N. Joshi, M. Mazzocco
The first five classical Painlevé equations are known to have solutions described by divergent asymptotic power series near infinity. Here we prove that such solutions also exist for the infinite hierarchy of equations associated with the second Painlevé equation. Moreover we prove that these are unique in certain sectors near infinity.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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.
We prove that for any autonomous 4-dimensional integral system of Painlevé type, the Jacobian of the generic spectral curve has a unique polarization, and thus by Torelli's theorem cannot be isomorphic as an unpolarized abelian surface to any other Jacobian. This enables us to identify the spectral curve and any irreducible genus two component of the boundary of an affine patch of the Liouville torus.
Generalization of the Painlevé Property and Existence and Uniqueness in Fractional Differential Equations
本文推广了Painlevé性质至分数阶微分方程，建立了线性与非线性分数阶微分方程的存在唯一性定理，推动了分数阶微积分的可积性研究，为建模记忆和遗传特性现象提供基础。
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…
Recently in a paper by Lin, Dai and Tibboel, it was shown that the third and fourth Painlevé equations have tronquée and tritronquée solutions. We obtain global information about these tronquée and tritronquée solutions. We find their sectors of analyticity, their Borel summed representations in these sectors as well as the asymptotic position of the singularities near the boundaries of the analyticity sectors. We also correct slight errors in the paper mentioned.
The existence, uniqueness and convergence of formal Puiseux series solutions of non-autonomous algebraic differential equations of first order at a nonsingular point of the equation is studied, including the case where the celebrated Painleve theorem cannot be applied explicitly for the study of convergence. Several examples illustrating relationships to the Painleve theorem and lesser-known Petrovic's results are provided.
Rational Solutions of the Fifth Painlevé Equation. Generalised Laguerre Polynomials
本文研究第五潘勒韦方程的有理解，用广义拉盖尔多项式表示解，确定其性质、微分-差分方程、根结构与判别式，并给出朗斯基拉盖尔多项式系数的显式表达式。
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…
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 present a constructive procedure to obtain the critical behavior of Painleve' VI transcendents and solve the connection problem. This procedure yields two and one parameter families of solutions, including trigonometric and logarithmic behaviors, and three classes of solutions with Taylor expansion at a critical point.
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 …
On the Riemann-Hilbert approach to asymptotics of tronquée solutions of Painlevé I
本文利用Riemann-Hilbert方法与最速下降法，通过构造含互补误差函数的局部参数化子，给出Painlevé I方程tronquée解超出庞加莱展开的非微扰贡献细节。
Resurgence of Tritronquées Solutions of the Deformed Painlevé I Equation
本文证明了变形 Painlevé I 方程的形式 $\hbar$ 幂级数解具有复苏性，揭示了其 Borel 变换在 Fermat 五次曲面上的全局多值全纯结构及五重覆盖映射下的奇点分布。
We prove that if y" = f(y,y',t,α, β,..) is a generic Painleve equation (i.e. an equation in one of the families PI-PVI but with the complex parameters α, β,.. algebraically independent) then any algebraic dependence over C(t) between a set of solutions and their derivatives (y_1,..,y_n,y_1',..,y_n') is witnessed by a pair of solutions and their derivatives (y_i,y_i',y_j,y_j'). The proof combines work by the Japanese school on "irreducibility" of the Painleve equations, with the trichomoty theorem for strongly minimal sets in differentially closed fields.
Full Asymptotic Expansion of Monodromy Data for the First Painlevé Transcendent: Applications to Connection Problems
本文通过改进复 WKB 方法，严格证明了第一类 Painlevé 超越函数在连接问题中单值数据的完全渐近展开，并给出了非线性特征值及极点参数的完整展开。
We construct algebro-geometric upper triangular solutions of rank two Schlesinger systems. Using these solutions we derive two families of solutions to the sixth Painlevé equation with parameters $({1}/{8}, -{1}/{8}, {1}/{8}, {3}/{8})$ expressed in simple forms using periods of differentials on elliptic curves. Similarly for every integer $n$ different from $0$ and $-1$ we obtain one family of solutions to the sixth Painlevé equation with parameters $(\frac{9n^2+12n+4}{8}, -\frac{n^2}{8}, \frac{n^2}{8}, \frac{4-n^2}{8})$.
In the first part of our paper we discuss linear 2nd order differential equations in the complex domain, especially Heun class equations, that is, the Heun equation and its confluent cases. The second part of our paper is devoted to Painlevé I-VI equations. Our philosophy is to treat these families of equations in a unified way. This philosophy works especially well for Heun class equations. We discuss its classification into 5 supertypes, subdivided into 10 types (not counting trivial cases). We also introduce in a unified way deformed Heun class equations, which contain an additional nonloga…
We study the fundamental rogue wave solutions of the focusing nonlinear Schrödinger equation in the limit of large order. Using a recently-proposed Riemann-Hilbert representation of the rogue wave solution of arbitrary order $k$, we establish the existence of a limiting profile of the rogue wave in the large-$k$ limit when the solution is viewed in appropriate rescaled variables capturing the near-field region where the solution has the largest amplitude. The limiting profile is a new particular solution of the focusing nonlinear Schrödinger equation in the rescaled variables --- the rogue wav…
We modify the classical Paley-Wiener spaces $PW_x$ of entire functions of finite exponential type at most $x>0$, which are square integrable on the real line, via the additional condition of vanishing at finitely many complex points $z_1, ..., z_n$. We compute the reproducing kernels and relate their variations with respect to $x$ to a Krein differential system, whose coefficient (which we call the $μ$-function) and solutions have determinantal expressions. Arguments specific to the case where the "trivial zeros" $z_1, ..., z_n$ are in arithmetic progression on the imaginary axis allow us to e…
We study solutions $(x_n)_{n \in \mathbb{N}}$ of nonhomogeneous nonlinear second order difference equations of the type $\ell_n = x_n ( σ_{n,1} x_{n+1} + σ_{n,0} x_n + σ_{n,-1} x_{n-1} ) + κ_n x_n$, with given initial data $x_0 \in \mathbb{R}$, $x_1 \in \mathbb{R}^+$ where $(\ell_n)_{n\in\mathbb{N}} \in \mathbb{R}^+$, $(σ_{n,0})_{n\in\mathbb{N}} \in \mathbb{R}^+$ and $(κ_n)_{n\in\mathbb{N}} \in \mathbb{R}$ and the left and right $σ$-coefficients satisfy either $(σ_{n,1})_{n\in\mathbb{N}} \in \mathbb{R}^+$ and $(σ_{n,-1})_{n\in \mathbb{N}} \in \mathbb{R}^+$ or $(σ_{n,1})_{n\in\mathbb{N}} \in \m…
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…
The equations governing one-dimensional, steady-state electrodiffusion are considered when there are arbitrarily many mobile ionic species present, in any number of valence classes, possibly also with a uniform distribution of fixed charges. Exact constant field solutions and new formulas of Goldman-Hodgkin-Katz type are found. All of these formulas are exact, unlike the usual approximate ones. Corresponding boundary conditions on the ionic concentrations are identified. The question of uniqueness of constant field solutions with such boundary conditions is considered, and is re-posed in terms…
A second order ordinary differential equation with a superlinear term $g(x,u)$ under radiation boundary conditions is studied. Using a shooting argument, all the results obtained in a previous work for a Painlevé II equation are extended. It is proved that the uniqueness or multiplicity of solutions depend on the interaction between the mapping $\frac {\partial g}{\partial u}(\cdot,0)$ and the first eigenvalue of the associated linear operator. Furthermore, two open problems regarding, on the one hand, the existence of sign-changing solutions and, on the other hand, exact multiplicity are solv…
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…
In this paper we use the Riemann-Hilbert problem, with jumps supported on appropriate curves in the complex plane, for matrix biorthogonal polynomials and apply it to find Sylvester systems of differential equations for the orthogonal polynomials and its second kind functions as well. For this aim, Sylvester type differential Pearson equations for the matrix of weights are shown to be instrumental. Several applications are given, in order of increasing complexity. First, a general discussion of non-Abelian Hermite biorthogonal polynomials in the real line, understood as those whose matrix of w…
Transition asymptotics for the real solutions of the sinh-Gordon Painlevé III equation
研究 sinh-Gordon Painlevé III 方程实解在参数和变量同时趋于无穷时的过渡行为，揭示奇异解与光滑解之间的渐近转换。
The theory of orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference equations of certain coefficient functions appearing in the theory. A natural formulation of the Riemann-Hilbert problem is presented which has as its solution the above system of orthogonal polynomials and associated functions. In particular for the case of regular semi-classical weights on the unit circle $ w(z) = \prod^m_{j=1}(z-z_j(t))^{ρ_j} $, consisting…
Painlevé-III Monodromy Maps Under the $D_6\to D_8$ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions
该研究通过Bäcklund变换生成Painlevé-III($D_6$)的解族，结合级数分析与Riemann-Hilbert方法证明其在大参数标度下收敛于Painlevé-III($D_8$)的解，并推导了相关有理解、Umemura多项式的渐近性质。
Asymptotics and total integrals of the $\mathrm{P}_{\rm I}^{2}$ tritronquée solution and its Hamiltonian
本文研究了 Painlevé I 层级第二成员 $\mathrm{P}_{\rm I}^{2}$ 方程的三截断解 $u(x,t)$，获得了其在 $x\to\pm \infty$ 时的一致完整渐近展开，并据此成功导出了该解及其相关哈密顿量的总积分。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。