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Abstract <p>The scheme of a continuation for solutions of evolutionary ODEs and PDEs, allowing the formation of singularities in a finite time, is presented. The proposed scheme is applied to various problems that were previously considered in the authors’ works. This paper is a review article, devoted to describing a unified approach to continuing solutions of differential equations. The developed approach combines many previous considerations of the authors. The continuation procedure is based on an extension of the phase space and prolongation of trajectories into the extended space. The scheme of trajectories continuation is applied to the finite dimensional Hamiltonian system, the focusing nonlinear Schrödinger equation, the linear Schrödinger equation with a degenerated Hamiltonian.</p>

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Continuation of Solutions of Differential Equations through the Formation of Singularities

  • 期刊论文

  • 发表日期：

    2025年6月1日

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  • V. A. Glazatov,
  • Yu. N. Orlov,
  • V. Zh. Sakbaev

摘要

Abstract

The scheme of a continuation for solutions of evolutionary ODEs and PDEs, allowing the formation of singularities in a finite time, is presented. The proposed scheme is applied to various problems that were previously considered in the authors’ works. This paper is a review article, devoted to describing a unified approach to continuing solutions of differential equations. The developed approach combines many previous considerations of the authors. The continuation procedure is based on an extension of the phase space and prolongation of trajectories into the extended space. The scheme of trajectories continuation is applied to the finite dimensional Hamiltonian system, the focusing nonlinear Schrödinger equation, the linear Schrödinger equation with a degenerated Hamiltonian.