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Abstract <p>In the case of a homogeneous medium, we describe the dualism of spaces of soliton solutions and solutions of an induced functional differential equation of pointwise type and formulate theorems on the existence and uniqueness of such dual solutions. This dualism is of the type of dualisms between various mathematical objects, for example, between a topological linear space and its dual. In the case of an inhomogeneous medium, we describe dualism of another type between the spaces of quasi-soliton solutions and solutions of an induced one-parameter family of functional differential equations of pointwise type and formulate theorems on the existence and uniqueness of such dual solutions. For a finite-difference analogue of the wave equation with a nonlinear potential, the entire family of soliton solutions (in the case of a homogeneous medium) and of quasi-soliton solutions (in the case of an inhomogeneous medium) is constructed.</p>

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Principles of Dualism in the Theory of Solutions to Infinite-Dimensional Differential Equations Depending on Existing Types of Symmetries

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • L. A. Beklaryan,
  • A. L. Beklaryan

摘要

Abstract

In the case of a homogeneous medium, we describe the dualism of spaces of soliton solutions and solutions of an induced functional differential equation of pointwise type and formulate theorems on the existence and uniqueness of such dual solutions. This dualism is of the type of dualisms between various mathematical objects, for example, between a topological linear space and its dual. In the case of an inhomogeneous medium, we describe dualism of another type between the spaces of quasi-soliton solutions and solutions of an induced one-parameter family of functional differential equations of pointwise type and formulate theorems on the existence and uniqueness of such dual solutions. For a finite-difference analogue of the wave equation with a nonlinear potential, the entire family of soliton solutions (in the case of a homogeneous medium) and of quasi-soliton solutions (in the case of an inhomogeneous medium) is constructed.