28 research outputs found

### Nonsingular solutions of Hitchin's equations for noncompact gauge groups

We consider a general ansatz for solving the 2-dimensional Hitchin's
equations, which arise as dimensional reduction of the 4-dimensional self-dual
Yang-Mills equations, with remarkable integrability properties. We focus on the
case when the gauge group G is given by a real form of SL(2,C). For G=SO(2,1),
the resulting field equations are shown to reduce to either the Liouville,
elliptic sinh-Gordon or elliptic sine-Gordon equations. As opposed to the
compact case, given by G=SU(2), the field equations associated with the
noncompact group SO(2,1) are shown to have smooth real solutions with
nonsingular action densities, which are furthermore localized in some sense. We
conclude by discussing some particular solutions, defined on R^2, S^2 and T^2,
that come out of this ansatz.Comment: 12 pages, 3 figures. To appear in Nonlinearit

### On the location of poles for the Ablowitz-Segur family of solutions to the second Painlev\'e equation

Using a simple operator-norm estimate we show that the solution to the second
Painlev\'e equation within the Ablowitz-Segur family is pole-free in a well
defined region of the complex plane of the independent variable. The result is
illustrated with several numerical examples.Comment: 8 pages, to appear in Nonlinearit

### Long-Time Asymptotics of the Toda Lattice for Decaying Initial Data Revisited

The purpose of this article is to give a streamlined and self-contained
treatment of the long-time asymptotics of the Toda lattice for decaying initial
data in the soliton and in the similarity region via the method of nonlinear
steepest descent.Comment: 41 page

### Inverse scattering transform for the Toda hierarchy with steplike finite-gap backgrounds

We provide a rigorous treatment of the inverse scattering transform for the
entire Toda hierarchy for solutions which are asymptotically close to (in
general) different finite-gap solutions as $n\to\pm\infty$.Comment: 10 page

### Stability of the periodic Toda lattice under short range perturbations

We consider the stability of the periodic Toda lattice (and slightly more
generally of the algebro-geometric finite-gap lattice) under a short range
perturbation. We prove that the perturbed lattice asymptotically approaches a
modulated lattice.
More precisely, let $g$ be the genus of the hyperelliptic curve associated
with the unperturbed solution. We show that, apart from the phenomenon of the
solitons travelling on the quasi-periodic background, the $n/t$-pane contains
$g+2$ areas where the perturbed solution is close to a finite-gap solution in
the same isospectral torus. In between there are $g+1$ regions where the
perturbed solution is asymptotically close to a modulated lattice which
undergoes a continuous phase transition (in the Jacobian variety) and which
interpolates between these isospectral solutions. In the special case of the
free lattice ($g=0$) the isospectral torus consists of just one point and we
recover the known result.
Both the solutions in the isospectral torus and the phase transition are
explicitly characterized in terms of Abelian integrals on the underlying
hyperelliptic curve.
Our method relies on the equivalence of the inverse spectral problem to a
matrix Riemann--Hilbert problem defined on the hyperelliptic curve and
generalizes the so-called nonlinear stationary phase/steepest descent method
for Riemann--Hilbert problem deformations to Riemann surfaces.Comment: 38 pages, 1 figure. This version combines both the original version
and arXiv:0805.384

### 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

### Hard loss of stability in Painlev\'e-2 equation

A special asymptotic solution of the Painlev\'e-2 equation with small
parameter is studied. This solution has a critical point $t_*$ corresponding to
a bifurcation phenomenon. When $t<t_*$ the constructed solution varies slowly
and when $t>t_*$ the solution oscillates very fast. We investigate the
transitional layer in detail and obtain a smooth asymptotic solution, using a
sequence of scaling and matching procedures

### INVERSE SCATTERING TRANSFORM ANALYSIS OF STOKES-ANTI-STOKES STIMULATED RAMAN SCATTERING

Zakharov-Shabat--Ablowitz-Kaup-Newel-Segur representation for
Stokes-anti-Stokes stimulated Raman scattering is proposed. Periodical waves,
solitons and self-similarity solutions are derived. Transient and bright
threshold solitons are discussed.Comment: 16 pages, LaTeX, no figure