We extend the technique of asymptotic series matching to exponential asymptotics expansions (transseries) and show that the extension provides a method of finding singularities of solutions of nonlinear differential equations, using asymptotic information. This transasymptotic matching method is applied to Painlevé's first equation. The solutions of P1 that are bounded in some direction towards infinity can be expressed as series of functions, obtained by generalized Borel summation of formal transseries solutions; the series converge in a neighborhood of infinity. We prove (under certain restrictions) that the boundary of the region of convergence contains actual poles of the associated solution. As a consequence, the position of these exterior poles is derived from asymptotic data. In particular, we prove that the location of the outermost pole $x_p(C)$ on $\RR^+$ of a solution is monotonic in a parameter $C$ describing its asymptotics on antistokes lines$^1$, and obtain rigorous bounds for $x_p(C)$. We also derive the behavior of $x_p(C)$ for large $C\in\CC$. The appendix gives a detailed classical proof that the only singularities of solutions of P1 are poles.
We analyze the one parameter family of tronquée solutions of the Painlevé equation \P1 in the pole-free sectors together with the region of the first array of poles. We find a convergent expansion for these solutions, containing one free parameter multiplying exponentially small corrections to the Borel summed power series. We link the position of the poles in the first array to the free parameter, and find the asymptotic expansion of the pole positions in this first array (in inverse powers of the independent variable). We show that the tritronquées are given by the condition that the paramet…
Using a simple operator-norm estimate we show that the solution to the second Painlevé equation within the Ablowitz-Segur family is pole-free in a well defined region of the complex plane of the independent variable. The result is illustrated with several numerical examples.
In this paper, we study the Painlevé VI equation with parameter $(\frac {9}{8},\frac{-1}{8},\frac{1}{8},\frac{3}{8})$. We prove (i) An explicit formula to count the number of poles of an algebraic solution with the monodromy group $D_{N}$, where $D_{N}$ is the dihedral group of order $2N$. (ii) There are only four solutions without poles in $\mathbb{C}\backslash \left \{ 0,1\right \} $. (iii) If the monodromy group of the associated linear ODE of a solution $λ\left( t\right) $ is unitary, then $λ( t) $ has no poles in $\mathbb{R}% \backslash \{ 0,1\} $.
For the Painlevé 6 transcendents, we provide a unitary description of the critical behaviours, the connection formulae, their complete tabulation, and the asymptotic distribution of the poles close to a critical point.
We show that the well-known Hastings-McLeod solution to the second Painlevé equation is pole-free in the region $\arg x \in [-\fracπ{3},\fracπ{3}]\cup [\frac{2π}{3},\frac{ 4 π}{3}]$, which proves an important special case of a general conjecture concerning pole distributions of Painlevé transcedents proposed by Novokshenov. Our strategy is to construct explicit quasi-solutions approximating the Hastings-McLeod solution in different regions of the complex plane, and estimate the errors rigorously. The main idea is very similar to the one used to prove Dubrovin's conjecture for the first Painlev…
We establish the existence of real pole-free solutions to all even members of the Painlevé I hierarchy. We also obtain asymptotics for those solutions and describe their relevance in the description of critical asymptotic behavior of solutions to the KdV equation in the small dispersion limit. This was understood in the case of a generic critical point, and we generalize it here to the case of non-generic critical points.
It is well-known that the first and second Painlevé equations admit solutions characterised by divergent asymptotic expansions near infinity in specified sectors of the complex plane. Such solutions are pole-free in these sectors and called tronquée solutions by Boutroux. In this paper, we show that similar solutions exist for the third and fourth Painlevé equations as well.
The paper is about a Painlevé III equation and its relation to isomonodromic families of vector bundles on P^1 with meromorphic connections. The purpose of the paper is two-fold: it offers a conceptual language for the geometrical objects underlying Painlevé equations, and it offers new results on a particular Painlevé III equation, which we denote by P_{III}(0,0,4,-4). This is equivalent to the radial sine (or sinh) Gordon equation and, as such, it appears very widely in geometry and physics. Complex multi-valued solutions on C^* are the natural context for most of the paper, but in the last …
We study the distribution of singularities (poles and zeros) of rational solutions of the Painlevé IV equation by means of the isomonodromic deformation method. Singularities are expressed in terms of the roots of generalised Hermite $H_{m,n}$ and generalised Okamoto $Q_{m,n}$ polynomials. We show that roots of generalised Hermite and Okamoto polynomials are described by an inverse monodromy problem for an anharmonic oscillator of degree two. As a consequence they turn out to be classified by the monodromy representation of a class of meromorphic functions with a finite number of singularities…
We consider a connection problem of the first Painlevé equation ($\mathrm{P_I}$), trying to connect the local behavior (Laurent series) near poles and the asymptotic behavior as the variable $t$ tends to negative infinity for real $\mathrm{P_I}$ functions. We get a classification of the real $\mathrm{P_I}$ functions in terms of $(p,H)$ so that they behave differently at the negative infinity, where $p$ is the location of a pole and $H$ is the free parameter in the Laurent series. Some limiting-form connection formulas of $\mathrm{P_I}$ functions are obtained for large $H$. Specifically, for th…
We prove that under a very general setting, a system of ODE passes the Painleve test if and only if there is a good change of variable, such that the pole singularity solutions are converted to regular power series, while the converted ODE system is still kept regular. A consequence is that all principal balances of an ODE system converge. We also prove that the results are natural with respect to Hamiltonian systems.
We establish the existence of a real solution y(x,T) with no poles on the real line of the following fourth order analogue of the Painleve I equation, x=Ty-({1/6}y^3+{1/24}(y_x^2+2yy_{xx})+{1/240}y_{xxxx}). This proves the existence part of a conjecture posed by Dubrovin. We obtain our result by proving the solvability of an associated Riemann-Hilbert problem through the approach of a vanishing lemma. In addition, by applying the Deift/Zhou steepest-descent method to this Riemann-Hilbert problem, we obtain the asymptotics for y(x,T) as x\to\pm\infty.
We obtain convergent representations (as Borel summed transseries) for the five one-parameter families of truncated solutions of the fifth Painlevé equation with nonzero parameters, valid in half planes, for large independent variable. We also find the position of the first array of poles, bordering the region of analyticity. For a special value of this parameter they represent tri-truncated solutions, analytic in almost the full complex plane, for large independent variable. A brief historical note, and references on truncated solutions of the other Painlevé equations are also included.
We study unitary invariant random matrix ensembles with singular potentials. We obtain asymptotics for the partition functions associated to the Laguerre and Gaussian Unitary Ensembles perturbed with a pole of order $k$ at the origin, in the double scaling limit where the size of the matrices grows, and at the same time the strength of the pole decreases at an appropriate speed. In addition, we obtain double scaling asymptotics of the correlation kernel for a general class of ensembles of positive-definite Hermitian matrices perturbed with a pole. Our results are described in terms of a hierar…
For the fifth Painlevé equation, we present families of convergent series solutions near the origin and the corresponding monodromy data for the associated isomonodromy linear system. These solutions are of complex power type, of inverse logarithmic type and of Taylor series type. It is also possible to compute the monodromy data in non-generic cases. Solutions of logarithmic type are derived from those of inverse logarithmic type through a Bäcklund transformation found by Gromak. In a special case the complex power type of solutions have relatively simple oscillatory expressions. For the comp…
We consider the second Painlevé equation $$ u"(x)=2u^3(x)+xu(x)-α, $$ where $α$ is a nonzero constant. Using the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems, we rigorously prove the asymptotics as $x \to \pm \infty$ for both the real and purely imaginary Ablowitz-Segur solutions, as well as the corresponding connection formulas. We also show that the real Ablowitz-Segur solutions have no real poles when $α\in (-1/2, 1/2)$.
We study the asymptotic behaviour of the solutions of the fifth Painlevé equation as the independent variable approaches zero and infinity in the space of initial values. We show that the limit set of each solution is compact and connected and, moreover, that any solution with the essential singularity at zero has an infinite number of poles and zeroes, and any solution with the essential singularity at infinity has infinite number of poles and takes value $1$ infinitely many times.
We consider a family of solutions to the Painlevé II equation $$ u''(x)=2u^3(x)+xu(x)-α\qquad \textrm{with } \a \in \mathbb{R} \cut \{0\}, $$ which have infinitely many poles on $(-\infty, 0)$. Using Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems, we rigorously derive their singular asymptotics as $x \to -\infty$. In the meantime, we extend the existing asymptotic results when $x\to +\infty$ from $\a-\frac{1}{2} \notin \mathbb{Z}$ to any real $\a$. The connection formulas are also obtained.
The increasing tritronquée solutions of the Painlevé-II equation with parameter $α$ exhibit square-root asymptotics in the maximally-large sector $|\arg(x)|<\tfrac{2}{3}π$ and have recently appeared in applications where it is necessary to understand the behavior of these solutions for complex values of $α$. Here these solutions are investigated from the point of view of a Riemann-Hilbert representation related to the Lax pair of Jimbo and Miwa, which naturally arises in the analysis of rogue waves of infinite order. We show that for generic complex $α$, all such solutions are asymptotically p…
The solutions of the perturbed first Painlevé equation $y"=6y^2-x^μ$, $μ>-4$, are uniquely determined by the free constant $C$ multiplying the exponentially small terms in the complete large $x$ asymptotic expansions. Full details are given, including the nonlinear Stokes phenomenon, and the computation of the relevant Stokes multipliers. We derive asymptotic approximations, depending on $C$, for the locations of the singularities that appear on the boundary of the sectors of validity of these exponentially-improved asymptotic expansions. Several numerical examples illustrate the power of the …
We consider the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions of the inhomogeneous Painlevé II equation $$ u"(x)=2u^3(x)+xu(x)-α\qquad \textrm{for } α\in \mathbb{R} \textrm{ and } |α| > \frac{1}{2}. $$ These solutions are obtained from the classical Ablowitz-Segur and Hastings-McLeod solutions via the Bäcklund transformation, and satisfy the same asymptotic behaviors when $x \to \pm \infty$. For $|α| > 1/2$, we show that the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions possess $[ \, |α| + \frac{1}{2} \, ]$ simple poles on the real axis, which rigorously justifies the num…
The Painleve-IV equation has three families of rational solutions generated by the generalized Hermite polynomials. Each family is indexed by two positive integers m and n. These functions have applications to nonlinear wave equations, random matrices, fluid dynamics, and quantum mechanics. Numerical studies suggest the zeros and poles form a deformed n by m rectangular grid. Properly scaled, the zeros and poles appear to densely fill certain curvilinear rectangles as m and n tend to infinity with r=m/n fixed. Generalizing a method of Bertola and Bothner used to study rational Painleve-II func…
The degenerate third Painleve' equation, $u"(t)=(u'(t))^2/u(t)-u'(t)/t+1/t(-8c u^2(t)+2ab)+b^2/u(t)$, where $c=+/-1$, $b>0$, and $a$ is a complex parameter, is studied via the Isomonodromy Deformation Method. Asymptotics of general regular and singular solutions $u(t)$ as $t -> +/-\infty$ and $t -> +/-i\infty$ are derived and parametrized in terms of the monodromy data of the associated 2X2 linear auxiliary problem introduced in the first part of this work [1]. Using these results, three-real-parameter families of solutions that have infinite sequences of zeroes and poles that are asymptotical…
We study dynamics of solutions in the initial value space of the sixth Painlevé equation as the independent variable approaches zero. Our main results describe the repeller set, show that the number of poles and zeroes of general solutions is unbounded, and that the complex limit set of each solution exists and is compact and connected.
We study factorizations of rational matrix functions with simple poles on the Riemann sphere. For the quadratic case (two poles) we show, using multiplicative representations of such matrix functions, that a good coordinate system on this space is given by a mix of residue eigenvectors of the matrix and its inverse. Our approach is motivated by the theory of discrete isomonodromic transformations and their relationship with difference Painlevé equations. In particular, in these coordinates, basic isomonodromic transformations take the form of the discrete Euler-Lagrange equations. Secondly we …
We show that there exists a rational change of coordinates of Painlevé's P1 equation $y''=6y^2+x$ and of the elliptic equation $y''=6y^2$ after which these two equations become analytically equivalent in a region in the complex phase space where $y$ and $y'$ are unbounded. The region of equivalence comprises all singularities of solutions of P1 (i.e. outside the region of equivalence, solutions are analytic). The Painlevé property of P1 (that the only movable singularities are poles) follows as a corollary. Conversely, we argue that the Painlevé property is crucial in reducing P1, in a singula…
We study the asymptotic behaviour of the solutions of the generic ($D_6^{(1)}$-type) third Painlevé equation in the space of initial values as the independent variable approaches infinity (or zero) and show that the limit set of each solution is compact and connected. Moreover, we prove that any solution with essential singularity at infinity has an infinite number of poles and zeroes, and similarly at the origin.
The Painlevé-III equation with parameters $Θ_0=n+m$ and $Θ_\infty=m-n+1$ has a unique rational solution $u(x)=u_n(x;m)$ with $u_n(\infty;m)=1$ whenever $n\in\mathbb{Z}$. Using a Riemann-Hilbert representation proposed in \cite{BothnerMS18}, we study the asymptotic behavior of $u_n(x;m)$ in the limit $n\to+\infty$ with $m\in\mathbb{C}$ held fixed. We isolate an eye-shaped domain $E$ in the $y=n^{-1}x$ plane that asymptotically confines the poles and zeros of $u_n(x;m)$ for all values of the second parameter $m$. We then show that unless $m$ is a half-integer, the interior of $E$ is filled with …
We study the asymptotic behaviour of solutions of the fourth Pain\-levé equation as the independent variable goes to infinity in its space of (complex) initial values, which is a generalisation of phase space described by Okamoto. We show that the limit set of each solution is compact and connected and, moreover, that any non-special solution has an infinite number of poles and infinite number of zeroes.
The Painlevé classification is one of the central problems in analytics theory of differential equations rooted in the XIX century. Although it saw many significant advances in analyzing certain classes of equations, the classification still remains an open problem especially for the higher-order equations. One of the main classical methods of Painlevé analysis is based on considering the resonance numbers corresponding to the possible indices of arbitrary coefficients in the Laurent expansion of the general solution in a neighborhood of a movable singularity. Complex and non-integer values of…
We show that the fourth order nonlinear ODE which controls the pole dynamics in the general solution of equation $P_I^2$ compatible with the KdV equation exhibits two remarkable properties: 1) it governs the isomonodromy deformations of a $2\times2$ matrix linear ODE with polynomial coefficients, and 2) it does not possesses the Painlevé property. We also study the properties of the Riemann--Hilbert problem associated to this ODE and find its large $t$ asymptotic solution for the physically interesting initial data.
The classical Painlevé equations are so well known that it may come as a surprise to learn that the asymptotic description of its solutions remains incomplete. The problem lies mainly with the description of families of solutions in the complex domain. Where asymptotic descriptions are known, they are stated in the literature as valid for large connected domains, which include movable poles of families of solutions. However, asymptotic analysis necessarily assumes that the solutions are bounded and so these domains must be punctured at locations corresponding to movable poles, leading to asymp…
We study the solutions of the second Painlevé equation in the space of initial conditions first constructed by Okamoto, in the limit as the independent variable, x, goes to infinity. Simultaneously, we study solutions of the related equation known as the thirty-fourth Painlevé equation. By considering degenerate cases of the autonomous flow, we recover the known special solutions, which are either rational functions or expressible in terms of Airy functions. We show that the solutions that do not vanish at infinity possess an infinite number of poles. An essential element of our construction i…
The Painleve-IV equation has two families of rational solutions generated respectively by the generalized Hermite polynomials and the generalized Okamoto polynomials. We apply the isomonodromy method to represent all of these rational solutions by means of two related Riemann-Hilbert problems, each of which involves two integer-valued parameters related to the two parameters in the Painleve-IV equation. We then use the steepest-descent method to analyze the rational solutions in the limit that at least one of the parameters is large. Our analysis provides rigorous justification for formal asym…
In a 1979 paper, K. Okamoto introduced the space of initial values for the six Painlevé equations and their associated Hamiltonian systems, showing that these define regular initial value problems at every point of an augmented phase space, a rational surface with certain exceptional divisors removed. We show that the construction of the space of initial values remains meaningful for certain classes of second-order complex differential equations, and more generally, Hamiltonian systems, where all movable singularities of all their solutions are algebraic poles (by some authors denoted the quas…
Poles of solutions to the Painleve-I equation are intimately related to the theory of the cubic anharmonic oscillator. In particular, poles of integrale tritronquee are in 1-1 correspondence with cubic oscillators that admit the simultaneous solutions of two quantization conditions. We analyze this pair of quantization conditions by means of a suitable version of the complex WKB method.
We obtain the double scaling asymptotic behavior of the recurrence coefficients and the partition function at the critical point of the $N\times N$ Hermitian random matrix model with cubic potential. We prove that the recurrence coefficients admit an asymptotic expansion in powers of $N^{-2/5}$, and in the leading order the asymptotic behavior of the recurrence coefficients is given by a Boutroux tronquée solution to the Painlevé I equation. We also obtain the double scaling limit of the partition function, and we prove that the poles of the tronquée solution are limits of zeros of the partiti…
The critical behavior of a three real parameter class of solutions of the sixth Painlevé equation is computed, and parametrized in terms of monodromy data of the associated $2\times 2$ matrix linear Fuchsian system of ODE. The class may contain solutions with poles accumulating at the critical point. The study of this class closes a gap in the description of the transcendents in one to one correspondence with the monodromy data. These transcendents are reviewed in the paper. Some formulas that relate the monodromy data to the critical behaviors of the four real (two complex) parameter class of…
Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III $({\rm D}_7)$ Equation
All of the six Painlevé equations except the first have families of rational solutions, which are frequently important in applications. The third Painlevé equation in generic form depends on two parameters $m$ and $n$, and it has rational solutions if and only if at least one of the parameters is an integer. We use known algebraic representations of the solutions to study numerically how the distributions of poles and zeros behave as $n\in\mathbb{Z}$ increases and how the patterns vary with $m\in\mathbb{C}$. This study suggests that it is reasonable to consider the rational solutions in the li…
For the Schlesinger-type equation related to the fifth Painlevé equation (V) via isomonodromy deformation, we present a three-parameter family of matrix solutions along the imaginary axis near the point at infinity, and also the corresponding monodromy data. Two-parameter solutions of (V) with their monodromy data immediately follow from our results. Under certain conditions, these solutions of (V) admit sequences of zeros and of poles along the imaginary axis. The monodromy data are obtained by matching techniques for a perturbed linear system.
We show that the tritronquée solution of the Painlevé equation $\P1$, $ y"=6y^2+z$ which is analytic for large $z$ with $ \arg z \in (-\frac{3π}{5}, π)$ is pole-free in a region containing the full sector ${z \ne 0, \arg z \in [-\frac{3π}{5}, π]}$ and the disk ${z: |z| < 37/20}$. This proves in particular the Dubrovin conjecture, an open problem in the theory of Painlevé transcendents. The method, building on a technique developed in Costin, Huang, Schlag (2012), is general and constructive. As a byproduct, we obtain the value of the tritronquée and its derivative at zero within less than 1/10…
We analyze the large-$n$ behavior of soliton solutions of the integrable focusing nonlinear Schrödinger equation with associated spectral data consisting of a single pair of conjugate poles of order $2n$. Starting from the zero background, we generate multiple-pole solitons by $n$-fold application of Darboux transformations. The resulting functions are encoded in a Riemann-Hilbert problem using the robust inverse-scattering transform method recently introduced by Bilman and Miller. For moderate values of $n$ we solve the Riemann-Hilbert problem exactly. With appropriate scaling, the resulting …
We try to obtain meromorphic solution of Time dependent Schroedinger equation which partially satisfies Painleve Integrable property. Our study and analysis exhibits meromorphic behavior of classical particle trajectory. In other words, particle is confined in punctured complex domain in singular fundamental domain. . We have explicitly developed solution from Jacobi Elliptic function and pole expansion approach in which solution remains meromorphic . Branch point analysis also shows solution branches out near such singular point. Meromorphic behavior is significant for a classical particle wi…
We study the semiclassical limit of the sine-Gordon (sG) equation with below threshold pure impulse initial data of Klaus-Shaw type. The Whitham averaged approximation of this system exhibits a gradient catastrophe in finite time. In accordance with a conjecture of Dubrovin, Grava and Klein, we found that in a $\mathcal{O}(ε^{4/5})$ neighborhood near the gradient catastrophe point, the asymptotics of the sG solution are universally described by the Painlevé I tritronquée solution. A linear map can be explicitly made from the tritronquée solution to this neighborhood. Under this map: away from …
