Related papers: Asymptotics for the noncommutative Painlev\'{e} II…
In this paper, we study the asymptotic behavior of a family of pole-free solutions to the noncommutative Painlev\'e II equation. These particular solutions can be expressed in terms of the Fredholm determinant of the matrix version of the…
It is shown that a generalization of the Painlev\'e-II equation (P-II) to a system of coupled equations with symmetry breaking terms is integrable. A Lax pair for this system is used to relate the asymptotic behavior of the solutions at…
We consider real-valued solutions $u=u(x|s),x\in\mathbb{R}$ of the second Painlev\'e equation $u_{xx}=xu+2u^3$ which are parametrized in terms of the monodromy data $s\equiv(s_1,s_2,s_3)\subset\mathbb{C}^3$ of the associated Flaschka-Newell…
We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form $\frac{1}{Z_n} \big|\det \big( M^2-tI \big)\big|^{\alpha} e^{-n\operatorname{Tr} V(M)}dM$, where $M$ is an $n\times…
In this work we propose a new method for investigating connection problems for the class of nonlinear second-order differential equations known as the Painlev{\'e} equations. Such problems can be characterized by the question as to how the…
We consider a family of solutions to the Painlev\'e II equation $$ u''(x)=2u^3(x)+xu(x)-\alpha \qquad \textrm{with } \a \in \mathbb{R} \cut \{0\}, $$ which have infinitely many poles on $(-\infty, 0)$. Using Deift-Zhou nonlinear steepest…
The third Painlev\'e equation in its generic form, often referred to as Painlev\'e-III($D_6$), is given by $$ \frac{{\rm d}^2u}{{\rm d}x^2} =\frac{1}{u}\left(\frac{{\rm d}u}{{\rm d}x}\right)^2-\frac{1}{x}\frac{{\rm d}u}{{\rm…
We consider two special cases of the connection problem for the second Painlev\'e equation (PII) using the method of uniform asymptotics proposed by Bassom et al.. We give a classification of the real solutions of PII on the negative…
The asymptotic solution for the Painleve-2 equation with small parameter is considered. The solution has algebraic behavior before point $t_*$ and fast oscillating behavior after the point $t_*$. In the transition layer the behavior of the…
In this paper, we revisit the asymptotic formulas of real Painlev\'e I transcendents as the independent variable tends to negative infinity, which were initially derived by Kapaev with the complex WKB method. Using the Riemann-Hilbert…
In this paper we discuss Airy solutions of the second Painlev\'e equation (\mbox{\rm P$_{\rm II}$}) and two related equations, the Painlev\'e XXXIV equation ($\mbox{\rm P}_{34}$) and the Jimbo-Miwa-Okamoto $\sigma$ form of \mbox{\rm P$_{\rm…
We obtain asymptotics of polynomials satisfying the orthogonality relations $$ \int_{\mathbb{R}} z^k P_n(z; t , N) \mathrm{e}^{-N \left(\frac{1}{4}z^4 + \frac{t}{2}z^2 \right)} \mathrm{d} z = 0 \quad \text{ for } \quad k = 0, 1, ..., n-1,…
In this paper, we consider Cauchy problem for the modified Korteweg-de Vries hierarchy on the real line with decaying initial data. Using the Riemann--Hilbert formulation and nonlinear steepest descent method, we derive a uniform asymptotic…
In this paper, we find all possible asymptotic behaviours of the solutions of the second Painlev\'e equation $y''=2y^3+xy +\alpha$ as the parameter $\alpha\to\infty$ in the local region $x\ll\alpha^{2/3}$. We prove that these are asymptotic…
We show that the alternative discrete Painlev\'e I equation (alt-dP$_{\rm I}$) has a unique solution which remains positive for all $n \geq 0$. Furthermore, we identify this positive solution in terms of a special solution of the second…
The rational solutions of the Painlev\'e-II equation appear in several applications and are known to have many remarkable algebraic and analytic properties. They also have several different representations, useful in different ways for…
We consider the Clarkson-McLeod solutions of the fourth Painlev\'e equation. This family of solutions behave like $\kappa D_{\alpha-\frac{1}{2}}^2(\sqrt{2}x)$ as $x\rightarrow +\infty$, where $\kappa $ is an arbitrary real constant and…
In this paper we study the asymptotic behavior for large argument of a family of solutions of the Painlev\'e equation P$_{\rm VI} arising in the context of Random Matrix Theory [1]. We show this family of solutions are uniquely determined…
The generalized Hastings-McLeod solutions to the inhomogeneous Painlev\'{e}-II equation arise in multi-critical unitary random matrix ensembles, the chiral two-matrix model for rectangular matrices, non-intersecting squared Bessel paths,…
Bilinear structure for the discrete Painlev\'e I equation is investigated. The solution on semi-infinite lattice is given in terms of the Casorati determinant of discrete Airy function. Based on this fact, the discrete Painlev\'e I equation…