pages
A Note on the Painlevé Property of Coupled KdV Equations
Institute of Physics, National Academy of Sciences of Belarus, 220072 Minsk, Belarus
Received 25 October 2013; Accepted 9 January 2014; Published 19 February 2014
Academic Editor: Michael Grinfeld
Copyright © 2014 Sergei Sakovich. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We prove that one system of coupled KdV equations, claimed by Hirota et al. to pass the Painlevé test for integrability, actually fails the test at the highest resonance of the generic branch and therefore must be nonintegrable.
In Section 6 of their paper , Hirota et al. reported that the system of coupled KdV equations passes the Painlevé test for integrability if and only if the parameter is equal to , or , or . The authors of  pointed out that the cases and of (1) correspond to integrable systems of coupled KdV equations, whereas the problem of integrability of (1) with remains open.
In the present short note, we show that the system (1) with actually does not pass the Painlevé test, and its integrability should not be expected therefore.
2. Singularity Analysis
First of all, let us notice that the -component system (1) can be studied in the form of the following triangular system of two coupled KdV equations: where the new dependent variable is defined by , the new dependent variable is any one of the components , and the subscripts and denote partial derivatives. Indeed, the system (1) is equivalent to the system consisting of the first equation of (2) along with copies of the second equation of (2) with (say). Therefore, in order to check whether or not the system (1) passes the Painlevé test, it is sufficient to consider the two equations in (2) and do not repeat the same calculations for the remaining copies of the second equation of (2).
Setting in (2) and starting the Weiss-Kruskal algorithm of singularity analysis [2, 3], we use the expansions and with and determine branches (i.e., admissible choices of ) together with corresponding positions of resonances (where arbitrary functions of can enter the expansions). The exponents and and the positions of resonances turn out to be integer in all branches. In what follows, we only consider the generic singular branch, where , , is arbitrary, and . This branch describes the singular behavior of generic solutions. There are also two nongeneric branches, but they correspond to the constraints and imposed on the generic branch and do not require any separate consideration therefore.
Substituting the expansions with into the system (2) with , we obtain the following recursion relations for the coefficients and of (3): where the subscript denotes the derivative with respect to , and for formally.
Now we have to check whether the recursion relations (4) are compatible at the resonances.
The resonance , as always, corresponds to the arbitrariness of the function in .
We have in (4) at , for the chosen branch. The function remains arbitrary, which corresponds to the resonance 0.
Setting in (4), we find that , while the function remains arbitrary, and the compatibility condition at the resonance 1 is satisfied.
At and , which are not resonances, we get from (4), respectively,
Setting in (4), we find that while the function remains arbitrary, and the compatibility condition at the resonance 4 is satisfied.
At , which is not a resonance, we find from (4) that
Setting in (4), we obtain while the function remains arbitrary, and the compatibility condition at the resonance 6 is satisfied.
At , which is not a resonance, we get from (4) the following:
The highest resonance in the chosen branch is 8. Setting in the recursion relations (4), we find that the function remains arbitrary, but the compatibility condition at the resonance 8 is not satisfied, and we obtain the following constraint imposed on some of arbitrary functions that appeared at lower resonances:
The appearance of the constraint (11) means that the Laurent type expansions (3) do not represent the general solution of the studied system, and we have to modify the expansion for by introducing logarithmic terms, starting from the term proportional to . This nondominant logarithmic branching of solutions is a clear symptom of nonintegrability. Consequently, the case of the system (2)—and of the system (1), equivalently—fails the Painlevé test.
Let us note, moreover, that the singularity analysis of coupled KdV equations has been addressed in the papers [4, 5], published prior to . In particular, the integrable cases and of the system (1) can be found in  as the systems (vi) and (vii), respectively, which have passed the Painlevé test, whereas the case in Section of  predicts that the system (1) with must fail the Painlevé test for integrability.
The obtained result that the system of coupled KdV equations (1) with actually does not pass the Painlevé test for integrability explains very well why no Lax representation has been proposed as yet for this case of coupled KdV equations.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author is grateful to the anonymous reviewer for valuable comments.
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