6 research outputs found
On universality of critical behavior in the focusing nonlinear Schr\uf6dinger equation, elliptic umbilic catastrophe and the Tritronqu\ue9e solution to the Painlev\ue9-I equation
We argue that the critical behavior near the point of "gradient catastrophe" of the solution to the Cauchy problem for the focusing nonlinear Schrodinger equation i epsilon Psi(t) + epsilon(2)/2 Psi(xx) + vertical bar Psi vertical bar(2)Psi = 0, epsilon << 1, with analytic initial data of the form Psi( x, 0; epsilon) = A(x)e(i/epsilon) (S(x)) is approximately described by a particular solution to the Painleve-I equation
On critical behaviour in systems of Hamiltonian partial differential equations
We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\ue9-I (PI) equation or its fourth-order analogue P2I. As concrete examples, we discuss nonlinear Schr\uf6dinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture
Uniformization and Constructive Analytic Continuation of Taylor Series
We analyze the general mathematical problem of global reconstruction of a
function with least possible errors, based on partial information such as n
terms of a Taylor series at a point, possibly also with coefficients of finite
precision. We refer to this as the "inverse approximation theory problem,
because we seek to reconstruct a function from a given approximation, rather
than constructing an approximation for a given function. Within the class of
functions analytic on a common Riemann surface Omega, and a common Maclaurin
series, we prove an optimality result on their reconstruction at other points
on Omega, and provide a method to attain it. The procedure uses the
uniformization theorem, and the optimal reconstruction errors depend only on
the distance to the origin. We provide explicit uniformization maps for some
Riemann surfaces Omega of interest in applications. One such map is the
covering of the Borel plane of the tritronquee solutions to the Painleve
equations PI-PV. As an application we show that this uniformization map leads
to dramatic improvement in the extrapolation of the PI tritronquee solution
throughout its domain of analyticity and also into the pole sector. Given
further information about the function, such as is available for the ubiquitous
class of resurgent functions, significantly better approximations are possible
and we construct them. In particular, any one of their singularities can be
eliminated by specific linear operators, and the local structure at the chosen
singularity can be obtained in fine detail. More generally, for functions of
reasonable complexity, based on the nth order truncates alone we propose new
efficient tools which are convergent as n to infty, which provide near-optimal
approximations of functions globally, as well as in their most interesting
regions, near singularities or natural boundaries.Comment: 39 pages, 9 figures; v2 some clarifications adde
Local Emergence of Peregrine Solitons: Experiments and Theory
It has been shown analytically that Peregrine solitons emerge locally from a universal mechanism in the so-called semiclassical limit of the one-dimensional focusing nonlinear Schrödinger equation. Experimentally, this limit corresponds to the strongly nonlinear regime where the dispersion is much weaker than nonlinearity at initial time. We review here evidences of this phenomenon obtained on different experimental platforms. In particular, the spontaneous emergence of coherent structures exhibiting locally the Peregrine soliton behavior has been demonstrated in optical fiber experiments involving either single pulse or partially coherent waves. We also review theoretical and numerical results showing the link between this phenomenon and the emergence of heavy-tailed statistics (rogue waves)
Analysis of Singular Solutions of Certain Painlevé Equations
The six PainlevĂ© equations can be described as the boundary between the non- integrable- and the trivially integrable-systems. Ever since their discovery they have found numerous applications in mathematics and physics. The solutions of the PainlevĂ© equations are, in most cases, highly transcendental and hence cannot be expressed in closed form. Asymptotic methods do better, and can establish the behaviour of some of the solutions of the PainlevĂ© equations in the neighbourhood of a singularity, such as the point at infinity. Although the quantitative nature of these neighbourhoods is not initially implied from the asymptotic analysis, some regularity results exist for some of the PainlevĂ© equations. In this research, we will present such results for some of the remaining PainlevĂ© equations. In particular, we will provide concrete estimates of the intervals of analyticity of a one-parameter family of solutions of the second PainlevĂ© equation, and estimate the domain of analyticity of a âtriply-truncatedâ solution of the fourth PainlevĂ© equation. In addition we will also deduce the existence of solutions with particular asymptotic behaviour for the discrete PainlevĂ© equations, which are discrete integrable nonlinear systems