ON THE ASYMPTOTICS OF THE REAL SOLUTIONS TO THE GENERAL SIXTH PAINLEVÉ EQUATION

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1 ON THE ASYMPTOTICS OF THE REAL SOLUTIONS TO THE GENERAL SIXTH PAINLEVÉ EQUATION HUIZENG QIN AND YOUMIN LU Received 9 May 006; Revised 3 September 006; Accepted October 006 We study the general sixth Painlevé equation, develop, and justify the existence of several groups of asymptotics of its real solutions. Our methods also justify the differentiability of the asymptotics. Particular attention is paid to the solutions between 0 and. We find the asymptotics of all real solutions between 0 and of the sixth Painlevé equationas x. Copyright 006. All rights reserved.. Introduction The mathematical and physical significance of the six Painlevé transcendents has been well established. In the last 0 to 30 years, many mathematicians have spent dramatic effort on studying the properties of these transcendents. Although it is the most complicated one among the six Painlevé equations, there have been many results about the sixth Painlevé transcendent. In fact, the asymptotics problem of the sixth Painlevé transcendent has been studied in many papers such as [,, 4 7, 9 ], and the connection problem is also studied in the papers [, 4 7, 0, ]. In this paper, we study the general sixth Painlevé equation d y dx dy dx y y yy y x x x y x α βx γx y y x x y x, δxx y x dy dx PVI where α, β, γ, andδ are parameters. Heuristically, if y is a small solution of PVI, the following equation truncated from PVI would be its major part as x : d y dx dy dx y dy βy y x dx x y γy x y.. International Mathematics and Mathematical Sciences Volume 006, Article ID 6956, Pages 0 DOI 0.55/IJMMS/006/6956

2 Asymptotics of the real solutions to PainlevéVI Letting t lnx and applying elementary techniques to., one may solve it to get three different solutions: y A B sinalnx b,. where A /β γ/a and B /β γ/a β/a ; y b xa B b x a A,.3 where A / β γ/a and B β/a / β γ/a ;and y β γ lnx b β β γ..4 It is reasonable to expect that some solutions of PVI take.,.3, or.4 as their asymptotics. Indeed, many authors [4 7, 9 ] have obtained the corresponding asymptotics as x 0 that can be used to obtain.,.3, and.4 by applying the wellknown symmetry transformations. In this paper, we will prove the following theorems. The differences of our results are pointed out following each theorem. Theorem.. Let β>0 and γ<0. Ifx 0 >, 0 <y 0 <, andy is a solution of PVI with yx 0 y 0, then 0 <y< for all x>x 0 and it satisfies, as x, y A B cosalnx bo x, y abx sinalnx bo x,.5 where A /β γ/a and B /β γ/a β/a. It is clear that the parameters need to satisfy the condition /β γ/a β/a. The complex form of the asymptotics as x 0 corresponding to this asymptotics has been obtained in many papers [7, 0], but our result provides the conditions on the coefficients of the equation for the real solutions to exist, together with the bound 0 <y<. Our proof of this theorem is also elementary and simple. Theorem.. Equation PVI has a group of solutions with the following asymptotics: y b xa A O x 3/a /, y ab xa O x 3/a 3/, as x,.6 where 0 <a</4 or 3/4 <a<,anda / β γ/a.

3 H. Qin and Y. Lu 3 We have noticed that it makes more sense for this result to be true when 0 <a<. But, it seems to be impossible to prove it using our method. This asymptotics is actually awell-knownresult[4 6, 9 ], but our result here also estimates the second term of the leading behavior of the solution as well as the differentiability of the asymptotics. Theorem.3. If γ β 0, then PVI has a group of solutions with the following asymptotics: y β γ lnx b β β γ, y β γx lnx b, as x..7 Various forms of this result occur in the literature. For example, in [7] Guzzetti has the following result for x 0: { θ x x θ [ 0 lnx 4r θ ] } 0 yx 4 θ0 θ 0 θx θ0, θ θx 0 ±θ x, xr ± θ 0 lnx, θ 0 ±θ x..8 The coefficients θ 0 and θ x come from the isomonodromy deformation theory and r is a free complex parameter. Applying the symmetry transformations yx xzt andx t to this result, our result in Theorem.3 can be obtained. It is well known that the transformation y x/z transforms PVI to itself with α,β,γ,δ changed to β, α,/ δ,/ γ. Hence, based on the previous theorems, one may easily obtain the following one. Theorem.4. Equation PVI has solutions with the following asymptotics: y x A B sinalnx b, as x, provided that α<0, δ>,.9 where A / α δ/a and B / α δ/a α/a,and y x / α δlnx b, as x, provided that α δ..0. Proof of Theorems. and.3 In this section, we use the classical successive approximation method to prove Theorem.. The proof of Theorem.3 is similar. In fact,shimomura [] studied a more general nonlinear ordinary differential equation, applied the successive approximation method to it, and obtained the result as an application. We first denote the functions.,.3,

4 4 Asymptotics of the real solutions to PainlevéVI and.4asy 0 and substitute t lnx and y y 0 y into PVI. The new equation is d y dy0 dy β y 0 y 0 y0 G y 0, y y H y 0, y y dy K y 0, y where G y 0, y y 0 y 0 γ y0 y dy dy0 y 0 y0 y e t I y 0, y dy e t F y 0, y,t, dy0 y0 y y 0 y0 y β γ y0 y y0 y0 y, H y 0, y y 0 y0 y K y 0, y y 0 y y0 dy0 y 0 y, y 0 y e t y0 y e t, I y 0, y y0 y e t dy 0 y0 y e t e t, F y 0, y,t y0 y e t dy0 y0 y e t e t dy 0. α y 0 y y0 y y 0 y e t e t β y 0 y y 0 y e t y0 y e t γ y 0 y e t ± δ y0 y y0 y e t y 0 y e t.. As a routine, we introducea new functionz by using the standard transformation y y 0 y0 z..3 Now,. ischangeo d z β 3/ y0 y0 γ y 0 / y0 z M y 0, y z N y 0, y z dz P y 0, y dz e t Q y 0, y z e t I dz y 0, y e y t 0 y0 F y 0, y,t,.4

5 H. Qin and Y. Lu 5 where M y 0, y y 0 y0 G y 0 y 0, y y 0 y0 H y 0, y y0 N y 0, y y 0 y0 H y 0, y y 0 y 0 y0 K y 0, y, 4y 3/ 0 y 0 3/ K y 0, y, P y 0, y y 0 y0 K y 0, y, Q y 0, y y 0 y 0 y0 dy 0 I y 0, y..5 To prove Theorem., we assume that y 0 takes the function in.3. Then, there exist constants L and t 0 such that, for a <, t>t 0,and y y 0, the following estimates are true: M y 0, y <L, P y 0, y <L, Q y 0, y <e at L, N y 0, y <L, F y 0, y,t y 0 y0 <eat L, I y 0, y <e at L..6 We convert.4 into the followingintegral equation: z z t t τ R τ,z τ,z τ dτ,.7 where z z, z z,and Rτ,z,z γ y0 / y0 β 3/ y 0 zm y 0, y z N dz y 0, y z P dz y 0, y y 0 e t Q y 0, y z e t I dz y 0, y e y t 0 y0 F y 0, y,t..8 In order to apply the successive approximation method to.7, we rewrite it as Zt L t,zt,.9

6 6 Asymptotics of the real solutions to PainlevéVI where Zt z z andlt,τ,z is the right-hand side of.7. Now, we can define the sequence Z t 0, Z n t L t,z n t, n 0,,,....0 We first take care of the case when a < /4. Let t 0 be large enough such that, when t t 0, y 0 y0 e / a/t <, 5L a e / a/t < 6, 4e at < 6, 8Le at < 6, 4L 3a ea t < 6, a e at < 6, 8L 9a ea t 6.. Then, Z t τ t e τ t y 0 y0 F y 0,0,τ dτ L a e at ea/ /t, fort t 0, Z t Z 0 t ea/ /t, fort t 0.. Assume that Z n t ea/ /t, Z n t Z n t n e a/ /t, fort t 0..3 Then, for t t 0, Z n t τ t 5Le a τ e a/τ Le 3/a τ dτ t ea/ /t..4

7 H. Qin and Y. Lu 7 Using the mean value theorem, we can also get Z n t Z n t τ t e aτ Le aτ 3Le a τ Z n τ Z n τ dτ t n τ t e a/τ Le a/τ 3Le 3/a τ dτ t n e a/ /t..5 Therefore, the sequence {Z n t} converges uniformly to Ztand Zt ea/ /t t t 0..6 Hence, we have proved that PVI has a solution satisfying y b eat A O e 3/a /t b xa A O x 3/a /, y ab xa O x 3/a 3/, as x,0<a<.7 4. Applying the transformation y x/z to PVI and using the result we have obtained, wecangettheasymptoticsfor3/4 <a< and finish the proof of Theorem.. 3. Proof of Theorem. We can easily prove Theorems. and.3 using the successive approximation method since the corresponding homogeneous equation is easy to solve. When y 0 takes the expression in., the corresponding homogeneous equation to.4 becomes one of the famous Hill equations [3] whose solutions are very hard to analyze. Thus, we have difficulties to apply the successive approximation method to this case. Fortunately, we can manage to manipulate PVI a littlebit and apply a method usedby Hastingsand McLeod [8]toit. We first prove the first part of the theorem. Suppose that yx 0forsomex >x 0. Since yx is analytic near x, we have the expansion yx c x x n O x x n, 3. where c 0andn>0. Substituting 3.intoPVI,wegettheequation cnn n n x x O x x c n x x n O x x n 3. β c x n n x x O x x.

8 8 Asymptotics of the real solutions to PainlevéVI Thus, we have n andc/β/cx 0. This is impossible when β>0 and therefore, yx > 0forallx>x 0. Similarly, we can prove that yx < forallx>x 0 when γ<0. This result enables us to assume that y is a solution between 0 and in this section. We first apply the transformation to PVI and obtain d y y dy y { e t ye t β y y dy t lnx 3.3 γy y y e t ye t dy αyy ye t e t βy y e t y e t γy e t δyy e t ye t }. 3.4 Since 0 <y<, we can rewrite 3.4into { d [y ] / dy y [ β y y y y } y / / dy y γy ] dy y e t y y Qy,tdy. 3.5 Integrating both sides of 3.5, we get dy β y y y γ y C y y O e t. 3.6 Since β>0, γ<0, and β/y γ/ y dominates /y yoe t whent is large, C a > 0, /y,/ y, and dy/ are all bounded as t goes to infinity. Multiplying both sides of 3.6byy y and letting y z r where r is a constant to be determined later, we get dz a z ra β γ a z β β γ a r a r O e t. 3.7 We select r /β γ/a A,then To solve 3.8, we let dz a z a B O e t. 3.8 zt ρtcosφt, z t aρtsinφt. 3.9

9 H. Qin and Y. Lu 9 We are using two functions ρt andφt inzt andz t to describe the relationship of the function zt and its derivative. Because of this relationship, one of ρt andφt should be depending on another as the product rule of derivatives prescribes. Substituting 3.9into3.8, one first gets ρ t B O e t. 3.0 Following 3.8, we may also get dφ a a z z a z ρ a O e t. 3. t Integrating 3. and combining the result with 3.9 and3.0, one gets zt B O e t cos at b O e t, z t a B O e t sin at b O e t, 3. and finishes the proof of Theorem.. Acknowledgment The authors are deeply grateful to the referee for his/her valuable comments and suggestions. References [] P. Boalch, From Klein to Painlevé via Fourier, Laplace and Jimbo, Proceedings of the London Mathematical Society. Third Series , no., [] A. D. Bruno and I. V. Goryuchkina, Expansions of solutions of the sixth Painlevé equation, Doklady Mathematics , no., [3] E. A. Coddington and R. Carlson, Linear Ordinary Differential Equations, SIAM, Pennsylvania, 997. [4] B. Dubrovin and M. Mazzocco, Monodromy of certain Painlevé-VI transcendents and reflection groups, Inventiones Mathematicae 4 000, no., [5] D. Guzzetti, On the critical behavior, the connection problem and the elliptic representation of a Painlevé VI equation, Mathematical Physics, Analysis and Geometry 4 00, no. 4, [6], The elliptic representation of the general Painlevé VI equation, Communications on Pure and Applied Mathematics 55 00, no. 0, [7], Matching procedure for the sixth Painlevé equation, Physics A: Mathematical and General , no. 39, [8] S. P. Hastings and J. B. McLeod, A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de equation, Archive for Rational Mechanics and Analysis , no., 3 5. [9] K. Iwasaki, H. Kimura, S. Shimomura, and M. Yoshida, From Gauss to Painlevé, AspectsofMathematics, vol. E6, Friedr. Vieweg & Sohn, Braunschweig, 99. [0] M. Jimbo, Monodromy problem and the boundary condition for some Painlevé equations, Publications of the Research Institute for Mathematical Sciences. Kyoto University 8 98, no. 3, 37 6.

10 0 Asymptotics of the real solutions to PainlevéVI [] M. Mazzocco, Picard and Chazy solutions to the Painlevé VI equation, Mathematische Annalen 3 00, no., [] S. Shimomura, A family of solutions of a nonlinear ordinary differential equation and its application to Painlevé equations III, V and VI, the Mathematical Society of Japan , no. 4, Huizeng Qin: Department of Mathematics and Information Science, Shandong University of Technology, ZiBo, Shandong 55049, China address: qinhz 000@63.com Youmin Lu: Department of Mathematics and Computer Science, Bloomsburg University, Bloomsburg, PA 785, USA address: ylu@bloomu.edu

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