We review results recently obtained for two-point boundary value problems for integrable evolution PDEs modelling the one-way propagation of small amplitude water waves. The main example is the Korteweg-deVries equation. We give an explicit representation of the solution of the corresponding linearised PDE. This representation differs from the one obtained for the solution of boundary value problems on an infinite interval because of the presence of a discrete sum contribution. The discrete spectrum associated in this way with the linearised equation is due to the boundary conditions. The same discrete structure appears in the representation of the solution of the analogous boundary value problem for the KdV equation; in this case however, the spectral transform has an additional discrete spectrum due to the nonlinearity, and present in the representation of the solution of KdV on any domain. In the case of a finite domain,the nature of the joint discrete spectrum seems to determine the nature of the interaction between the propagating solitons and the boundary, and in particular whether solitons can propagate after the interaction.
|Title of host publication||Mathematical and Numerical Aspects of Wave Propagation WAVES 2003|
|Subtitle of host publication||Proceedings of The Sixth International Conference on Mathematical and Numerical Aspects of Wave Propagation Held at Jyväskylä, Finland, 30 June – 4 July 2003|
|Number of pages||6|
|Publication status||Published - 2003|
Pelloni, B. (2003). The propagation of nonlinear water waves in a bounded domain. In Mathematical and Numerical Aspects of Wave Propagation WAVES 2003: Proceedings of The Sixth International Conference on Mathematical and Numerical Aspects of Wave Propagation Held at Jyväskylä, Finland, 30 June – 4 July 2003 (pp. 711-716). Springer. https://doi.org/10.1007/978-3-642-55856-6_115