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Dirichlet-to-Neumann map for evolution PDEs on the half-line with time-periodic boundary conditions

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CATW04 - Complex analysis: techniques, applications and computations - perspectives in 2023

For a well-posed boundary value problem, a certain number of boundary values must be prescribed as boundary conditions, while the rest of the boundary values are unknown. The task of determining the unknown boundary values in terms of the prescribed ones is called the computation of the “generalised Dirichlet-to-Neumann map”. Here we elaborate on a new approach for finding the Dirichlet-to-Neumann map in the large time limit for evolution PDEs on the half-line, for the physically significant case of time-periodic boundary conditions [1].The method is illustrated both for linear PDEs (including the heat equation, the convection-diffusion equation and the linearised KdV equation) and for integrable nonlinear PDEs, in particular for the focusing NLS equation. It is shown that the time-dependent part of the Lax pair is instrumental in yielding, via an elegant algebraic calculation, the large t asymptotics of the periodic unknown boundary values in terms of the prescribed periodic boundary data. This method is based on earlier work by Lenells and Fokas [2], in which the NLS equation was treated via a more complicated approach.This is joint work with Prof. A. S. Fokas.[1]   A. S. Fokas and M. C. van der Weele. The Unified Transform for Evolution Equations on the Half-Line with Time-Periodic Boundary Conditions. Stud. Appl. Math., 147(4):1339-1368, 2021.[2]   J. Lenells and A. S. Fokas. The Nonlinear Schrödinger Equation with t-Periodic Data: II. Perturbative Results. Proc. R. Soc. A, 471:20140926, 2015.

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