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.

中文翻译：

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圆上可积演化方程的一种新方法

我们提出了一种基于统一变换的周期设置中可积演化方程初值问题的求解新方法。使用非线性薛定谔方程作为模型示例，我们表明圆上初值问题的解可以表示为黎曼-希尔伯特问题的解，该问题的公式涉及根据初始数据定义的数量独自的。我们的方法为圆上的问题提供了有效的解决方案，这在概念上类似于通过逆散射变换解决直线上的问题。