544 research outputs found

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

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    Using the Riemann-Hilbert approach, we study the quasi-linear Stokes phenomenon for the second Painlev\'e equation yxx=2y3+xy−αy_{xx}=2y^3+xy-\alpha. The precise description of the exponentially small jump in the dominant solution approaching α/x\alpha/x as ∣x∣→∞|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

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    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

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    The degenerate third Painlev\'{e} equation, u′′=(u′)2u−u′τ+1τ(−8ϵu2+2ab)+b2uu^{\prime \prime} = \frac{(u^{\prime})^{2}}{u} - \frac{u^{\prime}}{\tau} + \frac{1}{\tau}(-8 \epsilon u^{2} + 2ab) + \frac{b^{2}}{u}, where ϵ,b∈R\epsilon,b \in \mathbb{R}, and a∈Ca \in \mathbb{C}, and the associated tau-function are studied via the Isomonodromy Deformation Method. Connection formulae for asymptotics of the general as τ→±0\tau \to \pm 0 and ±i0\pm i0 solution and general regular as τ→±∞\tau \to \pm \infty and ±i∞\pm i \infty solution are obtained.Comment: 40 pages, LaTeX2

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

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    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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