相关论文: One-Parameter Meromorphic Solution of the Degenera…
Various properties of algebroid solutions of the degenerate third Painlev\'e equation, \begin{equation*} u^{\prime \prime}(\tau) \! = \! \frac{(u^{\prime}(\tau))^{2}}{u(\tau)} \! - \! \frac{u^{\prime}(\tau)}{\tau} \! + \! \frac{1}{\tau} \!…
We prove that there exists the unique odd meromorphic solution of dP3, $u(\tau)$ such that $u(0)=0$, and study some of its properties, mainly: the coefficients of its Taylor expansion at the origin and asymptotic behaviour as…
We give a classification for the small-$\tau$ asymptotic behaviours of solutions to the degenerate third Painlev\'e equation, $u^{''}(\tau) = \frac{(u^{\prime}(\tau))^{2}}{u(\tau)} - \frac{u^{\prime}(\tau)}{\tau} + \frac{1}{\tau}\left(-8…
The degenerate third Painlev\'{e} equation, $u^{\prime \prime} = \frac{(u^{\prime})^{2}}{u} - \frac{u^{\prime}}{\tau} + \frac{1}{\tau}(-8 \epsilon u^{2} + 2ab) + \frac{b^{2}}{u}$, where $\epsilon,b \in \mathbb{R}$, and $a \in \mathbb{C}$,…
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…
The Painlev\'e-III equation with parameters $\Theta_0=n+m$ and $\Theta_\infty=m-n+1$ 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…
A one-parameter family of trans-series asymptotics of solutions to the Degenerate Painlev\'{e} III Equation (DP3E) are parametrised in terms of the monodromy data of an associated two-by-two linear auxiliary problem via the isomonodromy…
In this paper, we study the isomonodromy deformation equations for the $n\times n$ system of first order meromorphic linear ordinary differential equations with two second order poles. We analyze the asymptotic behaviour of the solutions at…
We study a twistor correspondence based on the Joukowski map reduced from one for stationary-axisymmetric self-dual Yang-Mills and adapt it to the Painlev\'e III equation. A natural condition on the geometry (axissimplicity) leads to…
The third Painlev\'e equation in its generic form, often referred to as Painlev\'e-III($D_6$), is given by $$ \frac{{\rm d}^2u}{{\rm d}x^2} =\frac{1}{u}\left(\frac{{\rm d}u}{{\rm d}x}\right)^2-\frac{1}{x}\frac{{\rm d}u}{{\rm…
We analyze the asymptotic properties a special solution of the $(3,4)$ string equation, which appears in the study of the multicritical quartic $2$-matrix model. In particular, we show that in a certain parameter regime, the corresponding…
It is known from \cite{LW} that the solvability of the mean field equation $\Delta u+e^{u}=8n\pi \delta_{0}$ with $n\in\mathbb{N}_{\geq 1}$ on a flat torus $E_{\tau}$ essentially depends on the geometry of $E_{\tau}$. A conjecture is the…
We study underdetermined-elliptic linear partial differential operators $P$ on asymptotically Euclidean manifolds, such as the divergence operator on 1-forms or symmetric 2-tensors. Suitably interpreted, these are instances of (weighted)…
We investigate the degenerate fractional Schr\"{o}dinger-Kirchhoff-Poisson equation in $\mathbb{R}^3$ with critical nonlinearity and electromagnetic fields $\varepsilon^{2s} M([u]_{s,A}^2)(-\Delta)_{A}^su + V(x)u + \phi u = k(x)|u|^{r-2}u +…
For the Schlesinger-type equation related to the fifth Painlev\'e equation (V) via isomonodromy deformation, we present a three-parameter family of matrix solutions along the imaginary axis near the point at infinity, and also the…
The increasing tritronqu\'ee solutions of the Painlev\'e-II equation with parameter $\alpha$ exhibit square-root asymptotics in the maximally-large sector $|\arg(x)|<\tfrac{2}{3}\pi$ and have recently appeared in applications where it is…
We consider a family of solutions to the Painlev\'e II equation $$ 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…
We consider the higher order asymptotics for the mKdV equation in the Painlev\'e sector. We first show that the solution admits a uniform expansion to all orders in powers of $t^{-1/3}$ with coefficients that are smooth functions of…
In this paper we construct explicit solutions and calculate the corresponding $\tau$-function to the system of Schlesinger equations describing isomonodromy deformations of $2\times 2$ matrix linear ordinary differential equation whose…
We study the nonlocal scalar field equation with a vanishing parameter \[ \left\{\begin{array}{lll} (-\Delta)^s u+\epsilon u &=|u|^{p-2}u -|u|^{q-2}u \quad\text{in}\quad\mathbb{R}^N \\ u >0, & u \in H^s(\mathbb{R}^N), \end{array} \right. \]…