A new approach to integrable evolution equations on the circle

Abstract

We propose a new approach for the solution of initial value problems for integrable evolution equations in the periodic setting based on the unified transform. Using the nonlinear Schrödinger equation as a model example, we show that the solution of the initial value problem on the circle can be expressed in terms of the solution of a Riemann-Hilbert problem whose formulation involves quantities which are defined in terms of the initial data alone. Our approach provides an effective solution of the problem on the circle which is conceptually analogous to the solution of the problem on the line via the inverse scattering transform.

29 pages

References in corpus (4)

• The Nonlinear Schrödinger Equation on the Interval
• The Unified Method: I Non-Linearizable Problems on the Half-Line
• The nonlinear Schrödinger equation with $t$-periodic data: I. Exact results
• The nonlinear Schrödinger equation with $t$-periodic data: II. Perturbative results

Cited by in corpus (3)

• The implementation of the unified transform to the nonlinear Schrödinger equation with periodic initial conditions
• The Robin problem for the nonlinear Schrödinger equation on the half-line
• Periodic finite-band solutions to the focusing nonlinear Schrödinger equation by the Fokas method: inverse and direct problems