Asymptotic approach for the rigid condition of appearance of the oscillations in the solution of the Painleve-2 equation

O. M. Kiselev

Home/Archive/arXiv

arXiv · solv-int/9902007

Asymptotic approach for the rigid condition of appearance of the oscillations in the solution of the Painleve-2 equation

O. M. Kiselev

View original source ↗Read source document ↗Export metadata ↓

Abstract

The asymptotic solution for the Painleve-2 equation with small parameter is considered. The solution has algebraic behavior before point $t_*$ and fast oscillating behavior after the point $t_*$. In the transition layer the behavior of the asymptotic solution is more complicated. The leading term of the asymptotics satisfies the Painleve-1 equation and some elliptic equation with constant coefficients, where the solution of the Painleve-1 equation has poles. The uniform smooth asymptotics are constructed in the interval, containing the critical point $t_*$.

Explore related subjects

solv-int nlin.SI

Keep this discovery

BibTeX RIS

O. M. Kiselev. 1999-02-05. Asymptotic approach for the rigid condition of appearance of the oscillations in the solution of the Painleve-2 equation. https://arxiv.org/abs/solv-int/9902007

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2013

On two aspects of the Painleve analysis

We use the Calogero equation to illustrate the following two aspects of the Painleve analysis of nonlinear PDEs. First, if a nonlinear equation passes the Painleve test for integrability, the singular expansions of its solutions around characteristic hypersurfaces can be neither single-valued functions of independent variables nor single-valued functionals of data. Second, if the truncation of singular expansions of solutions is consistent, the truncation not necessarily leads to the simplest, or elementary, auto-Backlund transformation related to the Lax pair.

solv-int↗

arXiv2009

The tetrahedral analog of Veneziano amplitude

In solv-int/9812016 it was shown that the Veneziano amplitude in string theory comes naturally from one of the simplest solutions of the functional pentagon equation (FPE). More generally, FPE is intimately connected with the duality condition for scattering processes. Here I find the amplitude that comes the same way from a solution of the functional tetrahedron equation, with the duality replaced by the local Yang - Baxter equation.

solv-int↗

arXiv2004

Equations of Geodesic Deviation and the Inverse Scattering Transform

Solutions of equations of geodesic deviation in three- and four- dimensional spaces obtained by the inverse scattering transform are considered. It is shown that in the case of three-dimensional space solutions of geodesic deviation equations are reduced to solutions of the well-known Zakharov-Shabat problem. In four- dimensional space system of geodesic deviation equations is associated with $3\times 3$ matrix Schrödinger equation, and dependence on parameters defined by the nonlinear equations of three-wave interaction.

solv-int↗