arXiv subjects

Kriti Sehgal

Publications and source records attributed to Kriti Sehgal.

2 recordsLinked to original sources

arXiv2024

Long time evolution of the Hénon-Heiles system for small energy

The Hénon-Heiles system, initially introduced as a simplified model of galactic dynamics, has become a paradigmatic example in the study of nonlinear systems. Despite its simplicity, it exhibits remarkably rich dynamical behavior, including the interplay between regular and chaotic orbital dynamics, resonances, and stochastic regions in phase space, which have inspired extensive research in nonlinear dynamics. In this work, we investigate the system's solutions at small energy levels, deriving asymptotic constants of motion that remain valid over remarkably long timescales -- far exceeding the range of validity of conventional perturbation techniques. Our approach leverages the system's inherent two-scale dynamics, employing a novel analytical framework to uncover these long-lived invariants. The derived formulas exhibit excellent agreement with numerical simulations, providing a deeper understanding of the system's long-term behavior.

math.CA↗

arXiv2024

On the pointwise existence of Cauchy $\rm{P.V.}$ integrals

The Riemann-Hilbert (RH) approach, whose origins can be traced back to Riemann's PhD thesis, is well known to be far-reaching. It provides a general framework for expressing solutions of integrable problems such as ODEs or PDEs. Its generalization concerning monodromy groups of Fuchsian systems is one of Hilbert's 23 problems. In this paper we extend a basic result in scalar RH theory, the Sokhotski-Plemelj formula. Classically, this formula is derived under assumptions of Hölder continuity, although it also holds true under weaker, Dini, continuity conditions. Dini continuity is still too restrictive. We prove the Sokhotski-Plemelj formula under weaker assumptions, namely continuity at a point and an $L^1$ condition. Furthermore, we provide sufficient conditions for the existence of the Cauchy $\rm{P.V.}$ integral which are in a precise sense also necessary: weakening them runs into obstructions of a mathematical foundations nature.

math.CV↗