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IJMMS 2003:13, 845–851 PII. S0161171203206220 http://ijmms.hindawi.com © Hindawi Publishing Corp. ON THE ASYMPTOTIC REPRESENTATION OF THE SOLUTIONS TO THE FOURTH GENERAL PAINLEVÉ EQUATION YOUMIN LU Received 5 June 2002 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. 2000 Mathematics Subject Classification: 34E99. 1. 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 1 dη 2 3 3 β d2 η = + η + 4ξη2 + ξ 2 − α η + , dξ 2 2η dξ 2 η (1.1) where α and β are parameters. In [2], 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 η(ξ) with the following asymptotics: (i) as ξ → eπ i/4 ∞, −2 ir β i 2 cos φ, η(ξ) = r − +O ξ ± ξ ξ 2 (1.2) −2 β sin φ, η (ξ) = ±2 r 2 − + O ξ 2 where φ = iξ 2 − (1/2)(3r + iα) log(−iξ 2 ) + θ0 + O(ξ −2 ), and r > 0 and θ0 are real parameters satisfying r 2 − β/2 > 0; (ii) as ξ → eπ i/4 0, η(ξ) = eπ i/4 a + be−π i/4 ξ + O ξ 2 , where a and b are real parameters with a > 0. (1.3) 846 YOUMIN LU Furthermore, any solution of (1.1) of the form η(ξ) = eπ i/4 W (ξ) with W (ξ) ≥ 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 η(ξ) = e−π i/4 y, ξ = e−π i/4 x, (1.4) 3 β y+ , − y 3 − 4xy 2 − 2 x 2 + α 2 y (1.5) (1.1) is related to the equation 1 d2 y = dx 2 2y dy dx 2 Applying the transformation where α = iα. y(x) = −xw 2 (t), t = −x 2 , (1.6) we can obtain the equation w + t −1 w = α 1 −2 3 5 1 3 1 β t w− w + w − w + t −1 w + 2 3 . 16 16 2 4 4 8t w (1.7) It is easy to prove that w, w , and 1/t 2 w 2 are all bounded as t → +∞ and ≤ 0 and β ≥ 0. Inspecting (1.7) carefully, we can conclude that there when α are possibly four cases for the behaviour of w as t → +∞ : (1) w(t) → 0 as t → +∞; √ (2) w(t) → 2 as t → +∞; (3) w(t) → 2/3 as t → +∞; (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 (2) and develop the asymptotic representation of η corresponding to this case as ξ → e5π i/4 (+∞). 2. Development of the asymptotic representation. From now on, we as√ ≤ 0, β ≥ 0, and w(t) → 2 as t → +∞. We first prove the following sume that α theorem. √ Theorem 2.1. The function W is bounded as t → +∞ if w = 2 + t −1/2 W . √ Proof. Applying the transformation w = 2 + t −1/2 W to (1.7), we obtain W = √ √ 3 −2 9 2 −1/2 2 13 −1 3 2 −3/2 t t W −W − t t W − W − 16 16 4 4 √ √ 2 −1/2 α 15 2 −3/2 4 3 −2 5 α βt −3/2 t t W + t − W − + t −1 W + . 16 16 4 4 8w 3 (2.1) ON THE ASYMPTOTIC REPRESENTATION OF THE SOLUTIONS . . . 847 Multiplying both sides of (2.1) by 2W and integrating it, we have √ W 2 + W 2 = √ 2 −3/2 3 2 t −5/2 3 −2 2 3 t −3 2 t t W − W+ t W dt − t W dt 8 16 t0 16 8 t0 √ √ t 3 2 −1/2 3 3 2 13 −1 4 13 t −2 4 t t W − − W − t −3/2 W 3 dt − t W dt 2 4 8 8 t0 t0 √ √ t t 3 2 −3/2 5 9 2 1 −2 6 1 t t W − − W − t −5/2 W 5 dt − t −3 W 6 dt 8 16 t0 16 8 t0 √ √ 2 −1/2 2 t −3/2 t −2 2 α α α α t + W+ t W dt + t −1 W 2 + t W dt 2 4 4 4 t0 t0 √ t β β 2 dt − − + C1 . 2 3 8tw 2 8 t0 t w (2.2) Clearly, W 2 (t) = o(t), and therefore C1 = o(t0 ). Because t −1/2 W = o(1) as t t → +∞, t0 t −5/2 W dt is bounded and the following inequalities are true for large t0 and t ≥ t0 : √ 9 2 t −5/2 5 1 t −2 4 t W dt + t W dt ≥ t W dt, 16 t0 2 t0 t0 √ √ 3 2 −1/2 3 3 2 −3/2 5 1 2 W2+ W + W ≥ W , t t 2 8 2 √ t √ t 3 2 α 2 t −5/2 |W | − W dt − t −3/2 |W | − W dt 32 t0 8 t0 √ 2 t −3/2 α t ≤− |W | − W . 4 t0 13 8 t −2 4 (2.3) Hence, there exists a constant C2 and a large t0 such that 1 3 −2 2 3 t W + W 2 + W 2 + 2 16 8 t 3 1 t −3 W 2 dt + t −1 W 4 + 8 2 t0 t t0 t −2 W 4 dt t −2 2 1 −2 6 1 t −3 6 α α t W + t W dt − t −1 W 2 − t W dt 16 8 t0 4 4 t0 √ √ 3 2 t −3/2 3 2 t −3/2 α |W | + W dt + |W | + W dt + t t 32 t0 8 t0 √ √ 3 2 t −3/2 α 3 2 t −3/2 t t ≤ |W | − W dt − |W | − W dt + C2 . 32 t0 4 t0 + (2.4) t From (2.1), we can claim that limt→+∞ W (t) ≠ −∞. If t0 t −3/2 W 3 dt ≥ 0 and t −3/2 W dt ≥ 0, then every term, especially W 2 , in inequality (2.4) is bounded. t0 t If W is bounded below, the same argument can be applied to obtain that W 2 is 848 YOUMIN LU bounded for large t. If W is not bounded below, from inequality (2.4), we have √ 3 2 1 3 2 t −3/2 1 t |W | − W dt |W | − W ≤ W 2 ≤ 8 2 32 t0 (2.5) √ 2 t −3/2 α − t |W | − W dt + C2 . 8 t0 Since (|W | − W )3 > −(2α/3)(|W | − W ) when (|W | − W ) is larger than a certain constant, there exists a constant C3 > 0 and C3 = o(t0 ) such that √ 3 2 3 2 t −3/2 t (2.6) |W | − W dt + C3 . |W | − W ≤ 2 t0 Now, let I be the right-hand side of (2.6). Then, √ √ 3 3 2 −3/2 3/2 3 2 −3/2 t t I . I = |W | − W ≤ 2 2 (2.7) √ 1/2 1/2 Integrating both sides of (2.7), we obtain 1/I 1/2 ≥ 1/C3 − 3 2/2t0 + √ 1/2 > 0 for large t0 . Thus, I is bounded, and we reach a contradiction. 3 2/2t Hence, W is bounded below and the theorem is proved. Theorem 2.2. As ξ → ∞ along arg ξ = 5π /4, η(ξ) = −2ξ ± 2eπ i/4 2d + O ξ −1 cos φ, η (ξ) = −2 ∓ 4e3π i/4 2dξ sin φ + O ξ −1 , (2.8) where φ = −iξ 2 + (3c 2 + iα) log(−iξ 2 ) + φ0 + O(ξ −1 ). Proof. We need to rewrite (2.2) to be as follows: √ √ 2 −3/2 3 2 t −5/2 3 −2 2 3 t −3 2 W 2 + W 2 = W+ t W dt − t W dt t t W − 8 16 16 8 √ t √ 13 −1 4 13 t −2 4 3 2 −1/2 3 3 2 W − t −3/2 W 3 dt − t W dt − t t W − 2 4 8 8 √ √ t t 3 2 −3/2 5 9 2 1 −2 6 1 − t t W − W − t −5/2 W 5 dt − t −3 W 6 dt 8 16 16 8 √ √ 2 t −3/2 t −2 2 2 −1/2 α α α α t W+ t W dt + t −1 W 2 + t W dt + 2 4 4 4 √ t dt β 2 β − + C. − 8tw 2 8 t2w 3 (2.9) Then, C = limt→∞ (W 2 (t)+W 2 (t)). If C = 0, then W → 0 as t → +∞. In this case, √ we can solve (2.1) to obtain that W (t) ∼ (α 2/4)t −1/2 , and W = O(t −3/2 ). √ √ 2/4)t −1 and w = −(1/2)t −3/2 W + t −1/2 W = O(t −2 ). If Thus, w ∼ 2 + (α C ≠ 0, we let W (t) = ρ(t) cos φ, W (t) = ρ(t) sin φ. (2.10) ON THE ASYMPTOTIC REPRESENTATION OF THE SOLUTIONS . . . 849 Then, ρ 2 (t) = C + O t −1/2 . (2.11) Now, we let C = c 2 and find the asymptotics of φ dφ dt W W + W 2 W 2 + W 2 √ √ −1 2/4)t−1/2 W+(α/4)t W 2+O t−3/2 −(9 2/4)t−1/2 W 3 −(13/4)t−1 W 4+(α √ √ = −1+ 2/2)t−1/2 W+O(t−1 ) c 2 −(3 2/2)t−1/2 W 3+(α √ √ 2 −1/2 13 α 9 2 −1/2 3 α t W + t W − 2 t −1 W 4 + 2 t −1 W 2 = −1 − 4c 2 4c 2 4c 4c 2 27 3α α − 4 t −1 W 6 + 4 t −1 W 4 − 4 t −1 W 2 + O t −3/2 . 4c c 4c (2.12) = −1 + This implies that W behaves like cos t 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 (2.12), we can find the asymptotic representation of dt in terms of dφ as follows: dt = 1 − √ √ −1 2 −1/2 9 2 −1/2 3 α (−dφ). t W + t W + O t 4c 2 4c 2 (2.13) Plugging (2.13) back to (2.12), we have √ √ √ 2 −1/2 9 2 −1/2 3 α 9 2 −1/2 3 dφ = −dt + − t W + t W 1 − t W 4c 2 4c 2 4c 2 √ −1 2 −1/2 α (−dφ) + t W + O t 4c 2 α 27 3α 13 + − 2 t −1 W 4 + 2 t −1 W 2 − 4 t −1 W 6 + 4 t −1 W 4 4c 4c 4c c 2 α − 4 t −1 W 2 + O t −3/2 dt 4c √ √ 2 −1/2 9 2 −1/2 3 α t W + t W (−dφ) = −dt + − 4c 2 4c 2 −3/2 −1 4 α 2 −1 2 81 −1 6 9α (−dφ) + t W − t W + t W + O t 8c 4 4c 4 8c 4 13 α 27 3α + − 2 t −1 W 4 + 2 t −1 W 2 − 4 t −1 W 6 + 4 t −1 W 4 4c 4c 4c c 2 α − 4 t −1 W 2 + O t −3/2 dt. 4c (2.14) 850 YOUMIN LU By (2.9) and (2.10), √ √ 1/2 2 −1/2 3 2 −1/2 3 α W = C− W + W + O t −1 cos φ t t 2 2 √ √ 2 −1/2 3 2 −1/2 3 α W + W + O t −1 cos φ. = c− t t 4c 4c (2.15) Plugging (2.15) back to (2.14), we obtain √ −1/2 9c 2 −1/2 α dφ = −dt + − t t cos3 φ + cos φ (−dφ) 4 4c 81 −1 3 27α −1 3α + t W cos3 φ − t W cos3 φ − 3 t −1 W 3 cos φ 8c 8c 8c 2 −1 α + 3 t W cos φ (−dφ) 8c −3/2 −1 4 α 2 −1 2 81 −1 6 9α (−dφ) + t W − t W + t W + O t 8c 4 4c 4 8c 4 α 27 3α 13 + − 2 t −1 W 4 + 2 t −1 W 2 − 4 t −1 W 6 + 4 t −1 W 4 4c 4c 4c c 2 α − 4 t −1 W 2 + O t −3/2 dt. 4c √ −1/2 9c 2 −1/2 α = −dt + − t t cos3 φ + cos φ (−dφ) 4 4c 27c 2 −1 13c 2 −1 6 −1 cos4 φ + O t −3/2 dφ t cos φ + t cos4 φ + 3αt + − 2 4 t −1 dt = −dt + 3c 2 − α √ −1/2 α 27c 2 −1 5 9c 2 −1/2 t t t (−dφ) cos3 φ + cos φ + cos6 φ − + − 4 4c 2 16 2 13c 3 1 α t −1 cos4 φ − + + 3α + t −1 cos2 φ − + O t −3/2 dφ. 4 8 4 2 (2.16) log t + φ0 + O(t −1/2 ). Therefore, φ = −t + (3c 2 − α) Using transformations (1.4), (1.6), (2.10), and the one in Theorem 2.1, we get the asymptotic representation in the theorem and finish the proof of the theorem. References [1] [2] 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. 143 (1998), no. 3, 241–271. P. A. Clarkson and J. 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. 278, Plenum, New York, 1992, pp. 1–31. ON THE ASYMPTOTIC REPRESENTATION OF THE SOLUTIONS . . . [3] [4] [5] [6] [7] 851 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, 1986. A. V. Kitaev, The method of isomonodromy deformations and the asymptotics of solutions of the “complete” third Painlevé equations, Math. USSR-Sb. 62 (1989), no. 2, 421–444. Y. Lu, Application of connection formula on eliminating plausible asymptotic representation of the Painlevé transcendents, submitted to Appl. Anal. , An asymptotic representation of the general fourth Painlevé transcendents, submitted to Int. J. Appl. Math. 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 17815, USA E-mail address: [email protected]