相关论文: Riemann-Hilbert characterisation of Painlev\'e 5 a…

The classical Painlev\'e 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…

Leading terms of asymptotic expansions for the general complex solutions of the fifth Painlev\'e equation as $t\to\imath\infty$ are found. These asymptotics are parameterized by monodromy data of the associated linear ODE. $$…

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…

For the fifth Painlev\'e transcendents an asymptotic representation by the Jacobi $\mathrm{sn}$-function is presented in cheese-like strips along generic directions near the point at infinity. Its elliptic main part may be understood to…

For the fifth Painlev\'e 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…

We study the sixth $q$-difference Painlev\'e equation ($q{\textrm{P}_{\textrm{VI}}}$) through its associated Riemann-Hilbert problem (RHP) and show that the RHP is always solvable for irreducible monodromy data. This enables us to identify…

We study the solutions of the sixth Painlev\'e equation with a logarithmic asymptotic behavior at a critical point. We compute the monodromy group associated to the solutions by the method of monodromy preserving deformations and we…

Motivated by the simplest case of tt*-Toda equations, we study the large and small $x$ asymptotics for $x>0$ of real solutions of the sinh-Godron Painlev\'e III($D_6$) equation. These solutions are parametrized through the monodromy data of…

The integrability (solvability via an associated single-valued linear problem) of a differential equation is closely related to the singularity structure of its solutions. In particular, there is strong evidence that all integrable…

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 the asymptotic behavior of the Ablowitz-Segur solutions for the second Painlev\'e equation using the Riemann-Hilbert approach and methods based on asymptotic expansions of classical special functions. Recent results show that the…

This is a sequel to papers by the last two authors making the Riemann-Hilbert correspondence and isomonodromy explicit. For the degenerate fifth Painlev\'e equation, the moduli spaces for connections and for monodromy are explicitly…

Explicit solutions to the Riemann-Hilbert problem will be found realising some irreducible non-rigid local systems. The relation to isomonodromy and the sixth Painleve equation will be described. Keywords: Riemann-Hilbert problem, Painleve…

We study moduli spaces of meromorphic connections (with arbitrary order poles) over Riemann surfaces together with the corresponding spaces of monodromy data (involving Stokes matrices). Natural symplectic structures are found and described…

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…

For the fifth Painlev\'e equation it is known that a general solution is represented asymptotically by an elliptic function in cheese-like strips near the point at infinity. We present an explicit asymptotic formula for the error term of…

We investigate the long-time asymptotic behavior of a class of solutions to the defocusing Manakov system under nonzero boundary conditions. These solutions are characterized by a $3 \times 3$ matrix Riemann Hilbert problem. We find that…

In the context of $q$-Painlev\'e VI with generic parameter values, the Riemann-Hilbert correspondence induces a one-to-one mapping between solutions of the nonlinear equation and points on an affine Segre surface. Upon fixing a generic…

These lectures introduce the method of nonlinear steepest descent for Riemann-Hilbert problems. This method finds use in studying asymptotics associated to a variety of special functions such as the Painlev\'{e} equations and orthogonal…

For a general solution of the third Painlev\'e equation of complete type we show the Boutroux ansatz near the point at infinity. It admits an asymptotic representation in terms of the Jacobi sn-function in cheese-like strips along generic…