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Partial differential systems with non-local nonlinearities: generation and solutions.

Margaret Beck1, Anastasia Doikou2, Simon J A Malham3

  • 1Department of Mathematics, Boston University, Boston, MA 02215, USA.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|March 7, 2018
PubMed
Summary
This summary is machine-generated.

Researchers developed a novel method to solve complex evolutionary partial differential systems with non-local nonlinearities using a Fredholm integral equation, enabling solutions for arbitrary initial data.

Keywords:
partial differential systemsGrassmannian flowsnon-local nonlinearity

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Area of Science:

  • Applied Mathematics
  • Nonlinear Dynamics
  • Mathematical Physics

Background:

  • Evolutionary partial differential equations (PDEs) with non-local nonlinearities present significant analytical challenges.
  • Existing methods often struggle with arbitrary initial conditions and complex nonlinear terms.

Purpose of the Study:

  • To introduce a new method for generating exact solutions to a broad range of evolutionary PDEs with non-local nonlinearities.
  • To demonstrate the method's applicability to diverse systems, including reaction-diffusion and nonlinear Schrödinger equations.

Main Methods:

  • Development of a Fredholm integral equation that links the linearized flow to an auxiliary linear flow.
  • Analogy drawn to the Marchenko integral equation, a known tool in integrable systems.
  • Application and verification of the method through numerical simulations on specific examples.

Main Results:

  • The proposed method successfully generates solutions for evolutionary PDEs with non-local nonlinearities from their linearized counterparts.
  • The Fredholm integral equation provides a direct link between the nonlinear system and a solvable auxiliary linear system.
  • Numerical simulations confirm the method's effectiveness for reaction-diffusion and nonlinear Schrödinger equations.

Conclusions:

  • The developed Fredholm integral equation approach offers a powerful tool for solving challenging nonlinear PDEs.
  • The method is versatile and can be extended to other classes of nonlinear systems.
  • This work contributes to the understanding of stability in nonlinear waves and patterns.