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Inverse scattering transform
About: Inverse scattering transform is a research topic. Over the lifetime, 3732 publications have been published within this topic receiving 134903 citations.
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875 citations
01 Jan 2012
TL;DR: In particular, the linear dispersive term in the Korteweg-de Vries equation prevents this from ever happening in its solution as discussed by the authors, and the instability and subsequent modulation of an initially uniform wave profile can be prevented by including dispersive effects in the shallow water theory.
Abstract: Dispersion and nonlinearity play a fundamental role in wave motions in nature. The nonlinear shallow water equations that neglect dispersion altogether lead to breaking phenomena of the typical hyperbolic kind with the development of a vertical profile. In particular, the linear dispersive term in the Korteweg–de Vries equation prevents this from ever happening in its solution. In general, breaking can be prevented by including dispersive effects in the shallow water theory. The nonlinear theory provides some insight into the question of how nonlinearity affects dispersive wave motions. Another interesting feature is the instability and subsequent modulation of an initially uniform wave profile.
864 citations
TL;DR: In this article, the conceptual analogy between Fourier analysis and exact solution to a class of nonlinear differential-difference equations is discussed in detail, and the dispersion relation of the associated linearized equation is prominent in developing a systematic procedure for isolating and solving the equation.
Abstract: The conceptual analogy between Fourier analysis and the exact solution to a class of nonlinear differential–difference equations is discussed in detail. We find that the dispersion relation of the associated linearized equation is prominent in developing a systematic procedure for isolating and solving the equation. As examples, a number of new equations are presented. The method of solution makes use of the techniques of inverse scattering. Soliton solutions and conserved quantities are worked out.
851 citations
TL;DR: In this paper, a new discrete eigenvalue problem has been introduced to obtain and solve certain classes of nonlinear differential-difference equations, which can be obtained by inverse scattering.
Abstract: A method is presented which enables one to obtain and solve certain classes of nonlinear differential−difference equations. The introduction of a new discrete eigenvalue problem allows the exact solution of the self−dual network equations to be found by inverse scattering. The eigenvalue problem has as its singular limit the continuous eigenvalue equations of Zakharov and Shabat. Some interesting differences arise both in the scattering analysis and in the time dependence from previous work.
818 citations
815 citations