• Corpus ID: 119313286

The Direct Monodromy Problem of Painleve-I

@article{Masoero2010TheDM,
  title={The Direct Monodromy Problem of Painleve-I},
  author={Davide Masoero},
  journal={arXiv: Classical Analysis and ODEs},
  year={2010}
}
  • D. Masoero
  • Published 9 July 2010
  • Mathematics
  • arXiv: Classical Analysis and ODEs
The Painleve first equation can be represented as the equation of isomonodromic deformation of a Schrodinger equation with a cubic potential. We introduce a new algorithm for computing the direct monodromy problem for this Schrodinger equation. The algorithm is based on the geometric theory of Schrodinger equation due to Nevanlinna 

Figures from this paper

Numerical Solution of Riemann–Hilbert Problems: Painlevé II

  • S. Olver
  • Mathematics
    Found. Comput. Math.
  • 2011
A new, spectrally accurate method for solving matrix-valued Riemann–Hilbert problems numerically is described and can be used to relate initial conditions with asymptotic behavior.

References

SHOWING 1-10 OF 16 REFERENCES

Explicit continued fractions and quantum gravity

This paper deals with the subject of completely integrable systems, particularly Painlevé equations, monodromy and Stokes parameters, complex analysis, approximation theory, computational

Poles of integrále tritronquée and anharmonic oscillators. A WKB approach

Poles of solutions to the Painlevé-I equations are intimately related to the theory of the cubic anharmonic oscillator. In particular, poles of integrále tritronquée are in bijection with cubic

Y-System and Deformed Thermodynamic Bethe Ansatz

We introduce a new tool, the Deformed TBA (Deformed Thermodynamic Bethe Ansatz), to analyze the monodromy problem of the cubic oscillator. The Deformed TBA is a system of five coupled nonlinear

Anharmonic oscillators, the thermodynamic Bethe ansatz and nonlinear integral equations

The spectral determinant D(E) of the quartic oscillator is known to satisfy a functional equation. This is mapped onto the A3-related Y-system emerging in the treatment of a certain perturbed

Analytic Continuation of Eigenvalues of a Quartic Oscillator

We consider the Schrödinger operator on the real line with even quartic potential x4 + αx2 and study analytic continuation of eigenvalues, as functions of parameter α. We prove several properties of

On universality of critical behaviour in Hamiltonian PDEs

Our main goal is the comparative study of singularities of solutions to the systems of first order quasilinear PDEs and their perturbations containing higher derivatives. The study is focused on the

A general framework for solving Riemann–Hilbert problems numerically

  • S. Olver
  • Mathematics
    Numerische Mathematik
  • 2012
A new, numerical framework for the approximation of solutions to matrix-valued Riemann–Hilbert problems is developed, based on a recent method for the homogeneous Painlevé II Riemann–Hilbert problem.

Poles of intégrale tritronquée and anharmonic oscillators. Asymptotic localization from WKB analysis

Poles of intégrale tritronquée are in bijection with cubic oscillators that admit the simultaneous solutions of two quantization conditions. We show that the poles are well approximated by solutions

Universality of the Break-up Profile for the KdV Equation in the Small Dispersion Limit Using the Riemann-Hilbert Approach

AbstractWe obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation $$u_t+6uu_x+\epsilon^{2}u_{xxx}=0,\quad u(x,t=0,\epsilon)=u_0(x),$$for

On Universality of Critical Behavior in the Focusing Nonlinear Schrödinger Equation, Elliptic Umbilic Catastrophe and the Tritronquée Solution to the Painlevé-I Equation

It is argued that the critical behavior near the point of “gradient catastrophe” of the solution of the Cauchy problem for the focusing nonlinear Schrödinger equation is approximately described by a particular solution to the Painlevé-I equation.