Rational Solutions of the Painlevé-II Equation Revisited
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2017), 065, 29 pages Rational Solutions of the Painlev´e-II EquationRevisited
Peter D. MILLER and Yue SHENGDepartment of Mathematics, University of Michigan,530 Church St., Ann Arbor, MI 48109, USA
E-mail: [email protected], [email protected]
URL: http://math.lsa.umich.edu/~millerpd/
Received April 18, 2017, in final form August 07, 2017; Published online August 16, 2017https://doi.org/10.3842/SIGMA.2017.065
Abstract.
The rational solutions of the Painlev´e-II equation appear in several applicationsand are known to have many remarkable algebraic and analytic properties. They also haveseveral different representations, useful in different ways for establishing these properties.In particular, Riemann–Hilbert representations have proven to be useful for extracting theasymptotic behavior of the rational solutions in the limit of large degree (equivalently thelarge-parameter limit). We review the elementary properties of the rational Painlev´e-IIfunctions, and then we describe three different Riemann–Hilbert representations of themthat have appeared in the literature: a representation by means of the isomonodromy theoryof the Flaschka–Newell Lax pair, a second representation by means of the isomonodromytheory of the Jimbo–Miwa Lax pair, and a third representation found by Bertola and Bothnerrelated to pseudo-orthogonal polynomials. We prove that the Flaschka–Newell and Bertola–Bothner Riemann–Hilbert representations of the rational Painlev´e-II functions are explicitlyconnected to each other. Finally, we review recent results describing the asymptotic behaviorof the rational Painlev´e-II functions obtained from these Riemann–Hilbert representationsby means of the steepest descent method.
Key words:
Painlev´e equations; rational functions; Riemann–Hilbert problems; steepestdescent method
Rational solutions of the second Painlev´e equation are important in several applied problems.It was discovered by Bass [3] that a certain Nernst–Planck model of steady electrolysis with twoions reduces to the Painlev´e-II equation, and in [36] the special role of rational solutions washighlighted in the context of this model (see also [4]). Johnson [24] notes that rational solutionsof the Painlev´e-II equation parametrize certain string theoretic models in high-energy physics.Clarkson [11] reviews the application of rational solutions of Painlev´e-II to the classification ofcertain equilibrium vortex configurations in ideal planar fluid flow (the poles of opposite residuedetermine the location of vortices of opposite circulation). More recently Buckingham and oneof the authors [7] found that the collection of rational solutions of the Painlev´e-II equationuniversally describes the space-time location of kinks in the semiclassical sine-Gordon equa-tion near a transversal crossing of the separatrix for the simple pendulum. Also, Shapiro andTater [37] have observed a connection between the rational Painlev´e-II solutions of large degreeand the characteristic polynomials for the quasi-exactly solvable part of the discrete spectrum a r X i v : . [ n li n . S I] A ug P.D. Miller and Y. Shengin a boundary-value problem for the stationary Schr¨odinger operator with a quartic potentialexhibiting PT-symmetry.In this paper, we review some of the well-known elementary properties of the rational solutionsof the Painlev´e-II equation, including three ways to represent them in terms of the solution ofa Riemann–Hilbert problem. We make a new contribution by showing that two of the threerepresentations, until now thought to be unrelated, are in fact connected with each other viaa simple and explicit transformation. Then we review results on the asymptotic behavior ofthe rational Painlev´e-II solutions in the large-degree limit that have been recently obtained byanalyzing these Riemann–Hilbert representations. Here our aim is to simplify the results andcorrect them where necessary as well as to report them in a unified context.
In this paper, we consider the Painlev´e-II equation with parameter m written in the formd p d y “ p ` yp ´ m, p “ p p y q , m P C . (1.1) Obviously if m “
0, one has p p y q “ p p y q would necessarily admit, for some p ‰ n P Z , a series expansion p p y q “ y n ` p ` p y ´ ` ¨ ¨ ¨ ˘ (1.2)convergent for sufficiently large | y | , and the a dominant balance in (1.1) with m “ n “ R Z .Conversely, if m P C is nonzero, (1.1) does not admit the zero solution. In this case, everyrational solution p p y q again has the form (1.2) for sufficiently large | y | and p ‰ n P Z ;it then follows from (1.1) that a dominant balance is achieved between the terms yp and ´ m yielding n “ ´ p “ m . Therefore if C is a counterclockwise-oriented circle of sufficientlylarge radius,12 π i ¿ C p p y q d y “ m. (1.3)Likewise if y P C is a finite pole of order n of the rational solution p p y q , then the substitutionof the Laurent series p p y q “ p y ´ y q ´ n p p ` p p y ´ y q ` ¨ ¨ ¨ q into (1.1) results in a dominant balance between p and 2 p yielding n “ p “ ˘
1, soevery pole of p p y q is simple and has residue ˘
1. If N ˘ p p q denotes the number of poles of p ofresidue ˘
1, then12 π i ¿ C p p y q d y “ N ` p p q ´ N ´ p p q P Z , so comparing with (1.3), we see that (1.1) admits a rational solution p p y q only if m P Z .ational Solutions of the Painlev´e-II Equation Revisited 3 In 1971, Lukashevich [31] discovered an explicit B¨acklund transformation for (1.1). Namely,supposing that p p y q is an arbitrary solution of (1.1), the related function p p p y q defined by p p p y q : “ p p y q ´ p p y q p p y q ´ p p y q ´ yp p y q ´ p p y q ´ p p y q ´ y “ ´ p p y q ´ m ` p p y q ´ p p y q ´ y , p m : “ m ` p p, m q replaced by p p p, p m q . Note that p p is a rational function whenever p is;therefore from the “seed solution” p p y q “ m “
0, the (iterated) B¨acklund transformationproduces a rational solution of (1.1) for every positive integer m . From the elementary symmetry p p p y q , m q ÞÑ p´ p p y q , ´ m q of (1.1) one then has the existence of a rational solution of (1.1) forevery m P Z , a fact that also follows from the earlier work of Yablonskii [39] and Vorob’ev [38](see Section 1.1.3 below). Hence, as pointed out by Airault [1], the condition m P Z is bothnecessary and sufficient for the existence of rational solutions to (1.1).The B¨acklund transformation (1.4) also yields a proof of uniqueness of the rational solutionfor given m P Z , a fact that was first noted by Murata [32]. Indeed, combining (1.4) with thesymmetry p p p y q , m q ÞÑ p´ p p y q , ´ m q yields a second B¨acklund transformation q p p y q : “ ´ p p y q ´ p p y q p p y q ` p p y q ` yp p y q ´ p p y q ` p p y q ` y “ ´ p p y q ` m ´ p p y q ` p p y q ` y q m : “ m ´ p p y q solving (1.1) to q p p y q solving (1.1) with p p, m q replaced by p q p, q m q . Importantly, anelementary computation shows that the B¨acklund transformations (1.4) and (1.5) are inversesof each other on the space of solutions of (1.1). In particular, both are injective maps. Sup-pose p p y q and r p p y q are distinct rational solutions of (1.1) for some m P Z zt u . Iterativelyapplying either (1.4) (if m ă
0) or (1.5) (if m ą | m | times, by injectivity we arrive at twodistinct rational solutions of (1.1) with m “ p p y q “ m “
0. See [17] for further information aboutB¨acklund transformations for Painlev´e equations.Explicitly, the first several rational Painlev´e-II solutions are p p y q “ , p ˘ p y q “ ˘ y , p ˘ p y q “ ˘ y ´ y p y ` q ,p ˘ p y q “ ˘ y p y ` y ` qp y ` qp y ` y ´ q , (1.6)where the subscript indicates the value of m in (1.1). It was first observed by Yablonskii [39] and Vorob’ev [38] that rational solutions of the Painlev´e-IIequation (1.1) can be represented in terms of special polynomials having an explicit recurrence The definition apparently fails if p p y q is such that 3 p p y q ´ p p y q ´ y vanishes identically. It is easy to checkthat this condition is consistent with (1.1) only if m “ ´ (so the numerator in (1.4) vanishes also). In thecase m “ ´ , the Riccati equation 3 p p y q ´ p p y q ´ y “ p p y q “ ´ φ p y q{ φ p y q where φ p y q ` yφ p y q “
0; in other words, φ is an Airy function and p p y q is a known special function (Airy) solutionof (1.1). A similar remark applies to the inverse transformation (1.5) which gives m “ and the denominatorvanishes for the Airy solutions. Note that both transformations (1.4) and (1.5) make sense whenever m P Z exceptpossibly for isolated values of y P C . P.D. Miller and Y. Shengrelation. The
Yablonskii–Vorob’ev polynomials are defined recursively by Q p z q : “ Q p z q : “ z ,and then Q n ` p z q : “ zQ n p z q ´ ` Q n p z q Q n p z q ´ Q n p z q ˘ Q n ´ p z q , n “ , , , . . . . (1.7)Then the rational solution of (1.1) can be expressed as p p y q “ p m p y q “ dd y ln ˆ Q m p z q Q m ´ p z q ˙ , z “ ˆ ˙ { y, m “ , , , . . . . (1.8)Of course the first surprise regarding the recursion formula (1.7) is that t Q n p z qu n “ is indeeda sequence of polynomials in z . Indeed this is the case, as can be seen by the alternative formuladue to Kajiwara and Ohta [26] expressing Q n p z q as a constant multiple of the Wronskian of 2 n ´ z (see also [10, Section 2.4]), and the first several iterates of (1.7) produce: Q p z q “ z ` , Q p z q “ z ` z ´ , Q p z q “ z ` z ` z. It has been shown that the polynomials t Q n p z qu n “ all have simple roots and that Q m and Q m ´ can have no roots in common [20], facts that are consistent via (1.8) with the fact that allpoles of p p y q are simple with residues ˘
1. Real roots of the Yablonskii–Vorob’ev polynomialscorrespond to real poles of p p y q , and these have been studied extensively by Roffelsen who hasshown that all nonzero real roots are all irrational [34] and that there are precisely t p n ` q{ u negative roots of Q n and t p n ` q{ u total real roots of Q n , and Q n p q “ n “ p mod 3 q [35]. Also, the real roots of Q n ` and Q n ´ interlace, as was proven by Clarkson [10]. The fact that all rational solutions of the Painlev´e-II equation (1.1) can be iteratively con-structed, either via the direct B¨acklund transformations (1.4) and (1.5) or via the recurrencerelation for the Yablonskii–Vorob’ev polynomials (1.7), is quite remarkable and indicative ofdeeper integrable structure underlying the Painlev´e-II equation. However, it must also bepointed out that the use of these iterative constructions is limited in practice, because theformulae generated become increasingly complicated as | m | increases. The situation is similarto that encountered when studying orthogonal polynomials, which in general can be constructedsystematically by a Gram–Schmidt orthogonalization algorithm, but the number of steps of thisalgorithm increases with the degree of the polynomial desired, making it difficult to appeal tothis approach to deduce properties of the general polynomial in the family.Therefore, if our interest is to understand the analytic properties of the rational Painlev´e-IIfunctions, it is necessary to have an alternative representation that admits the possibility ofasymptotic analysis for large | m | . In Section 2 we describe three such representations of therational Painlev´e-II solutions, two coming directly from the isomonodromic integrable structureunderlying the Painlev´e-II equation, and one related to a recently discovered representation ofthe squares of the Yablonskii–Vorob’ev polynomials in terms of the integrable structure behindorthogonal polynomials (which provides a work-around for the Gram–Schmidt procedure allo-wing large-degree asymptotics of general orthogonal polynomials to be computed). One of thecontributions of our paper is then to establish a new identity relating the orthogonal polynomialapproach to one of the isomonodromic approaches; see Section 2.4.These representations of the rational Painlev´e-II solutions have indeed proven to be useful incharacterizing the rational functions p m p y q in the limit of large | m | . In Section 3 we review someof the results that have been proven with their help, outlining some of the methods of proof.Below we will make frequent use of the Pauli spin matrices defined by σ : “ „ , σ : “ „ ´ ii 0 , σ : “ „ ´ . ational Solutions of the Painlev´e-II Equation Revisited 5 In 1980, Flaschka and Newell [16] showed how a self-similar reduction of the Lax pair repre-sentation of the modified Korteweg–de Vries equation reveals the Painlev´e-II equation in theform (1.1) to be an isomonodromic deformation of the linear equation B v B λ “ A FN p λ, y q v , A FN p λ, y q : “ „ ´ λ ´ p ´ i y pλ ` p ` mλ ´ pλ ´ p ` mλ ´ λ ` p ` i y (2.1)in which p , p , y , and m are regarded as numerical parameters. Indeed, (2.1) is compatible withthe auxiliary linear equation B v B y “ B FN p λ, y q v , B FN p λ, y q : “ „ ´ i λ pp i λ (2.2)only if the compatibility condition B A B y ´ B B B λ ` r A , B s “ (2.3)holds with A “ A FN and B “ B FN . This forces p to depend on y by the Painlev´e-II equationin the form (1.1) and forces p “ p p y q . The equation (2.2) then implies that the monodromydata associated with solutions of (2.1) depends trivially on y .Let us describe the monodromy data associated with rational solutions p “ p m p y q of (1.1)for m P Z . It is pointed out in [16] that whenever p p, p q “ p p m p y q , p m p y qq in (2.1) for therational solution p m p y q , the irregular singular point at λ “ 8 for (2.1) exhibits only trivialStokes phenomenon. This implies the existence of a fundamental solution matrix of (2.1) of theform V p λ, y q “ « I ` ÿ n “ K n p y q λ ´ n ff e ´ i θ p λ,y q σ (2.4)for some matrix coefficients K p y q , K p y q , and so on, where θ p λ, y q : “ λ ` yλ, and where the infinite series in (2.4) is convergent for | λ | sufficiently large, which in view of λ “ λ ‰
0. Assuming compatibility, i.e.,that p “ p p y q solves (1.1) with p “ p p y q , it can be shown that V p λ, y q is also a fundamentalsolution matrix for (2.2), and then it follows by substitution into the latter system that p p y q isrecovered from the subleading term of the expansion (2.4) by the formula p p y q “ K p y q “ ´ K p y q . (2.5)On the other hand, λ “ λ “ V p λ, y q “ « ? „ ´
11 1 h p y q σ ` ÿ n “ H n p y q λ n ff λ mσ (2.6) P.D. Miller and Y. Shengfor some scalar function h p y q ‰ H p y q , H p y q , and so on. Theabsence of logarithms in spite of the fact that the Frobenius exponents ˘ m differ by an integerfollows from the fact that, due to the triviality of the Stokes phenomenon at λ “ 8 , themonodromy matrix for (2.1) corresponding to any loop about the origin is the identity, hencediagonalizable. However the same fact implies an ambiguity in the formula (2.6) in whichthe dominant column in the limit λ Ñ h p y q ,and from the recurrence relations determining the higher-order terms from the preceding termsa predictable pattern emerges in which consecutive terms are alternating scalar multiples of thevectors p , q J and p´ , q J . A similar well-defined alternating pattern holds for the dominantcolumn, but only through the terms with n ď | m | ´
1, with the term for n “ | m | satisfying anequation that is consistent but indeterminate. Here a choice is made: the term for n “ | m | istaken to continue the alternating pattern of vectors p , q J and p´ , q J . Once this choice hasbeen made, the alternating pattern again continues to all orders of the dominant column. Inother words, Flaschka and Newell take V p λ, y q in the more specific form V p λ, y q “ ? „ ´
11 1 ˜ ÿ n “ σ n „ h n p y q h n p y q λ n ¸ λ mσ ,h p y q “ h p y q “ h p y q ´ . (2.7)There is then exactly one matrix solution of (2.1) of this form for a given scalar h p y q , andmoreover, assuming compatibility, h p y q can be chosen up to a constant scalar multiple so that V p λ, y q simultaneously solves (2.2). Again, the infinite series appearing in (2.7) is convergentnear λ “
0, and since there are no other finite singular points it is actually convergent forall λ P C . By taking the limits λ Ñ λ Ñ 8 respectively, Abel’s theorem implies theidentities det p V p λ, y qq “ p V p λ, y qq “ A FN p λ, y q in (2.1) has zero trace. Therefore, as both V p λ, y q and (for a suitable choice of h p y q ) V p λ, y q are simultaneous fundamental solution matrices for (2.1) and (2.2) defined in a common domain0 ă | λ | ă 8 , there exists a constant unimodular matrix G m such that V p λ, y q “ V p λ, y q G m , ă | λ | ă 8 . (2.8)The connection matrix G m is the monodromy data for the linear problem (2.1) in the casethat p “ p m p y q is a rational solution of (1.1). For more general solutions given m P Z , or fornon-integral values of m , the monodromy data becomes augmented with six Stokes matrices of alternating triangularity connecting solutions each having the form (2.4) (but only as anasymptotic series, with no convergence properties implied) in six overlapping sectors of theirregular singular point at λ “ 8 .It is easy to see that V p λ, y q : “ σ V p´ λ, y q σ is a fundamental solution matrix for thesystem (2.1) whenever V p λ, y q is. This substitution also leaves (2.2) invariant. Since V p λ, y q is uniquely determined from (2.1) and the leading term of its large- λ asymptotic expansion(convergent in the trivial-monodromy case at hand for rational solutions p “ p m p y q ), we deducethe identity V p λ, y q “ V p λ, y q . (2.9)Similarly, given the scalar h p y q , it follows from (2.7) that V p λ, y q “ V p λ, y q „ p´ q m p´ q m ` . (2.10)ational Solutions of the Painlev´e-II Equation Revisited 7Therefore, conjugating by σ and replacing λ ÞÑ ´ λ in (2.8), the use of the identities (2.9)–(2.10)shows that also V p λ, y q “ V p λ, y qp´ q m σ G m σ , ă | λ | ă 8 , and hence comparing again with (2.8) one sees that G m “ p´ q m σ G m σ . This matrix identityalong with the condition that det p G m q “ G m necessarily has the form G m “ „ α p´ q m α p´ q m ` p α q ´ p α q ´ , (2.11)where only the nonzero constant α is undetermined by symmetry.We may now formulate a Riemann–Hilbert problem to recover V p λ, y q and V p λ, y q , andhence also the rational Painlev´e-II function p m p y q , from the monodromy data, i.e., from theconnection matrix G m . To this end, we define a matrix M m p λ, y q by M m p λ, y q “ V p λ, y q e i θ p λ,y qq σ λ ´ mσ , | λ | ą , V p λ, y q e i θ p λ,y q σ λ ´ mσ , | λ | ă . It is then clear that M m p λ, y q solves the following Riemann–Hilbert problem. Riemann–Hilbert Problem 2.1 (Flaschka–Newell representation) . Let m P Z and y P C be given. Seek a ˆ matrix-valued function M m p λ, y q defined for λ P C , | λ | ‰ , with thefollowing properties: • Analyticity. M m p λ, y q is analytic for | λ | ‰ , taking continuous boundary values M m ` p λ, y q and M m ´ p λ, y q for | λ | “ from the interior and exterior respectively of theunit circle. • Jump condition.
The boundary values are related by M m ` p λ, y q “ M m ´ p λ, y q λ mσ e ´ i θ p λ,y q σ G ´ m e i θ p λ,y q σ λ ´ mσ , | λ | “ . • Normalization.
The matrix M m p λ, y q is normalized at λ “ 8 as follows: lim λ Ñ8 M m p λ, y q λ mσ “ I , where the limit may be taken in any direction. The solution of this Riemann–Hilbert problem exists precisely for those values of y P C that are not poles of p m p y q . Given the solution M m p λ, y q , one extracts the rational Painlev´e-IIfunction p m p y q from the limit (cf. (2.5)) p m p y q “
2i lim λ Ñ8 λ ` m M m p λ, y q “ ´
2i lim λ Ñ8 λ ´ m M m p λ, y q . (2.12)Note also that without loss of generality one may take the constant α in (2.11) to be α “ M m p λ, y q within the unit circle by multiplication on the right by α σ .Such a re-definition clearly does not affect M m p λ, y q for | λ | ą p m p y q .Flaschka and Newell observe that Riemann–Hilbert Problem 2.1 can be solved by reductionto finite-dimensional linear algebra, resulting in determinantal formulae for p m p y q equivalent toiterated B¨acklund transformations studied by Airault [1]. To see this, note that uniqueness ofsolutions of Riemann–Hilbert Problem 2.1 is an elementary consequence of Liouville’s theorem, P.D. Miller and Y. Shengso it is sufficient to construct a solution by any means. Now, M m p λ, y q necessarily has a con-vergent Laurent expansion about λ “ 8 , suggesting to seek M m p λ, y q as a suitable Laurentpolynomial. In fact, assuming without loss of generality that m ě
0, we may suppose that inthe domain | λ | ą M m p λ, y q has the form M m p λ, y q “ λ ´ m ` a p y q λ ´ m ´ ` ¨ ¨ ¨ ` a m ´ p y q λ ´ m ` a m p y q λ ´ m ,M m p λ, y q “ b p y q λ m ´ ` b p y q λ m ´ ` ¨ ¨ ¨ ` b m ´ p y q λ ` b m p y q . (2.13)This ansatz clearly satisfies the necessary analyticity condition for | λ | ą λ “ 8 . The jump condition can then be reinterpreted as requiring thatthe linear combinations M m ` p λ, y q : “ α “ M m ´ p λ, y q ` p´ q m e θ p λ,y q λ ´ m M m ´ p λ, y q ‰ ,M m ` p λ, y q : “ α “ p´ q m ` e ´ θ p λ,y q λ m M m ´ p λ, y q ` M m ´ p λ, y q ‰ , where the boundary values M m ´ p λ, y q and M m ´ p λ, y q are given by the ansatz (2.13), both beanalytic functions within the unit disk, where the only potential singularity is λ “
0. The formof the ansatz automatically guarantees that this is the case for M m ` p λ, y q , but M m ` p λ, y q hasprecisely 2 m negative powers of λ whose coefficients are required to vanish. It is easily seenthat this amounts to a square inhomogeneous linear system of equations, explicit in terms ofthe Taylor coefficients of e ˘ θ p λ,y q , on the 2 m unknowns a p y q , . . . , a m p y q and b p y q , . . . , b m p y q .The solution of this linear system by Cramer’s rule gives the rational Painlev´e-II function p m p y q in the form p m p y q “ b p y q . For example, in the case m “ M ` p λ, y q “ λ ´ ` a p y q λ ´ ` a p y q λ ´ ` e θ p λ,y q ` b p y q λ ´ ` b p y q λ ´ ˘ “ p a p y q ` b p y qq λ ´ ` p a p y q ` b p y q ` yb p y qq λ ´ ` ` ` yb p y q ´ y b p y q ˘ λ ´ ` ` ´ y b p y q ` ` ´ y ˘ b p y q ˘ λ ´ ` O p q , where the last term represents a function analytic at λ “
0, be analytic at λ “ p p y q “ b p y q “ p y ´ q{p y p y ` qq as expected (cf. (1.6)). In 1981, Jimbo and Miwa [23] found a representation of the Painlev´e-II equation as the com-patibility condition for a Lax pair different from that found by Flaschka and Newell. We takeJimbo and Miwa’s linear equations in the form B v B ζ “ A JM p ζ, y q v , A JM p ζ, y q : “ „ ´ ζ ´ U V ´ y U ζ ` W ´ V ζ ´ Z ζ ` U V ` y (2.14)and B v B y “ B JM p ζ, y q v , B JM p ζ, y q : “ „ ´ ζ U ´ V ζ (2.15)For this system, the compatibility condition (2.3) with A “ A JM and B “ B JM is equivalent tothe following first-order system of equations: U p y q “ ´ W p y q , V p y q “ Z p y q , W p y q “ U p y q V p y q ` y U p y q , Z p y q “ ´ U p y q V p y q ´ y V p y q . (2.16)ational Solutions of the Painlev´e-II Equation Revisited 9This system admits a first integral m : “ U p y q Z p y q ` V p y q W p y q ` “ const , (2.17)and then with p p y q “ U p y q{ U p y q the system (2.16) yields the Painlev´e-II equation for p p y q inthe form (1.1).As with the Flaschka–Newell approach, it is the problem (2.14) whose analysis for fixed y determines the monodromy data, which is then independent of y for simultaneous solutions of(2.14)–(2.15). However, the direct monodromy problem (2.14) has a different character than inthe Flaschka–Newell approach because (2.14) has only one singular point, an irregular singularpoint at infinity, while (2.1) has in addition a regular singular point at the origin if m ‰ ζ , and all monodromy data is generated onlyfrom the Stokes phenemonon about the singular point at infinity. In particular, it is the casethat for the rational solution p “ p m p y q for m P Z , solutions of (2.14) exhibit nontrivial Stokesphenomenon in contrast to the situation in Flaschka–Newell theory.The Stokes multipliers for (2.14) when p “ p m p y q is the rational solution of (1.1) for m P Z can be inferred from the following Riemann–Hilbert problem, which arises naturally in the studyof solutions of the sine-Gordon equation (cid:15) u tt ´ (cid:15) u xx ` sin p u q “ p x, t q “ p x crit , q ; see [7, Section 5]. Riemann–Hilbert Problem 2.2 (Jimbo–Miwa representation) . Let m P Z and y P C be given.Seek a ˆ matrix-valued function Z m p ζ, y q be defined for ζ P C z Σ , where Σ is the union of sixrays Σ : “ R Y e i π { R Y e ´ i π { R , and having the following properties: • Analyticity. Z m p ζ, y q is analytic for ζ P C z Σ , taking continuous boundary values alongthe boundary of each component of this domain. • Jump condition.
Taking each ray of Σ to be oriented in the direction away from theorigin and given a point ζ on one of the rays using the notation Z m ` p ζ, y q p resp. Z m ´ p ζ, y qq to denote the boundary value taken at ζ P Σ from the left p resp. right q , the boundary valuesare related by Z m ` p ζ, y q “ Z m ´ p ζ, y q e ´ φ p ζ,y q σ V e φ p ζ,y q σ , ζ P Σ zt u , φ p ζ, y q : “ ζ ` yζ, where V is constant along each ray and is as shown in Fig. . • Normalization.
The matrix Z m p ζ, y q is normalized at ζ “ 8 as follows: lim ζ Ñ8 Z m p ζ, y qp´ ζ q p ´ m q σ { “ I , where the limit can be taken in any direction except the positive real axis, which is thebranch cut for the principal branch of p´ ζ q p ´ m q σ { . From the solution of Riemann–Hilbert Problem 2.2 one obtains the rational Painlev´e-II func-tion p m p y q from the coefficients in the large- ζ expansion of Z m p ζ, y q : Z m p ζ, y qp´ ζ q p ´ m q σ { “ I ` A m p y q ζ ´ ` B m p y q ζ ´ ` O ` ζ ´ ˘ , ζ Ñ 8 , (2.18)by the formula p m p y q “ A m p y q ´ B m p y q A m p y q . (2.19)In [7], it was deduced that Riemann–Hilbert Problem 2.2 encodes the Stokes multipliers for theLax pair (2.14)–(2.15) associated with the rational Painlev´e-II function p m p y q as follows. Firstly,0 P.D. Miller and Y. Sheng ζ (cid:27) (cid:45)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:94)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74) (cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:93) (cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:30)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10) (cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:29) (cid:115) „ „ „ „ ´ ´ i0 ´ „ „ Figure 1.
The jump contour Σ and the value of the constant matrix V on each ray of Σ for Riemann–Hilbert Problem 2.2. by considering L m p ζ, y q : “ Z m p ζ, y q e ´ φ p ζ,y q σ , one shows that partial derivatives of L m p ζ, y q withrespect to ζ and y satisfy exactly the same jump conditions on the rays of Σ as does L m p ζ, y q itself, a fact that along with some local analysis near ζ “ ζ “ 8 shows that L m p ζ, y q is a simultaneous fundamental solution matrix of the two Lax pair equations (2.14)–(2.15),provided that the coefficients U , V , W , and Z are defined from the expansion (2.18) by theformulae U p y q : “ A m p y q , V p y q : “ A m p y q , W p y q : “ B m p y q ´ A m p y q A m p y q , Z p y q : “ B m p y q ´ A m p y q A m p y q . Then, by reexamination of the asymptotic behavior of L m p ζ, y q for large ζ one finds that theparameter m P Z appearing in Riemann–Hilbert Problem 2.2 is related to these functions bythe identity (2.17), identifying it with the parameter m appearing in the Painlev´e-II equation(1.1). It remains therefore to deduce that p m p y q defined now by the expression (2.19) is the rational solution of (1.1). This can be accomplished by first noting that in the case m “ p p y q “
0, at which point one can leveragethe y -part (2.15) of the Lax pair to construct Z p ζ, y q explicitly in terms of Airy functions ofargument 6 ´ { ` y ` ζ ˘ . Then, one can apply iterated discrete isomondromic Schlesingertransformations (also known in the integrable systems literature as Darboux transformations ;see [6, Section 2] and [23] for further information on these notions) to explicitly increment ordecrement the value of m in integer steps, with the corresponding effect on the coefficient p m p y q defined by (2.19) being given by the B¨acklund transformations (1.4) or (1.5) respectively. Asthese preserve rationality, one concludes that p m p y q given by (2.19) is precisely the rationalsolution of (1.1) when Z m p ζ, y q is the solution of Riemann–Hilbert Problem 2.2 for arbitrary m P Z . See [7, Section 5] for full details of these arguments. In [5], Bertola and Bothner derived a new Hankel determinant representation of the squares ofthe Yablonskii–Vorob’ev polynomials t Q n p z qu n “ defined by the recurrence relation (1.7) withinitial conditions Q p z q “ Q p z q “ z . This new identity leads to a formula expressingational Solutions of the Painlev´e-II Equation Revisited 11the rational Painlev´e-II function p m p y q in terms of pseudo-orthogonal polynomials (i.e., polyno-mials orthogonal with respect to an indefinite inner product involving contour integration witha complex-valued weight), and this in turn leads to a Riemann–Hilbert representation.The main theorem reported and proved in [5] is the following. Theorem 2.3 (Bertola and Bothner, [5]) . Given z P C , let t µ k p z qu k “ denote the Taylorcoefficients of the generating function f p t q : “ e tz ´ t : e tz ´ t “ ÿ k “ µ k p z q t k , p z, t q P C . Then, for any n ě , Q n ´ p z q “ p´ q t n { u D n p z q n ´ n ´ ź k “ „ p k q ! k ! , where t u u denotes the greatest integer less than or equal to u and D n p z q is the Hankel determinant D n p z q : “ det r µ l ` j ´ p z qs nl,j “ . The coefficients µ k p z q are polynomials with numerous special properties, some of which areenumerated in [5]. Similar determinantal representations of the Yablonskii–Vorob’ev polyno-mials themselves (not the squares) had been previously known [26], including one represen-ting Q n p z q via a non-Hankel determinant involving the scaled functions µ k ` { z ˘ and onerepresenting Q n p z q as a Hankel determinant built from functions that can be extracted froma generating function via a non-convergent asymptotic series [21]. However, it is the combinationof the Hankel structure of the determinant with the convergent nature of the generating functionexpansion that leads to a Riemann–Hilbert representation of p m p y q as we will now explain.When combined with Theorem 2.3, the representation (1.8) of p m p y q in terms of the Yab-lonskii–Vorob’ev polynomials gives p m p y q “
12 dd y ln p η m pp q { y qq , η m p z q : “ D m ` p z q D m p z q , m “ , , , . . . . (2.20)Now, since the polynomials t µ k p z qu k “ are Taylor coefficients of the entire function f p t q “ e tz ´ t , they may be written as contour integrals using the Cauchy integral formula: µ k p z q “ k ! d k d t k e tz ´ t ˇˇˇˇ t “ “ π i ¿ C t ´ k ´ e tz ´ t d t, k “ , , , , . . . . Here C is a simple contour encircling the origin in the counterclockwise direction; without lossof generality we will take it to coincide with the unit circle. Setting t “ ξ ´ in the integrandputs the formula in the equivalent form µ k p z q “ ¿ C ξ k d ν p ξ ; z q , k “ , , , , . . . , where C may be taken to be the same contour, and whered ν p ξ ; z q : “ e ´ ξ ´ ` ξ ´ z π i ξ d ξ. Thus, t µ k p z qu k “ are revealed as the monomial moments of a complex-valued weight paramet-rized by z P C and defined on the unit circle. This fact immediately gives an interpretation to2 P.D. Miller and Y. Shengthe ratio η m p z q of consecutive Hankel determinants (cf. (2.20)); it is the norming constant of themonic pseudo-orthogonal polynomial ψ m p ξ ; z q “ ξ m ` c m,m ´ p z q ξ m ´ ` ¨ ¨ ¨ ` c m, p z q ξ ` c m, p z q defined given z P C by the pseudo-orthogonality conditions ¿ C ψ m p ξ ; z q ξ j d ν p ξ ; z q “ , j “ , , , . . . , m ´ . (2.21)Indeed, if ψ m p ξ ; z q exists for the given value of z P C then it follows that η m p z q “ ¿ C ψ m p ξ ; z q ξ m d ν p ξ ; z q . (2.22)The points y P C where either ψ m ` ξ ; ` ˘ { y ˘ fails to exist or η m `` ˘ { y ˘ “ p m p y q .Now, it is well-known that given any complex measure on a suitable contour, the corre-sponding pseudo-orthogonal polynomial of degree m can be characterized via the solution ofa matrix Riemann–Hilbert problem of Fokas–Its–Kitaev type [18]. In the present context, thatRiemann–Hilbert problem is the following. Riemann–Hilbert Problem 2.4 (Bertola–Bothner representation) . Let m ě be an integer,and let z P C be given. Seek a ˆ matrix-valued function Y m p ξ, z q defined for ξ P C , | ξ | ‰ ,with the following properties: • Analyticity. Y m p ξ, z q is analytic for | ξ | ‰ , taking continuous boundary values Y m ` p ξ, z q and Y m ´ p ξ, z q for | ξ | “ from the interior and exterior respectively of the unitcircle. • Jump condition.
The boundary values are related by Y m ` p ξ, z q “ Y m ´ p ξ, z q „ ν p ξ ; z q , | ξ | “ ,ν p ξ ; z q : “ d ν p ξ ; z q d ξ “ e ´ ξ ´ ` zξ ´ π i ξ . (2.23) • Normalization.
The matrix Y m p ξ, z q is normalized at ξ “ 8 as follows: lim ξ Ñ8 Y m p ξ, z q ξ ´ mσ “ I , where the limit may be taken in any direction. Indeed, all of the relevant quantities associated with the pseudo-orthogonal polynomials forthe weight d ν p ξ ; z q are encoded in the solution of this problem. In particular, Y m p ξ, z q “ ψ m p ξ ; z q and Y m p ξ, z q “ π i ¿ C ψ m p w ; z q d ν p w ; z q w ´ ξ , from which it follows (cf. (2.21)–(2.22)) that η m p z q “ ´ π i lim ξ Ñ8 ξ m ` Y m p ξ, z q . Existence is not guaranteed for every z P C because integration against d ν p ξ ; z q does not define a definite innerproduct, nor does (2.21) represent Hermitian orthogonality which would require replacing ξ j with its complexconjugate. Hence the terminology of “pseudo-orthogonality”. ational Solutions of the Painlev´e-II Equation Revisited 13Asymptotic analysis of the pseudo-orthogonal polynomials ψ m p ξ ; z q in the limit of large m can therefore be carried out by applying steepest descent techniques to Riemann–Hilbert Prob-lem 2.4, as was first done in the case of true orthogonality on the real line in [14] and in thecase of true orthogonality on the unit circle in [2]. However, noting that the expression (2.20)involves differentiation with respect to the parameter z , a limit process that cannot be assumedto commute with the limit m Ñ 8 , Bertola and Bothner show how to obtain the relevantderivatives directly from the solution Y m p ξ, z q of Riemann–Hilbert Problem 2.4. The essence ofthe argument is as follows. The related matrix N m p ξ, z q : “ Y m p ξ, z q e zξ ´ σ { must be analyticfor ξ P C zt u and satisfies jump condition across the unit circle of exactly the form (2.23) inwhich z has been replaced by z “
0. As the parameter z no longer appears in the jump mat-rix for N m p ξ, z q , it follows that the partial derivative N mz p ξ, z q also satisfies exactly the samejump condition, and therefore the matrix ratio N mz p ξ, z q N m p ξ, z q ´ has no jump and so extendsto an analytic function on the punctured complex plane C zt u . The asymptotic behavior of N mz p ξ, z q N m p ξ, z q ´ for large and small ξ is easily expressed in terms of Y m p ξ, z q : N mz p ξ, z q N m p ξ, z q ´ “ Y m p z q ` σ ˘ ξ ´ ` O ` ξ ´ ˘ , ξ Ñ 8 , Y m p , z q σ Y m p , z q ´ ξ ´ ` O p q , ξ Ñ , where Y m p ξ, z q ξ ´ mσ “ I ` Y m p z q ξ ´ ` O p ξ ´ q as ξ Ñ 8 . Therefore N mz p ξ, z q N m p ξ, z q ´ isa z -dependent multiple of ξ ´ given by two equivalent formulae: N mz p ξ, z q N m p ξ, z q ´ “ ` Y m p z q ` σ ˘ ξ ´ “ Y m p , z q σ Y m p , z q ´ ξ ´ . From the p , q -entry in this matrix identity one obtains η m p z q “ ´ π i Y m , p z q “ π i Y m p , z q Y m p , z q , m “ , , , . . . , where we have used the fact that the necessarily unique solution of Riemann–Hilbert Problem 2.4has unit determinant. Therefore, from the solution of Riemann–Hilbert Problem 2.4 the rationalPainlev´e-II function p m p y q can be expressed without differentiation with respect to z as p m p y q “ ´ Y m p , z q Y m p , z q { Y m , p z q , z “ ˆ ˙ { y, Y m p z q : “ lim ξ Ñ8 ξ ` Y m p ξ, z q ξ ´ mσ ´ I ˘ , (2.24)for m “ , , , . . . . The Riemann–Hilbert representations of the rational Painlev´e-II functions appearing in theisomonodromy approaches of Flaschka–Newell (cf. Section 2.1) and Jimbo–Miwa (cf. Section 2.2)are known to be related. Indeed, Joshi, Kitaev, and Treharne found an explicit integral transformrelating simultaneous solutions of the corresponding Lax pairs [25, Corollary 3.2]. This inte-gral transform provides another explanation for the fact that the solution of Riemann–HilbertProblem 2.1 is rational in λ while that of Riemann–Hilbert Problem 2.2 is transcendental in ζ ,being built from Airy functions [7]. The approach of Bertola–Bothner also leads to a Riemann–Hilbert representation of the rational Painlev´e-II functions, but the approach is not motivatedby isomonodromy theory for any Lax pair, so it seems more mysterious from this point of view.In this section we show that the Riemann–Hilbert problem appearing in the Bertola–Bothnerapproach is in fact explicitly connected to that arising in the Flaschka–Newell isomonodromytheory:4 P.D. Miller and Y. Sheng Theorem 2.5.
Let m ě be an integer, suppose that y P C is not a pole of the ratio-nal Painlev´e-II function p m p y q , and let z “ ` ˘ { y . Then the unique solution M m p λ, y q ofRiemann–Hilbert Problem arising from Flaschka–Newell theory is related to the unique so-lution Y m p ξ, z q of Riemann–Hilbert Problem arising from the Bertola–Bothner approach byan explicit elementary transformation with an explicit elementary inverse p cf. equations (2.25) – (2.27) , (2.29) , (2.30) , (2.34) , and (2.36) in the proof below q . Proof .
We start with the Flaschka–Newell approach and Riemann–Hilbert Problem 2.1. Sup-pose without loss of generality that m “ , , , . . . . We begin by noting that the matrix G ´ m defined by (2.11) has the lower-upper factorization G ´ m “ „ p α q ´ p´ q m ` α p´ q m p α q ´ α “ „ p´ q m „ p α q ´ p´ q m ` α α , and therefore the jump matrix in Riemann–Hilbert Problem 2.1 is λ mσ e ´ i θ p λ,y q σ G ´ m e i θ p λ,y q σ λ ´ mσ “ „ p´ q m λ ´ m e θ p λ,y q „ p α q ´ p´ q m ` αλ m e ´ θ p λ,y q α , and the right-hand factor is obviously analytic within the unit disk and has unit determinant.Therefore, defining a new matrix P m p λ, y q in terms of the unknown M m p λ, y q by P m p λ, y q : “ $’’&’’% M m p λ, y q , | λ | ą , M m p λ, y q « p α q ´ p´ q m ` αλ m e ´ θ p λ,y q α ff ´ , | λ | ă , (2.25)we see that P m p λ, y q satisfies exactly the same conditions as specified in Riemann–HilbertProblem 2.1 except that the jump condition across the unit circle becomes instead P m ` p λ, y q “ P m ´ p λ, y q „ p´ q m λ ´ m e θ p λ,y q , | λ | “ . (2.26)This triangular jump matrix already suggests the Fokas–Its–Kitaev form that appears in theapproach of Bertola and Bothner, but we require two more steps to complete the identification.Firstly, we make the simple substitution Q m p ξ, z q : “ k σ σ P m ` , ` ˘ { z ˘ ´ P m ` cξ ´ , ` ˘ { z ˘ ξ ´ mσ σ k ´ σ , (2.27)where c : “ ´ i ¨ ´ { and k : “ i m ` c m e i π { ? π . Now observe that the following Riemann–Hilbert problem captures at the same time the matrix Q m p ξ, z q and the matrix Y m p ξ, z q appearing in the Bertola–Bothner approach, for differentvalues of the auxiliary parameter j P Z . Riemann–Hilbert Problem 2.6.
Let m P Z , j P Z , and z P C be given. Seek a ˆ matrix-valued function C m,j p ξ, z q defined for ξ P C , | ξ | ‰ , with the following properties: • Analyticity. C m,j p ξ, z q is analytic for | ξ | ‰ , taking continuous boundary values C m,j ` p ξ, z q and C m,j ´ p ξ, z q for | ξ | “ from the interior and exterior respectively of theunit circle. ational Solutions of the Painlev´e-II Equation Revisited 15 • Jump condition.
The boundary values are related by C m,j ` p ξ, z q “ C m,j ´ p ξ, z q „ ξ j ν p ξ ; z q , | ξ | “ ,ν p ξ ; z q “ e ´ ξ ´ ` zξ ´ π i ξ . (2.28) • Normalization.
The matrix C m,j p ξ, z q is normalized at ξ “ 8 as follows: lim ξ Ñ8 C m,j p ξ, z q ξ ´ mσ “ I , where the limit may be taken in any direction. Indeed, it is easy to check that Q m p ξ, z q “ C m, p ξ, z q and, for m ě Y m p ξ, z q “ C m, p ξ, z q (2.29)by comparison with the conditions of Riemann–Hilbert Problems 2.1 and 2.4. We complete theconnection between the Flaschka–Newell and Bertola–Bothner approaches by next establishingthe relation between solutions C m,j p ξ, z q for consecutive values of j P Z .The solution C m,j p ξ, z q of Riemann–Hilbert Problem 2.6 has a convergent Laurent expansionfor large | ξ | of the form C m,j p ξ, z q “ ` I ` R m,j p z q ξ ´ ` O ` ξ ´ ˘˘ ξ mσ , ξ Ñ 8 (2.30)for some residue matrix R m,j p z q . Noting that if it exists for a given z P C , the unique solutionof Riemann–Hilbert Problem 2.6 has unit determinant, consider the matrix q E p ξ, z q defined by q E p ξ, z q : “ C m,j p ξ, z q „ ξ C m,j ` p ξ, z q ´ , | ξ | ‰ . (2.31)It is straightforward to check from (2.28) that the boundary values taken by q E p ξ, z q on the unitcircle satisfy the trivial jump condition q E ` p ξ, z q “ q E ´ p ξ, z q for | ξ | “
1; hence q E p ξ, z q extends tothe whole complex plane as an entire function of ξ . Moreover, using (2.30) it follows that q E p ξ, z q has the following asymptotic expansion for large ξ : q E p ξ, z q “ ` I ` R m,j p z q ξ ´ ` O ` ξ ´ ˘˘ „ ξ ` I ´ R m,j ` p z q ξ ´ ` O ` ξ ´ ˘˘ “ « R m,j p z q´ R m,j ` p z q ξ ` R m,j p z q ´ R m,j ` p z q ff ` O ` ξ ´ ˘ , ξ Ñ 8 . (2.32)It then follows by Liouville’s theorem that all negative power terms in the Laurent expansionof q E p ξ, z q vanish, i.e., q E p ξ, z q is the linear function of ξ given by the explicit matrix on thesecond line of (2.32). Returning to (2.31), we have established the identity C m,j p ξ, z q „ ξ “ « R m,j p z q´ R m,j ` p z q ξ ` R m,j p z q ´ R m,j ` p z q ff C m,j ` p ξ, z q , | ξ | ‰ . (2.33)If we can express the second column of R m,j p z q in terms of C m,j ` p ξ, z q , then this becomes anexplicit formula for C m,j p ξ, z q in terms of the latter.6 P.D. Miller and Y. ShengTo this end, consider the second column of (2.33) evaluated at ξ “
0, which reads „ “ « C m,j ` p , z q ` R m,j p z q C m,j ` p , z q´ R m,j ` p z q C m,j ` p , z q ` p R m,j p z q ´ R m,j ` p z qq C m,j ` p , z q ff because C m,j p ξ, z q and C m,j ` p ξ, z q are analytic at z “
0. Therefore, R m,j p z q “ ´ C m,j ` p , z q C m,j ` p , z q and R m,j p z q “ R m,j ` p z q ` C m,j ` p , z q C m,j ` p , z q R m,j ` p z q , so substituting into (2.33) we recover the explicit formula for decrementing the value of j : C m,j p ξ, z q “ q E p ξ, z q C m,j ` p ξ, z q „ ξ ´ , where q E p ξ, z q “ « ´ C m,j ` p , z q C m,j ` p , z q ´ ´ R m,j ` p z q ξ ` R m,j ` p z q C m,j ` p , z q C m,j ` p , z q ´ ff . (2.34)In a similar way, the matrix p E p ξ, z q : “ C m,j ` p ξ, z q „ ξ
00 1 C m,j p ξ, z q ´ is an entire function that equals the polynomial part of its Laurent expansion for large ξ , andhence p E p ξ, z q “ « ξ ` R m,j ` p z q ´ R m,j p z q ´ R m,j p z q R m,j ` p z q ff , leading to the following analogue of (2.33): C m,j ` p ξ, z q „ ξ
00 1 “ « ξ ` R m,j ` p z q ´ R m,j p z q ´ R m,j p z q R m,j ` p z q ff C m,j p ξ, z q . (2.35)From the first column of (2.35) evaluated at ξ “ R m,j ` p z q “ ´ C m,j p , z q C m,j p , z q and R m,j ` p z q “ R m,j p z q ` C m,j p , z q C m,j p , z q R m,j p z q , so substituting into (2.35) we recover the explicit formula for incrementing the value of j : C m,j ` p ξ, z q “ p E p ξ, z q C m,j p ξ, z q „ ξ ´
00 1 , where p E p ξ, z q “ « ξ ` R m,j p z q C m,j p , z q C m,j p , z q ´ ´ R m,j p z q´ C m,j p , z q C m,j p , z q ´ ff . (2.36)Note that equations (2.34) and (2.36) can be interpreted as discrete Schlesinger/Darboux trans-formations (see [6, Section 2] and [23]) for Riemann–Hilbert Problem 2.6.Taking into account the explicit and obviously invertible transformations (2.25)–(2.27) rela-ting M m p λ, y q solving Riemann–Hilbert Problem 2.1 to Q m p ξ, z q “ C m, p ξ, z q via P m p λ, y q , theformulae (2.34) and (2.36) establish the connection with Riemann–Hilbert Problem 2.4 havingsolution Y m p ξ, z q “ C m, p ξ, z q . (cid:4) ational Solutions of the Painlev´e-II Equation Revisited 17We remark that although Theorem 2.5 provides an explicit relation between the solutions ofRiemann–Hilbert Problems 2.1 and 2.4, it can happen that for given z P C one of these problemsis solvable and the other is not. This occurs precisely when one of the denominators C m, p , z q in (2.34) or C m, p , z q in (2.36) vanishes. Indeed, we have mentioned before (and it actuallyfollows from the formula (2.12)) that the points z where Riemann–Hilbert Problem 2.1 failsto be solvable correspond precisely to the poles of p m . On the other hand, the formula (2.24)shows that it is possible that some poles of p m can arise from the well-defined function Y m , vanishing at a point z where Riemann–Hilbert Problem 2.4 has a solution; hence Riemann–Hilbert Problem 2.4 is solvable while Riemann–Hilbert Problem 2.1 is not. It can also happenthat Riemann–Hilbert Problem 2.4 fails to be solvable at a point z corresponding to a regularpoint of p m and hence a point of solvability of Riemann–Hilbert Problem 2.1, in which case theformula (2.24) retains sense locally via a limit process (i.e., l’Hˆopital’s rule). In this section, we assume without loss of generality that m ě
0. There have been severalstudies of the rational solutions p m p y q of the Painlev´e-II equation from the numerical point ofview, mostly concerned with looking for patterns in the distribution of poles of p m p y q in thecomplex y -plane as m varies. The earliest work in this direction that we are aware of is the1986 paper of Kametaka et al. [27] in which numerical methods were brought to bear on theproblem of finding roots of the Yablonskii–Vorob’ev polynomials for m as large as m “ m display features suggesting the breakdown of thenumerical method. A figure such as those from [27] also appears in the 1991 monograph [22].These studies show the poles of p m p y q being contained for reasonably large m within a roughlytriangular-shaped region of size increasing with m and therein organized in an apparently regular,crystalline pattern. Plots of poles of p m p y q obtained by similar methods also appear in [12],a paper that includes in addition a study of corresponding phenomena in higher-order equationsin the Painlev´e-II hierarchy. More recently, general numerical methods for the study of solutionswith many poles in differential equations have been advanced based on such techniques as Pad´eapproximation, and these methods have been shown to be capable of accurately reproducing thepole pattern of p m p y q , treating the Painlev´e-II equation (1.1) as an initial-value problem to besolved numerically taking as initial conditions the exact values of p m p q and p m p q [19, 33]. InFig. 2 we give our own plots of poles of p m p y q for m “ m “
30, and m “
60, which we madeby symbolically constructing the relevant Yablonskii–Vorob’ev polynomials in
Mathematica andusing
NSolve with the option
WorkingPrecision->50 to find the roots.These numerical observations suggest structure that should be explained, and yet the large- m limit in which the structural features of interest appear to become clear in the numerics isfundamentally out of reach of exact methods like iterated B¨acklund transformations or explicitdeterminantal formulae, the study of which becomes combinatorially prohibitive in this limit.Therefore one may consider instead methods of asymptotic analysis. A formal approach may bebased upon the observation that the modulus of the poles or zeros of p m p y q most distant from theorigin scales roughly like m { [22], which suggests examining p m p y q in a small neighborhood ofa point y “ m { x ; dominant balance arguments suggest that the size of the neighborhood shouldthen be proportional to m ´ { . So, letting x P C be fixed, consider the change of independentvariable y ÞÑ w in (1.1) given by (the relatively small shifts by 1 { y “ ` m ´ ˘ { x ` ` m ´ ˘ ´ { w. - -
20 0 20 40 - - - -
20 0 20 40 - - - -
20 0 20 40 - - Figure 2.
The poles of residue 1 (blue) and ´ p p y q (left), p p y q (center), and p p y q (right).Superimposed is the theoretical boundary of the elliptic region (cf. Section 3.2). Substituting this into (1.1) along with the scaling of the independent variable by p “ ` m ´ ˘ { P ,one arrives at the equivalent equationd P d w “ P ` x P ´ ` w P ´ ` m ` ˘ , which for large m appears to be a perturbation of an autonomous equation for an approximatingfunction r P p w q :d r P d w “ r P ` x r P ´ . (3.1)Multiplying by d r P { d w and integrating gives ˜ d r P d w ¸ “ r P ` x r P ´ r P ` Π , (3.2)where Π is an integration constant. If Π and x are related in such a way that the quarticpolynomial on the right-hand side of (3.2) has a double root r P , then r P p w q “ r P is an equilibriumsolution of (3.1). Double roots r P are necessarily related to x via the cubic equation3 r P ` x r P ´ “ x guaranteeing the existence of the double root can beexpressed in terms of a solution r P “ r P p x q of (3.3) byΠ “ Π p x q : “ r P p x q ´ x r P p x q . (3.4)It turns out (see Section 3.3.1 below) that this approximation of P p w q by the equilibriumsolution r P p x q accurately describes the rational Painlev´e-II function p m p y q in the pole-free region,provided that one selects the (unique) solution r P p x q of (3.3) with the asymptotic behavior r P p x q “ x ´ ` O ` x ´ ˘ as x Ñ 8 . This solution has branch points at x “ x c and x “ x c e ˘ π i { for x c : “ ´p { q { , which correspond to the corners of the triangular-shaped region containingthe poles. More general solutions of (3.1) can be expressed as elliptic functions of w with ellipticmodulus depending on the parameters x and Π. These also turn out to be important in describingthe rational Painlev´e-II functions in the interior of the triangular region. Indeed, if one fixesa value of x P C sufficiently small to correspond to y in the triangular region and views theational Solutions of the Painlev´e-II Equation Revisited 19 - - - - - - - - - - - - Figure 3.
The poles of residue 1 (blue) and ´ p m p y q for m “
15 (left), m “
30 (center), and m “
60 (right), plotted in the w -plane, a zoomed-in coordinate near y “ p m ´ q { x for x “ ´ { rational Painlev´e-II functions p m p y q as functions of the variable w , one sees increasingly regularpatterns of poles in the limit m Ñ 8 suggestive of the period parallelogram of an elliptic functionof w . See Fig. 3. A similar formal scaling argument can be applied to study the asymptoticbehavior of p m p y q near the corner points of the triangular region. For example, to zoom in onthe corner point on the negative real axis, we may make the scalings p “ ´ ´ m ¯ { ´ ˆ m ˙ { Y and y “ x c m { ` ˆ m ˙ { t, after which one sees that the Painlev´e-II equation (1.1) takes the formd Y d t “ Y ` t ` O ` m ´ { ˘ for t and Y bounded, i.e., a perturbation of the Painlev´e-I equation. This is a well-knowndegeneration of the Painlev´e-II equation [28, 30], and it suggests that particular solutions of thePainlev´e-I equation may play a role in the asymptotic description of p m p y q near the three cornerpoints. This also turns out to be true (see Section 3.3.4). Let r P p x q denote the solution of the cubic equation (3.3) with r P p x q “ x ´ ` O ` x ´ ˘ as x Ñ 8 ,which can be analytically continued to a maximal domain D consisting of the complex x -planeomitting three line segments connecting the three points x c , e ˘ π i { x c with the origin. For x P D ,let r p κ ; x q denote the function defined to satisfy r p κ ; x q “ κ ` r P p x q κ ` r P p x q ´ r P p x q ´ and r p κ ; x q “ κ ` O p q as κ Ñ 8 , defined on a maximal domain of analyticity in the κ -plane omitting only the segment connecting the roots of r p κ ; x q , one of which we denote by a p x q . Wedefine a function c p x q by c p x q : “ ż r P p x q a p x q p κ ´ r P p x qq r p κ ; x q d κ, x P D , (3.5)where the path of integration is arbitrary within the domain of analyticity of r p κ ; x q . The complex variable κ (written as z in [8, 9]) is a rescaling of the variable ζ from Riemann–Hilbert Prob-lem 2.2. It can be checked that the value of c p x q is unchanged by adding loops around the branch cut of r p κ ; x q to thepath of integration because r P p x q satisfies (3.3). m Ñ 8 , the region of the complex plane that contains thepoles of p m p y q is y P m { T , where T is the bounded component of the set of x P C for whichRe p c p x qq ‰
0. The boundary B T consists of points for which Re p c p x qq “
0. The integralin (3.5) can be evaluated in terms of elementary functions, taking appropriate care of branchesof multivalued functions; expressions can be found in [5, 8]. The exact formula is less importantthan the basic property that c p x q is analytic for x P D with algebraic branch points at the points x “ x c and x “ x c e ˘ π i { . This implies that B T is a union of three analytic arcs joining thebranch points pairwise, with reflection symmetry in the real axis and rotation symmetry aboutthe origin by integer multiples of 2 π {
3. The curve m { B T is superimposed on each of the poleplots in Fig. 2. We call T the elliptic region , the three branch points of r P p x q its corners , andthe three smooth arcs of B T its edges . Local analysis of c p x q shows [9, Section 2.3] that theinterior angles of B T at the three corners are all 2 π {
5, so that B T is a “curvilinear triangle” atbest. p m p y q by steepest descent We now present several results on the asymptotic behavior of the rational Painlev´e-II func-tion p m p y q , all of which have been obtained by the application of variants of the Deift–Zhousteepest descent method [15] to either Riemann–Hilbert Problem 2.2 (see [8, 9]) or Riemann–Hilbert Problem 2.4 (see [5]). Regardless of which Riemann–Hilbert problem is the startingpoint, the basic steps of the method are the same:1. Introduce a diagonal matrix multiplier built from exponentials of a scalar function fre-quently called a “ g -function” with the aim of simultaneously obtaining normalization tothe identity matrix at infinity and stabilizing the jump matrices of the problem so thatthey are alternately exponentially small perturbations of either constant matrices or purelyoscillatory matrices along different contour arcs. Frequently this step also requires somedeformation of the contour of the original Riemann–Hilbert problem by means of analyticcontinuation of the jump matrices.2. Use explicit matrix factorizations to algebraically separate oscillatory factors in the jumpmatrices having phase derivatives of opposite signs. Splitting the jump contour into sep-arate arcs for each factor, a subsequent deformation to either side of the original jumpcontour ensures that the oscillatory factors now become exponentially small in the limit m Ñ 8 .3. Construct an explicit model of the solution called a “parametrix” by considering only thoseremaining jump matrices that are not exponentially small perturbations of the identitymatrix.4. By comparing the unknown matrix obtained after the second step with the parametrix,obtain an equivalent Riemann–Hilbert problem for the matrix quotient. The aim of themethod is to ensure that the resulting Riemann–Hilbert problem is of “small-norm” type,meaning that it can be solved by a convergent iterative procedure that also allows forthe rigorous estimation of the solution. This analysis proves the accuracy of approximateformulae for the unknowns of interest, such as p m p y q , which are extracted from the explicitparametrix.The steepest descent method gets its name from the second step in the procedure, which resem-bles the type of contour deformations that one carries out in implementing the steepest descentmethod for the asymptotic expansion of exponential integrals.The form of the parametrix that one obtains is determined in most of the complex plane bythe number of contour arcs on which the g -function induces oscillations. This number is relatedational Solutions of the Painlev´e-II Equation Revisited 21to the genus of a hyperelliptic Riemann surface whose function theory is exploited to constructthe parametrix. As the original Riemann–Hilbert problem depends on a complex parameter y ,it is to be expected that the genus may be different for different values of y P C , leading tothe phenomenon of phase transitions. Indeed, the boundary of the elliptic region turns out tobe exactly such a phase transition. In particular the hyperelliptic curve that characterizes therational Painlev´e-II function p m p y q for large m when y lies outside of the elliptic region hasgenus zero. An interesting difference between the application of the steepest descent method tothe Jimbo–Miwa problem [8, 9] and its application to the Bertola–Bothner problem [5] is that inthe former case the curve corresponding to the elliptic region has genus 1 (hence the terminology“elliptic”) while in the latter case it instead has genus 2 (with some symmetries that allow itsfunction theory to be reducible to elliptic functions after all, see [5, Section 4.6]).We give no further details of the proofs of the following results, leading the reader to theoriginal references [5, 8, 9] for complete information. We also note that some of the resultsbelow have also been captured by the isomonodromy method, a WKB-ansatz based asymptoticapproach to Riemann–Hilbert problems [28]. p m in the exterior region The simplest result to state is the following.
Theorem 3.1 (Buckingham & Miller [8, Theorem 1], Bertola & Bothner [5, Corollary 6.1]) . Given a sufficiently large integer m ą , let K m be a set of points x in the exterior of T uniformly bounded away from the corners but otherwise with dist p x, T q ą ln p m q{ m . Then therational Painlev´e-II function p m p y q satisfies m ´ { p m ` m { x ˘ “ r P p x q ` O ` m ´ ˘ , m Ñ 8 with the error term being uniform for x P K m . In particular, p m ` m { x ˘ is pole free for x P K m and m sufficiently large. Recall that the limiting function r P p x q also has an interpretation as an equilibrium (“fast”variable w -independent) solution of the formal model differential equation (3.1). In [8] this resultis reported with an unimportant shift of the scaling parameter m ÞÑ m ´ in the argument of p m ,as this was convenient for the Riemann–Hilbert analysis used to prove the theorem. Once x moves into the elliptic region T and wild oscillations develop, this shift will have to be retainedto ensure full accuracy. p m in the elliptic region Now considering x P T , we define the integration constant Π in (3.2) no longer via (3.4) butrather via the following Boutroux conditions :Re ˜¿ a d r P d w d r P ¸ “ ˜¿ b d r P d w d r P ¸ “ , (3.6)where p a , b q is a basis of homology cycles on the elliptic curve Γ p x q determined as a subvarietyof C with coordinates p r P , d r P { d w q given by (3.2). In [8, Proposition 5] it is shown that theseconditions determine Π “ Π p x q uniquely as a continuous function on T with Π p q “
0. Moreover,the four roots of the polynomial on the right-hand side of (3.2) are then distinct for x P T , withtwo roots degenerating when x approaches an edge point of B T and all four roots degeneratingwhen x approaches a corner point of B T . The function Π p x q determined from the Boutrouxconditions (3.6) is smooth but decidedly non-analytic in x (cf. [8, equation (4.31)]).2 P.D. Miller and Y. Sheng κ (cid:45)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65) (cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:75) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:11) (cid:115) A p x q »– ´ ie ´p m ´ q u ` p x q e ´ wκ ´ ie p m ´ q u ` p x q e wκ fifl (cid:115) B p x q »– ´ ie ´p m ´ q u ´ p x q e ´ wκ ´ ie p m ´ q u ´ p x q e wκ fifl (cid:115) C p x q (cid:115) D p x q „ ´ ie ´ wκ ´ ie wκ Figure 4.
The branch cuts of R p κ ; x q for x “ W p κ ; x, w q for Riemann–HilbertProblem 3.2. Given a point x P T , we let A p x q , B p x q , C p x q , and D p x q denote the roots of the quartic R p κ ; x q “ κ ` xκ ´ κ ` Π p x q , observing that the notation is well-defined by continuity in x given that when x “ R p κ ; x q as an analyticfunction satisfying R p κ ; x q “ κ ` O p κ q as κ Ñ 8 and with branch cuts along line segmentsconnecting the four branch points as illustrated in Fig. 4. Now define u ` p x q : “ ż A p x q D p x q R p κ ; x q d κ and u ´ p x q : “ ż B p x q D p x q R p κ ; x q d κ, where the path of integration is in each case assumed to be a straight line. In order to presentthe results for x P T , we first formulate an auxiliary Riemann–Hilbert problem: Riemann–Hilbert Problem 3.2.
Let x P T and w P C be given and let m ě be an integer.Seek a ˆ matrix-valued function X m p κ ; x, w q defined for κ in the same domain where R p κ ; x q is analytic, with the following properties: • Analyticity. X m p κ ; x, w q is analytic in κ in its domain of definition, taking continuousboundary values X m ` p κ ; x, w q and X m ´ p κ ; x, w q from the left and right respectively on eachoriented arc of its jump contour as shown in Fig. , except at the four branch pointswhere ´ { power singularities are admitted. • Jump condition.
The boundary values are related by X m ´ p κ ; x, w q “ X m ` p κ ; x, w q W p κ ; x, w q , where the jump matrix W p κ ; x, w q is defined on each arc of the jump contour as shown inFig. . • Normalization.
The matrix X m p κ ; x, w q is normalized at κ “ 8 as follows: lim κ Ñ8 X m p κ ; x, w q “ I , where the limit may be taken in any direction. ational Solutions of the Painlev´e-II Equation Revisited 23The matrix X m p¨ ; x, w q is denoted O p out q p¨q in [8]. From the Laurent coefficients X m p x, w q : “ lim κ Ñ8 κ ` X m p κ ; x, w q ´ I ˘ , X m p x, w q : “ lim κ Ñ8 κ ` X m p κ ; x, w q ´ I ´ X m p x, w q κ ´ ˘ we then define a function r P m p x, w q by r P m p x, w q : “ X m , p x, w q ´ X m , p x, w q X m , p x, w q . Then we have the following result.
Theorem 3.3 (Buckingham & Miller [8, Proposition 7 & Theorem 2]) . For each x P T andinteger m ě , r P m p x, w q is an elliptic function of w that satisfies the model equation (3.1) p moreprecisely, with Π “ Π p x q defined as above, equation (3.2) q . Defining χ m p x, w q : “ , | r P m p x, w q| ď , ´ , | r P m p x, w q| ą , the asymptotic condition m ´ χ m p x,w q{ p m p y q χ m p x,w q “ r P m p x, w q χ m p x,w q ` O ` m ´ ˘ ,y “ ` m ´ ˘ { x ` ` m ´ ˘ ´ { w, (3.7) holds as m Ñ 8 uniformly for p x, w q in compact subsets of T ˆ C . The statement (3.7) says that m ´ { p m p y q and r P m p x, w q are uniformly close where r P m p x, w q is bounded, while their reciprocals are uniformly close where r P m p x, w q is bounded away fromzero. The fact that the approximating function r P m p x, w q depends on two variables deservessome explanation. Since w should be bounded for the indicated error estimate to be valid,variation of w amounts to the exploration of a small neighborhood of radius m ´ { of the point y “ ` m ´ ˘ { x . Thus fixing x P T and varying w one obtains a local approximation whosevalidity fails if w becomes large. It is on the w -scale that m ´ { p m p y q is well-approximated byan elliptic function of w , the meromorphic nature of which mirrors that of the original rationalPainlev´e-II function p m p y q . On the other hand, the same approximating formula (3.7) alsoallows x to vary within T ; here one may fix arbitrarily, say, w “ T that avoid poles, but that has an essentiallynon-meromorphic character due to the nonanalyticity of Π p x q . Geometrically, we may view T as a manifold with base coordinate x , while w plays the role of a coordinate on the tangentspace to T at x . Thus (3.7) approximates p m p y q with a function r P m p x, w q defined on thetangent bundle to T . We also can call x a macroscopic variable and w a microscopic variableto distinguish their different roles in (3.7).Numerous auxiliary results can be obtained from Theorem 3.3. Perhaps the main quantityof interest is the distribution of poles of residues ˘
1, which by (3.7) form regular lattices ofspacing proportional to m ´ { in the y -variable that slowly vary over distances proportionalto m { (the macroscopic x -scale) in the same variable. Bertola and Bothner characterize eachlattice globally via a pair of quantization conditions giving the lattice points as the intersectionsof two distinct families curves over T . In [8, Proposition 14] it is shown that, while the periodparallelograms of the lattices have limits in the w -plane as m Ñ 8 for given x P T , the offset ofthe lattices in the w -plane can fluctuate with m , accumulating a fixed shift with each increment4 P.D. Miller and Y. Sheng - - - - - - - - - - - - Figure 5.
The poles of residue 1 (blue) and ´ p m p y q for m “
58 (left), m “
59 (center), and m “
60 (right), plotted in the w -plane for x “
0. Note the shift of the lattices with m ; when x “
0, threeconsecutive shifts make up a lattice vector, so the asymptotic pattern has period 3 with respect to m .This dependence of the microscopic pattern near x “ m p mod 3 q has also been noted in a relatedproblem by Shapiro and Tater [37]. of m by a vector depending on the base point x P T ; see Fig. 5. As for how accurately the latticepoints approximate the poles of p m , it can be proved that the true poles of p m `` m ´ ˘ { x ˘ lying in any compact subset of T all move within the union of disks of radius of radius O ` { m ˘ centered at the lattice points (whose spacing in x is proportional to 1 { m ) if m is sufficientlylarge [8, Corollary 1]. See also [5, Theorem 1.6], where this result is formulated for disks ofradius o p { m q .In [8], formulae are also given for the asymptotic density of poles of p m `` m ´ ˘ { x ˘ asa function of x P T . Here, density is measured in terms of the microscopic coordinate w , andone may define both a planar density: r σ P p x q : “ lim M Ò8 t residue ´ w of r P m p x, w q with | w | ă M u πM , x P T, and a linear density of real poles for x P T X R : r σ L p x q : “ lim M Ò8 t real residue ´ w of r P m p x, w q in p´ M, M qu M , x P T X R . Since there are precisely two simple poles of opposite residue within each fundamental periodparallelogram of the elliptic function r P m p x, ¨q , the planar density is the reciprocal of the enclosedarea, which is readily calculated as a function of x (see [8, equation (4.144)]). The linear densityis similarly the reciprocal of the length of the period interval, since for x P T X R all poles arereal (modulo the period lattice). This leads to the explicit formula r σ L p x q “ « ż A p x q D p x q d κR p κ ; x q ` ż B p x q D p x q d κR p κ ; x q ff ´ ą , x P T X R . While the planar and linear densities are defined here from the known approximation r P m p x, w q ,they indeed capture the true local densities of poles of p m p m { x q [8, Theorem 5] in the limit oflarge m .Another type of result aims to capture the “local average” behavior of p m p y q . Here one notesthat as p m p y q has simple poles only, it is locally integrable with respect to area measure in the This statement corrects a mistake in equation (4.219) of [8]. Equations (4.217), (4.218), and (4.220) of thatreference should be similarly reformulated. ational Solutions of the Painlev´e-II Equation Revisited 25plane. Similarly, integrals of p m p y q with respect to Lebesgue measure on R are well-defined ifinterpreted in the principal-value sense. Thus, the following local averages are well-defined for x P T and x P T X R respectively: @ r P D p x q : “ ť p p x q r P m p x, w q d A p w q ť p p x q d A p w q , x P T, where p p x q denotes a period parallelogram and d A p w q is area measure in the w -plane, and @ r P D R p x q : “ L P . V . ż w ` Lw r P m p x, w q d w, x P T X R , where L is the length of a real period interval and w is not a pole of the integrand. Remarkably,as shown in [8, Proposition 11], these two quite different definitions actually agree where bothare defined: @ r P D R p x q “ @ r P D p x q , x P T X R . Also, x r P yp x q can be expressed in terms of basic quantities associated with the Riemann sur-face Γ p x q . It is furthermore shown in [8, Proposition 12] that x r P yp x q may be extended to thewhole complex x -plane as a continuous function by defining x r P yp x q : “ r P p x q (the distinguishedsolution of the cubic equation (3.3)) for x P C z T . This extended function is analytic in x outsideof T but fails to be analytic within T . Then we have the following result. Theorem 3.4 (Buckingham & Miller [8, Corollary 3 & Theorem 4]) . lim m Ñ8 m ´ { p m ` m { ˛ ˘ “ @ r P p˛q D , where the convergence is in the sense of the distributional topology on D p C zB T q . Also if ϕ P D pp C zB T q X R q is a smooth test function with compact real support avoiding B T , then lim m Ñ8 P . V . ż R m ´ { p m ` m { x ˘ ϕ p x q d x “ ż R @ r P D p x q ϕ p x q d x, expressing a similar distributional convergence where the integrals have to be interpreted in theprincipal value sense. p m near edges The function d p x q : “ c p x q ´ i π { B T that crosses the positive real x -axis, and it maps thisedge onto the imaginary segment with endpoints ˘ i π {
2. Also recalling the function r p κ ; x q fromSection 3.2, let r ˚ p x q : “ r ` r P p x q ; x ˘ and define (cid:96) p x q : “ ´
12 log ` r ˚ p x q r P p x q ˘ to be real for x P B T X R ` and analytically continued to the neighborhood of the sub-arc inquestion. Denoting by h n the leading coefficient of the normalized Hermite polynomial: h n : “ n { π { ? n ! , n “ , , , , . . . , d p x q ) by X mn p x q : “ d p x q ` ` n ` ˘ log ` m ´ ˘ m ´ ´ n ` m ´ (cid:96) p x q ` log ` ? πh n ˘ m ´ , n “ , , , , . . . . Finally, define the trigonometric functions T mn p x q by T mn p x q : “ ` coth `` m ´ ˘ X mn p x q ˘ , n ” m p mod 2 q , ` tanh `` m ´ ˘ X mn p x q ˘ , n ı m p mod 2 q , n “ , , , , . . . . Then we have the following result.
Theorem 3.5 (Buckingham & Miller [9, Theorem 2]) . Let arbitrarily small constants δ ą and σ ą , and an arbitrarily large constant M ą be given. Suppose that Re p d p x qq ě ´ M log p m q{ m and | arg p x q| ď π { ´ σ p this puts x in the sector containing the edge of B T of interest andprevents x from penetrating the elliptic region T by a distance greater than O p log p m q{ m qq .Suppose also that x is of distance at least δ { m from every pole of the functions T mn p x q , n “ , , , , . . . . Then m ´ { p m `` m ´ ˘ { x ˘ “ r P p x q ` ÿ n “ « ´ r ˚ p x q T mn p x q ` r P p x q r ˚ p x qp r ˚ p x q ´ r P p x qq T mn p x q r P p x q r ˚ p x qp r ˚ p x q ´ r P p x qq T mn p x q ´ ff ` O ` m ´ ˘ holds as m Ñ 8 uniformly for the indicated x . Note that the infinite series is easily seen to be convergent, and the whole series decays rapidlyto zero as m Ñ 8 if x lies outside of T , in which case this result agrees with Theorem 3.1.As x enters T , the terms in the series “turn on” one at a time, producing the curves of polesroughly parallel to the edge as can be seen in Fig. 2. Note that T mn p x q “ H mn p x q ` r P p x q “ ´ S p x q in the notation of [9]. One can observe from Theorem 3.5 that the curves ofpoles roughly correspond to the straight vertical lines Re p d p x qq “ ´ ` n ` ˘ log p m q{ m in the d -plane. There is also an interesting vertical “staggering” effect of the pole lattice as m varies.Indeed, given a value of α P ` ´ , ˘ , the poles of the approximation formula near the line indexedby n with | Im p d p x qq ´ πα | “ O p m ´ q form an approximate vertical lattice in the d -plane withspacing i π { m . The lattice is offset from the point d “ i πα ´ ` n ` ˘ log p m q{ m by a complexshift proportional to m ´ (i.e., proportional to the spacing) and depending on m , n , and α .Holding m fixed, one can observe that near the real axis this offset changes by approximatelyhalf of the lattice spacing with each consecutive value of n , and as x moves along the edgetoward the corner in the upper half-plane, this change in the offset with n gradually increasesto approximately 3 { n fixed and therefore lookingjust at the poles along the n th line from the edge, the change in offset with m is again half ofthe spacing near the real axis, but now the effect diminishes to zero as one moves along the edgetoward a corner of B T . This latter effect implies, in as much as one can draw conclusions fromTheorem 3.5 in the situation that x approaches a corner point along an edge, the pattern ofpoles of p m p y q should become independent of m near a corner point, even though it fluctuateswildly near typical points of T . A more precise version of this observation will be discussed inSection 3.3.4. p m near corners The Painlev´e-I equation Y p t q “ Y p t q ` t has a unique tritronqu´ee solution with the propertythat Y p t q “ ´ ˆ t ˙ { ` O ` t ´ ˘ , t Ñ 8 , | arg p´ t q| ď π ´ δ (3.8)ational Solutions of the Painlev´e-II Equation Revisited 27 - - - - - - - - - Figure 6.
The poles of residue 1 (blue) and ´ p p y q (left), p p y q (center), and p p y q (right),plotted in the complex t -plane, along with the boundary | arg p t q| “ π { Y p t q . Note how as m increases pairs of poles of opposite residues coalesce (each pairmoving toward a double pole of Y p t q ). for every δ ą
0; see Kapaev [29]. Thus the tritronqu´ee solution Y p t q is asymptotically pole-freein a sector of opening angle 4 π {
5. It has recently been proven [13] that in fact Y p t q is exactlypole-free for | arg p´ t q| ď π { | t | . This is the particular solution of thePainlev´e-I equation appearing in the formal analysis described in Section 3.1 that is needed todescribe the rational Painlev´e-II functions near corner points of T as the following result shows.Recall that x c : “ ´p { q { is the corner point of T on the negative real axis. Theorem 3.6 (Buckingham & Miller [9, Theorem 3]) . Let Y p t q be the tritronqu´ee solution ofthe Painlev´e-I equation determined by the asymptotic expansion (3.8) . If K is any compact setin the complex t -plane that does not contain any poles of Y p t q , then m ´ { p m `` m ´ ˘ { x ˘ “ ´ ´ { ´ m { ˆ ˙ { Y p t q ` O ` m ´ { ˘ holds as m Ñ 8 uniformly for t : “ ˆ ˙ { m { p x ´ x c q P K . This result is interesting in part because p m p y q is a function with simple poles only, and theapproximating function Y p t q is known to have double poles only. What actually happens in thelimit m Ñ 8 near the corner points is that pairs of simple poles of opposite residue for p m p y q merge into the “holes” excluded from K located near the double poles of Y . This phenomenoncan be clearly observed in the plots shown in [9]. The “pairing” of poles of opposite residuesnear the corners can also be seen in Fig. 6.Finally, we remark that the careful reader will observe that the various domains of thecomplex y -plane in which the asymptotic behavior of p m is now known actually do not overlap,so the whole complex plane has not been covered. The uniform asymptotic description of p m in neighborhoods of the edges and corners of T sufficiently large to achieve overlap remains anopen technical problem. Acknowledgements
P.D. Miller was supported during the preparation of this paper by the National Science Foun-dation under grant DMS-1513054. The authors are grateful to Thomas Bothner for many usefuldiscussions.8 P.D. Miller and Y. Sheng
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