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A new approach to integrable evolution equations on the circle.

A S Fokas1,2, J Lenells3

  • 1Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK.

Proceedings. Mathematical, Physical, and Engineering Sciences
|February 26, 2021
PubMed
Summary

We present a novel method for solving integrable evolution equations on a circle. This approach, using the unified transform, offers an effective solution analogous to the inverse scattering transform on the line.

Keywords:
Fokas methodRiemann–Hilbert problemfinite-gap solutionintegrable evolution equationinverse scatteringunified transform method

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Area of Science:

  • Mathematical Physics
  • Nonlinear Dynamics
  • Spectral Theory

Background:

  • Integrable evolution equations are crucial in various scientific fields.
  • Solving initial value problems in a periodic setting presents unique challenges.
  • The inverse scattering transform is a powerful tool for the line but not directly applicable to the circle.

Purpose of the Study:

  • To develop a new, effective method for solving initial value problems of integrable evolution equations in the periodic setting.
  • To demonstrate the applicability of the unified transform for problems on a circle.
  • To establish a conceptual link between solutions on the line and on the circle.

Main Methods:

  • The unified transform method is employed.
  • The nonlinear Schrödinger equation serves as a model example.
  • The solution is formulated via a Riemann-Hilbert problem.

Main Results:

  • A novel approach for solving initial value problems on the circle is proposed.
  • The solution is expressed using quantities solely dependent on initial data.
  • The method is shown to be effective for the nonlinear Schrödinger equation.

Conclusions:

  • The unified transform provides an effective solution for integrable evolution equations in the periodic setting.
  • This method offers a conceptually analogous framework to the inverse scattering transform on the line.
  • The approach simplifies the solution by relying only on initial data.