No abstract

and the coefficients A, . . . , Q are analytic functions of x and t in some domain. We supplement the paper [1] by analyzing Eq. (1) from the viewpoint of the Painlevé property. According to [2], an equation has the Painlevé property if all movable singular points of its general solution (if any) are poles.Consider the simplified equation for (1):If Eq.(2) has the Painlevé property, then its solution can be represented by the serieswhereand the equation ϕ(x, t) = 0 specifies a noncharacteristic singular manifold; moreover, ϕ t and two resonance coefficients should be arbitrary functions of t. By substituting the series (3) into Eq. (2) and by matching the coefficients of ϕ r−4 , we obtain the equationwhence we obtain the resonance numbers r = −1, 1, 6. Therefore, u 0 and u 5 are the resonance coefficients. (The resonance number r = −1 corresponds to the arbitrary choice of the function ϕ t .) Let us show that there is no functional relationship between u 0 , u 5 , and ϕ t . By matching the coefficients of like powers of ϕ on the left-and right-hand sides in Eq.(2), we find that the first resonance condition is valid automatically. Next, we obtainwhich implies the relation u 4 − u 1 u 2 = 0.The second resonance condition has the form 3 (u 4 ) t + u 1 ((u 2 ) t + 3u 3 ϕ t ) + 2u 2 ((u 1 ) t + 2u 2 ϕ t ) + 3u 3 ((u 0 ) t + u 1 ϕ t ) = 0,

No abstract

No abstract

No abstract