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Solitons and the Inverse Scattering Transform
Abstract
Abstract : Under appropriate conditions, ocean waves may be modeled by certain nonlinear evolution equations that admit soliton solutions and can be solved exactly by the inverse scattering transform (IST). The theory of these special equations is developed in five lectures. As physical models, these equations typically govern the evolution of narrow-band packets of small amplitude waves on a long (post-linear) time scale. This is demonstrated in Lecture I, using the Korteweg-deVries equation as an example. Lectures II and III develop the theory of IST on the infinite interval. The close connection of aspects of this theory to Fourier analysis, to canonical transformations of Hamiltonian systems, and to the theory of analytic functions is established. Typical solutions, including solitons and radiation, are discussed as well. With periodic boundary conditions, the Korteweg-deVries equation exhibits recurrence, as discussed in Lecture IV. The fifth lecture emphasizes the deep connection between evolution equations solvable by IST and Painleve transcendents, with an application to the Lorenz model.
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Cited by
Lederer, F., Stegeman, G. I., Christodoulides, D. N., Assanto, G., Segev, M., & Silberberg, Y. (2008) · Physics Reports · doi:10.1016/j.physrep.2008.04.004 · Cited by 1,180
Optical spatial solitons: historical overview and recent advances
Chen, Z., Segev, M., & Christodoulides, D. N. (2012) · Reports on Progress in Physics · doi:10.1088/0034-4885/75/8/086401 · Cited by 514
Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques
Carretero-González, R., Frantzeskakis, D. J., & Kevrekidis, P. G. (2008) · Nonlinearity · doi:10.1088/0951-7715/21/7/r01 · Cited by 339
Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators
Gendelman, O. V. (2001) · Nonlinear Dynamics · doi:10.1023/a:1012967003477 · Cited by 274
Metadata
- Authors
- Ablowitz, M. J., & Segur, H.
- Venue
- Society for Industrial and Applied Mathematics eBooks
- Published
- 1981-01
- OpenAlex
- W2118575947
- Cited by
- 4,870
- References
- 0
- FWCI
- 23.58
- Type
- book
- Language
- en
- Open access
- closed
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- Ablowitz, M. J., & Segur, H. (1981). Solitons and the Inverse Scattering Transform. Society for Industrial and Applied Mathematics eBooks. https://doi.org/10.1137/1.9781611970883
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