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This study analyzes the integrability of nonlinear differential systems. Researchers derived simple numerical conditions for complete and partial integrability using differential Galois group analysis.

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Area of Science:

  • Differential equations
  • Mathematical physics
  • Dynamical systems

Background:

  • Nonlinear differential systems are crucial in modeling complex phenomena.
  • Analyzing the integrability of these systems is essential for understanding their behavior.
  • Previous methods for integrability analysis can be complex and computationally intensive.

Purpose of the Study:

  • To develop a simplified method for analyzing the integrability of nonlinear three-dimensional differential systems.
  • To derive explicit conditions for complete and partial integrability.
  • To demonstrate the applicability of the derived conditions through examples.

Main Methods:

  • The study focuses on systems where right-hand sides are linear in one variable.
  • Explicit particular solutions are found, and variational equations are calculated along these solutions.
  • Properties of the differential Galois group of variational equations are analyzed to determine integrability conditions.

Main Results:

  • Conditions for B-integrability (complete integrability with two functionally independent rational first integrals) and partial integrability were obtained.
  • These conditions are expressed in a simple numerical form, facilitating verification.
  • The method was successfully applied to exemplary nonlinear three-dimensional differential systems.

Conclusions:

  • The derived conditions offer a straightforward approach to assessing the integrability of a specific class of nonlinear differential systems.
  • The findings contribute to the theoretical understanding of differential equations and their applications.
  • This work provides a practical tool for researchers working with complex dynamical systems.