ON THE ASYMPTOTIC REPRESENTATION OF THE SOLUTIONS TO THE FOURTH GENERAL PAINLEVÉ EQUATION

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1 IJMMS 003:13, PII. S Hindawi Publishing Corp. ON THE ASYMPTOTIC REPRESENTATION OF THE SOLUTIONS TO THE FOURTH GENERAL PAINLEVÉ EQUATION YOUMIN LU Received 5 June 00 There have been many results on the asymptotics of the Painlevé transcendents in recent years, but the asymptotics of the fourth Painlevé transcendent has not been studied much. In this note, we study the general fourth Painlevé equation and develop an asymptotic representation of a group of its solutions as the independent variable approaches infinity along a straight line. 000 Mathematics Subject Classification: 34E Introduction. Asymptotic behaviour of the second and the third Painlevé transcendents has been much studied [1, 3, 4, 7]. But there are very few results about the fourth Painlevé equation d η dξ = 1 dη 3 η dξ η3 4ξη ξ α η β η, 1.1 where α and β are parameters. In [], Clarkson and McLeod studied the special case of 1.1 for the parameter β = 0 and obtained the complete asymptotic representation of its solution with η = 0. In [6], we studied the general case of 1.1 and proved the following theorem. Theorem 1.1. Under the assumption that iα 0 and β>0, 1.1 has a solution ηξ with the following asymptotics: i as ξ e πi/4, ηξ = ir ξ ± i r β ξ O ξ cosφ, 1. η ξ =± r β O ξ sinφ, where φ = iξ 1/3r iαlog iξ θ 0 Oξ, and r>0 and θ 0 are real parameters satisfying r β/ > 0; ii as ξ e πi/4 0, ηξ = e πi/4 abe πi/4 ξ O ξ, 1.3 where a and b are real parameters with a>0.

2 46 YOUMIN LU Furthermore, any solution of 1.1 of the form ηξ = e πi/4 Wξ with Wξ 0 has the above asymptotics. A natural question following this result is how this solution behaves when ξ goes to infinity in the opposite direction of the line. As we know, through the transformations ηξ = e πi/4 y, ξ = e πi/4 x, is related to the equation d y dx = 1 dy 3 y dx y 3 4xy x α y β y, 1.5 where α = i α. Applying the transformation yx = xw t, t = x, 1.6 we can obtain the equation w t 1 w = 1 16 t w 3 16 w5 1 w3 1 4 w α 4 t 1 w β t w It is easy to prove that w, w,and1/t w are all bounded as t and when α 0 and β 0. Inspecting 1.7 carefully, we can conclude that there are possibly four cases for the behaviour of w as t : 1 wt 0ast ; wt ast ; 3 wt /3 ast ; 4 w does not have a limit as t. Case 1 has been studied and eliminated in [5]. In this note, we pay our attention to case and develop the asymptotic representation of η corresponding to this case as ξ e 5πi/4.. Development of the asymptotic representation. From now on, we assume that α 0, β 0, and wt ast. We first prove the following theorem. Theorem.1. The function W is bounded as t if w = t 1/ W. Proof. Applying the transformation w = t 1/ W to 1.7, we obtain W = 16 t 3/ 3 16 t W W 9 4 t 1/ W 13 4 t 1 W t 3/ W t W 5 α 4 t 1/ α 4 t 1 W βt 3/ w. 3.1

3 ON THE ASYMPTOTIC REPRESENTATION OF THE SOLUTIONS Multiplying both sides of.1 byw and integrating it, we have W W = t 3/ W 3 t t 5/ W t W 3 t 3 W 3 t 1/ W 3 3 t t 3/ W t 1 W 4 13 t W 4 3 t 3/ W 5 9 t t 5/ W t W 6 1 t 3 W 6 α t 1/ W α t t 3/ W α 4 4 t 1 W α t W 4 β tw β t t w C Clearly, W t = ot, and therefore C 1 = o. Because t 1/ W = o1 as t, t 5/ W is bounded and the following inequalities are true for large and t : 13 t t W 4 9 t t 5/ W 5 1 t W 4, 16 W 3 t 1/ W 3 3 t 3/ W 5 1 W, 3 t t 5/ W W α t t 3/ W W 3 α 4 t 3/ W W..3 Hence, there exists a constant C and a large such that W 1 W 3 16 t W 3 t 3 W 3 t 1 W 4 1 t W t W 6 1 t 3 W 6 α 4 t 1 W α t W 4 3 t t 3/ W W 3 α t t 3/ W W 3 3 t t 3/ W W 3 α t t 3/ W W C From.1, we can claim that lim t Wt. If t 3/ W 3 0 and t 3/ W 0, then every term, especially W, in inequality.4 is bounded. If W is bounded below, the same argument can be applied to obtain that W is

4 4 YOUMIN LU bounded for large t. If W is not bounded below, from inequality.4, we have 1 1 W W W 3 3 α t 3/ W W 3 t 3/ W W C..5 Since W W 3 > α/3 W W when W W is larger than a certain constant, there exists a constant C 3 > 0andC 3 = o such that 3 t W W t 3/ W W 3 C3..6 Now, let I be the right-hand side of.6. Then, I = 3 t 3/ W W 3 3 t 3/ I 3/..7 Integrating both sides of.7, we obtain 1/I 1/ 1/C 1/ 3 3 /t 1/ 0 3 /t 1/ > 0 for large. Thus, I is bounded, and we reach a contradiction. Hence, W is bounded below and the theorem is proved. Theorem.. As ξ along arg ξ = 5π/4, ηξ = ξ ± e πi/4 do ξ 1 cosφ, η ξ = 4e 3πi/4 dξ sinφo ξ 1,. where φ = iξ 3c iαlog iξ φ 0 Oξ 1. Proof. We need to rewrite. to be as follows: W W = t 3/ W 3 t t 5/ W t W 3 t 3 W 3 t 1/ W 3 3 t t 3/ W t 1 W 4 13 t W 4 3 t 3/ W 5 9 t t 5/ W t W 6 1 t 3 W 6 α t 1/ W α t t 3/ W α 4 4 t 1 W α t W 4 β tw β t t w C. 3.9 Then, C = lim t W tw t.ifc = 0, then W 0ast.Inthiscase, we can solve.1 to obtain that Wt α /4t 1/,andW = Ot 3/. Thus, w α /4t 1 and w = 1/t 3/ W t 1/ W = Ot.If C 0, we let Wt= ρtcosφ, W t = ρtsinφ..10

5 ON THE ASYMPTOTIC REPRESENTATION OF THE SOLUTIONS Then, ρ t = C O t 1/..11 Now, we let C = c and find the asymptotics of φ dφ = 1 W W W W W = 1 9 /4t 1/ W 3 13/4t 1 W 4 α /4t 1/ W α/4t 1 W O t 3/ c 3 /t 1/ W 3 α /t 1/ WOt 1 = 1 9 4c t 1/ W 3 α 4c t 1/ W 13 4c t 1 W 4 α 4c t 1 W 7 4c 4 t 1 W 6 3 α c 4 t 1 W 4 α 4c 4 t 1 W O t 3/..1 This implies that W behaves like cost roughly. Thus, we need to pay attention to the terms with even power of W which will give significant contribution to the expression of φ when we use integration by parts later. Using.1, we can find the asymptotic representation of in terms of dφ as follows: = 1 9 4c t 1/ W 3 α 4c t 1/ W O t 1 dφ..13 Plugging.13 back to.1, we have dφ = 9 4c t 1/ W 3 α 13 = 9 1 4c t 1/ W 1 9 4c t 1/ W 3 α 4c t 1/ W O t 1 4c t 1 W 4 α 4c t 1 W 7 4c 4 t 1 W 6 3 α c 4 t 1 W 4 α 4c 4 t 1 W O t 3/ 4c t 1/ W 3 α 4c t 1/ W dφ c 4 t 1 W 6 9 α 4c 4 t 1 W 4 α c 4 t 1 W O t 3/ 13 4c t 1 W 4 α 4c t 1 W 7 4c 4 t 1 W 6 3 α c 4 t 1 W 4 dφ α 4c 4 t 1 W O t 3/. dφ.14

6 50 YOUMIN LU By.9 and.10, W = C 3 = c 3 t 1/ W 3 α t 1/ W O t 1 1/ cosφ 4c t 1/ W 3 α 4c t 1/ W O t 1 cosφ..15 Plugging.15 back to.14, we obtain dφ = 9c 1 4 t 1/ cos 3 φ α 4c t 1/ cosφ dφ 7 α c t 1 W cos 3 φ 3 α c 3 t 1 W 3 cosφ c t 1 W 3 cos 3 φ α c 3 t 1 W cosφ dφ 1 c 4 t 1 W 6 9 α 4c 4 t 1 W 4 α c 4 t 1 W O t 3/ dφ 13 4c t 1 W 4 α 4c t 1 W 7 4c 4 t 1 W 6 3 α c 4 t 1 W 4 α 4c 4 t 1 W O t 3/. = 9c 4 t 1/ cos 3 φ α 4c t 1/ cosφ dφ 7c t 1 cos 6 φ 13c 4 t 1 cos 4 φ3 αt 1 cos 4 φo t 3/ dφ = 3c α t 1 9c 4 t 1/ cos 3 φ α 4c t 1/ cosφ 7c t 1 cos 6 φ 5 dφ 16 13c 3 α t 1 cos 4 φ 3 α4 4 t 1 cos φ 1 O t 3/ dφ..16 Therefore, φ = t 3c αlog t φ 0 Ot 1/. Using transformations 1.4, 1.6,.10, and the one in Theorem.1, we get the asymptotic representation in the theorem and finish the proof of the theorem. References [1] A. P. Bassom, P. A. Clarkson, C. K. Law, and J. B. McLeod, Application of uniform asymptotics to the second Painlevé transcendent, Arch. Rational Mech. Anal , no. 3, [] P.A.ClarksonandJ.B.McLeod,Integral equations and connection formulae for the Painlevé equations, Painlevé Transcendents Sainte-Adèle, PQ, 1990, NATO Adv. Sci. Inst. Ser. B Phys., vol. 7, Plenum, New York, 199, pp

7 ON THE ASYMPTOTIC REPRESENTATION OF THE SOLUTIONS [3] A. R. Its and V. Yu. Novokshenov, The Isomonodromic Deformation Method in the Theory of Painlevé Equations, Lecture Notes in Mathematics, vol. 1191, Springer-Verlag, Berlin, 196. [4] A. V. Kitaev, The method of isomonodromy deformations and the asymptotics of solutions of the complete third Painlevé equations, Math. USSR-Sb , no., [5] Y. Lu, Application of connection formula on eliminating plausible asymptotic representation of the Painlevé transcendents, submitted to Appl. Anal. [6], An asymptotic representation of the general fourth Painlevé transcendents, submitted to Int. J. Appl. Math. [7] J. B. McLeod and C. B. Wang, General asymptotics for the Painlevé III transcendent, personal communication. Youmin Lu: Department of Mathematics, Computer Science and Statistics, College of Science and Technology, Bloomsburg University of Pennsylvania, Bloomsburg, PA 1715, USA address: ylu@bloomu.edu

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