2020/05/09 by Kevin Burrage, Burrage, Kevin, Pamela Burrage +3
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Nonlinear Waves and Solitons #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2005.04561
openalex publication_date 2020/05/09 · openalex created_date 2022/07/18 · openalex updated_date 2026/07/28
We study reflectionless properties at the boundary for the wave equation in\none space dimension and time, in terms of a well-known matrix that arises from\na simple discretisation of space. It is known that all matrix functions of the\nfamiliar second difference matrix representing the Laplacian in this setting\nare the sum of a Toeplitz matrix and a Hankel matrix. The solution to the wave\nequation is one such matrix function. Here, we study the behaviour of the\ncorresponding waves that we call Toeplitz waves and Hankel waves. We show that\nthese waves can be written as certain linear combinations of even Bessel\nfunctions of the first kind. We find exact and explicit formulae for these\nwaves. We also show that the Toeplitz and Hankel waves are reflectionless on\neven, respectively odd, traversals of the domain. Our analysis naturally\nsuggests a new method of computer simulation that allows control, so that it is\npossible to choose -- in advance -- the number of reflections. An attractive\nresult that comes out of our analysis is the appearance of the well-known shift\nmatrix, and also other matrices that might be thought of as Hankel versions of\nthe shift matrix. By revealing the algebraic structure of the solution in terms\nof shift matrices, we make it clear how the Toeplitz and Hankel waves are\nindeed reflectionless at the boundary on even or odd traversals. Although the\nsubject of the reflectionless boundary condition has a long history, we believe\nthe point of view that we adopt here in terms of matrix functions is new.\n