Publication Abstracts

Chu and Chou 1989

Chu, C.K., and R.L. Chou, 1989: Solitons induced by boundary conditions. Adv. Appl. Mech., 27, 283-302, doi:10.1016/S0065-2156(08)70198-4.

Since the sighting of the solitary wave by Russell (1844) over a century ago, the discovery of the remarkable properties of solitons by Zabusky and Kruskal (1965) over two decades ago, and the invention of the inverse scattering transform by Gardner, Greene, Kruskal, and Miura (1967) shortly afterwards, the study of solitons has mushroomed into an active field of research. Considerable interest has also been shown in the generation of solitons.

In water, solitary waves can be generated by three different methods: an initial profile evolving into one or many solitary waves, a moving ship or equivalent pressure source on the surface of water, or boundary excitation, such as a sluice opening or a wall pushing. In other media, solitons can be produced by analogous situations.

In theoretical work, the first method corresponds to a pure initial value problem (for some governing differential equation or system of equations), the second to a driving term or an inhomogeneous differential equation (or equations), and the third to a mixed initial boundary value problem. Indeed, the most exciting advances in soliton theory, such as soliton interaction and collision and the inverse scattering transform, have all come from pure initial value problems. Only recently have steps been taken in the theoretical and numerical studies of the mixed initial boundary value problems.

The differential equations used in the theoretical and numerical works vary. The simplest and best studied is the Korteweg-deVries equation (KdV equation), either in the ordinary form or in the regularized form (the latter replaces the third x-derivative in the former by a mixed xxI-derivative). For water, the Boussinesq systems (also in different forms), and for nonlinear optics, the nonlinear Schrodinger equations have been used extensively.

In this paper, we summarize briefly experiments and then describe numerical results, mainly for water waves. We then describe some theoretical results obtained to date for mixed initial boundary value problems for the KdV equation and for the nonlinear Schrodinger equation.

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BibTeX Citation

  author={Chu, C. K. and Chou, R. L.},
  title={Solitons induced by boundary conditions},
  journal={Advances in Applied Mechanics},

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RIS Citation

ID  - ch08900n
AU  - Chu, C. K.
AU  - Chou, R. L.
PY  - 1989
TI  - Solitons induced by boundary conditions
JA  - Adv. Appl. Mech.
JO  - Advances in Applied Mechanics
VL  - 27
SP  - 283
EP  - 302
DO  - 10.1016/S0065-2156(08)70198-4
ER  -

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