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On the Algorithmic Verification of Nonlinear Superposition for Systems of First Order Ordinary Differential Equations

Veronika Treumova, Dmitry A. Lyakhov, Dominik L. Michels

TL;DR

The algorithmic verification of nonlinear superposition properties and its implementation is introduced, which provides the basis for the identification of nonlinear superpositions within a given system and for the construction of numerical methods which preserve important algebraic properties at the numerical level.

Abstract

This paper belongs to a group of work in the intersection of symbolic computation and group analysis aiming for the symbolic analysis of differential equations. The goal is to extract important properties without finding the explicit general solution. In this contribution, we introduce the algorithmic verification of nonlinear superposition properties and its implementation. More exactly, for a system of nonlinear ordinary differential equations of first order with a polynomial right-hand side, we check if the differential system admits a general solution by means of a superposition rule and a certain number of particular solutions. It is based on the theory of Newton polytopes and associated symbolic computation. The developed method provides the basis for the identification of nonlinear superpositions within a given system and for the construction of numerical methods which preserve important algebraic properties at the numerical level.

On the Algorithmic Verification of Nonlinear Superposition for Systems of First Order Ordinary Differential Equations

TL;DR

The algorithmic verification of nonlinear superposition properties and its implementation is introduced, which provides the basis for the identification of nonlinear superpositions within a given system and for the construction of numerical methods which preserve important algebraic properties at the numerical level.

Abstract

This paper belongs to a group of work in the intersection of symbolic computation and group analysis aiming for the symbolic analysis of differential equations. The goal is to extract important properties without finding the explicit general solution. In this contribution, we introduce the algorithmic verification of nonlinear superposition properties and its implementation. More exactly, for a system of nonlinear ordinary differential equations of first order with a polynomial right-hand side, we check if the differential system admits a general solution by means of a superposition rule and a certain number of particular solutions. It is based on the theory of Newton polytopes and associated symbolic computation. The developed method provides the basis for the identification of nonlinear superpositions within a given system and for the construction of numerical methods which preserve important algebraic properties at the numerical level.
Paper Structure (14 sections, 5 theorems, 42 equations, 1 figure, 2 algorithms)

This paper contains 14 sections, 5 theorems, 42 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical Formulation
  3. Related Work
  4. Riccati Equation
  5. Algorithmics
  6. One-dimensional Case
  7. General Case
  8. Implementation
  9. Non-trivial Example
  10. Difference Thomas Decomposition
  11. Matrix Ricatti Equation
  12. Generalization and Open Questions
  13. Conclusion
  14. Nonlinear superposition function

Key Result

Theorem 2.2

The system of first-order ODEs with a vector of dependent variables $\textbf{x} = (x_1, \dots, x_n)$ admits a nonlinear superposition if and only if it has the form of generalized separation of variables with $r < \infty$ members: where the operatorsWe use the common summation rule in repeated indices. satisfy the commutator relations in $C_{\alpha \beta}^{\gamma}$, which are called the structur

Figures (1)

  • Figure 1: Illustration of the exact (blue curve) and the numerical (red rhombi) solution of the Ricatti equation with coefficients (\ref{['coeffs']}).

Theorems & Definitions (9)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 5.1
  • Theorem 5.2
  • Theorem 8.1
  • Definition 9.1
  • Definition 9.2
  • Definition 9.3
  • Theorem 9.4