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Integrable nonlinear evolution partial differential equations in 4 + 2 and 3 + 1 dimensions.

A S Fokas1

  • 1Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK. t.fokas@damtp.cam.ac.uk

Physical Review Letters
|June 29, 2006
PubMed
Summary

Researchers have discovered integrable analogs of the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations in three spatial dimensions, solving a long-standing problem in the field of integrability.

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Area of Science:

  • Mathematical Physics
  • Integrability Theory
  • Nonlinear Partial Differential Equations

Background:

  • The search for integrable nonlinear evolution partial differential equations in three spatial dimensions has been a significant challenge since the 1970s.
  • Established integrable equations like Korteweg-de Vries and nonlinear Schrödinger exist in one spatial dimension, while Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations are known in two spatial dimensions.

Purpose of the Study:

  • To determine if integrable analogs of established equations exist in three spatial dimensions.
  • To present novel integrable generalizations of the KP and DS equations.

Main Methods:

  • Formulation of integrable generalizations in four spatial dimensions with complex time.
  • Imposition of a real-time requirement to reduce the system to three spatial dimensions.

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  • Development of a method for solving the derived equations.

    • Main Results:

    • Positive answer to the existence of integrable analogs in three spatial dimensions.
    • Presentation of novel integrable generalizations of KP and DS equations.
    • Demonstration of a solution method for these new equations.

    Conclusions:

    • The study successfully derives and solves integrable nonlinear evolution partial differential equations in three spatial dimensions.
    • This work provides a significant advancement in the field of integrability, offering new tools and insights.