The unified transform method (UTM) provides a novel approach to the analysis of initial boundary value problems for linear as well as for a particular class of nonlinear partial differential equations called integrable. If the latter equations are formulated in two dimensions (either one space and one time, or two space dimensions), the UTM expresses the solution in terms of a matrix Riemann–Hilbert (RH) problem with explicit dependence on the independent variables. For nonlinear integrable evolution equations, such as the celebrated nonlinear Schrödinger (NLS) equation, the associated jump matrices are computed in terms of the initial conditions and all boundary values. The unknown boundary values are characterized in terms of the initial datum and the given boundary conditions via the analysis of the so-called global relation. In general, this analysis involves the solution of certain nonlinear equations. In certain cases, called linearizable, it is possible to bypass this nonlinear step. In these cases, the UTM solves the given initial boundary value problem with the same level of efficiency as the well-known inverse scattering transform solves the initial value problem on the infinite line. We show here that the initial boundary value problem on a finite interval with x-periodic boundary conditions (which can alternatively be viewed as the initial value problem on a circle) belongs to the linearizable class. Indeed, by employing certain transformations of the associated RH problem and by using the global relation, the relevant jump matrices can be expressed explicitly in terms of the so-called scattering data, which are computed in terms of the initial datum. Details are given for NLS, but similar considerations are valid for other well-known integrable evolution equations, including the Korteweg–de Vries (KdV) and modified KdV equations.

统一变换方法（UTM）提供了一种新颖的方法来分析线性以及特定类别的非线性偏微分方程的初边值问题，这些问题称为可积。如果后一个方程是在两个维度（一个空间和一个时间，或两个空间维度）中制定的，则UTM用矩阵Riemann-Hilbert（RH）问题表示解决方案，并且显式依赖于独立变量。对于非线性可积演化方程，例如著名的非线性薛定ding（NLS）方程，将根据初始条件和所有边界值来计算关联的跳跃矩阵。通过分析所谓的整体关系，根据初始基准和给定的边界条件来表征未知的边界值。大体，该分析涉及某些非线性方程的解。在某些情况下，称为线性化，可以绕过该非线性步骤。在这些情况下，UTM可以以与已知的逆散射变换解决无限线上的初始值问题相同的效率来解决给定的初始边界值问题。我们在这里显示出有限区间上的初始边界值问题x周期边界条件（可以替换为圆上的初始值问题）属于线性化类。实际上，通过采用相关联的RH问题的某些变换并通过使用全局关系，可以根据所谓的散射数据明确表示相关的跳跃矩阵，这些散射数据是根据初始基准计算的。NLS的细节已给出，但类似的考虑对其他著名的可积分演化方程式也有效，包括Korteweg-de Vries（KdV）和修正的KdV方程式。