544 research outputs found

### Quasi-linear Stokes phenomenon for the second Painlev\'e transcendent

Using the Riemann-Hilbert approach, we study the quasi-linear Stokes
phenomenon for the second Painlev\'e equation $y_{xx}=2y^3+xy-\alpha$. The
precise description of the exponentially small jump in the dominant solution
approaching $\alpha/x$ as $|x|\to\infty$ is given. For the asymptotic power
expansion of the dominant solution, the coefficient asymptotics is found.Comment: 19 pages, LaTe

### An Alternative Approach to Algebro-Geometric Solutions of the AKNS Hierarchy

We develop an alternative systematic approach to the AKNS hierarchy based on
elementary algebraic methods. In particular, we recursively construct Lax pairs
for the entire AKNS hierarchy by introducing a fundamental polynomial formalism
and establish the basic algebro-geometric setting including associated
Burchnall-Chaundy curves, Baker-Akhiezer functions, trace formulas,
Dubrovin-type equations for analogs of Dirichlet and Neumann divisors, and
theta function representations for algebro-geometric solutions.Comment: LaTeX, submitted to Reviews in Mathematical Physic

### Connection Formulae for Asymptotics of Solutions of the Degenerate Third Painlev\'{e} Equation. I

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 $a \in \mathbb{C}$, and the associated tau-function are studied via the
Isomonodromy Deformation Method. Connection formulae for asymptotics of the
general as $\tau \to \pm 0$ and $\pm i0$ solution and general regular as $\tau
\to \pm \infty$ and $\pm i \infty$ solution are obtained.Comment: 40 pages, LaTeX2

### Isomonodromy aspects of the tt* equations of Cecotti and Vafa I. Stokes data

We describe all smooth solutions of the two-function tt*-Toda equations (a
version of the tt* equations, or equations for harmonic maps into
SL(n,R)/SO(n)) in terms of (i) asymptotic data, (ii) holomorphic data, and
(iii) monodromy data. This allows us to find all solutions with integral Stokes
data. These include solutions associated to nonlinear sigma models (quantum
cohomology) or Landau-Ginzburg models (unfoldings of singularities), as
conjectured by Cecotti and Vafa.Comment: 35 pages, 3 figures. Minor revisions for compatibility with the
recently posted Part II (arXiv:1312.4825

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