They can be given explicitly as.

Since the Lagrangian eq 1 contains second derivatives, the Euler—Lagrange equation of motion for this field is. It can be shown that any sufficiently fast decaying smooth solution will eventually split into a finite superposition of solitons travelling to the right plus a decaying dispersive part travelling to the left. The history of the KdV equation started with experiments by John Scott Russell in , followed by theoretical investigations by Lord Rayleigh and Joseph Boussinesq around and, finally, Korteweg and De Vries in Moreover, the solitons seems to be almost unaffected in shape by passing through each other though this could cause a change in their position.

Development of the analytic solution by means of the inverse scattering transform was done in by Gardner, Greene, Kruskal and Miura.

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The KdV equation is now seen to be closely connected to Huygens' principle. The KdV equation has several connections to physical problems. In addition to being the governing equation of the string in the Fermi—Pasta—Ulam—Tsingou problem in the continuum limit, it approximately describes the evolution of long, one-dimensional waves in many physical settings, including:.

Many different variations of the KdV equations have been studied. Some are listed in the following table. From Wikipedia, the free encyclopedia.

- Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equation!
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Mathematical model of waves on a shallow water surface. Derivation of Euler—Lagrange equations. This section does not cite any sources.

- Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014).
- Korteweg–de Vries equation - Wikipedia!
- Geometric Methods for Discrete Dynamical Systems.
- Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equations!
- Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrodinger equations.
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Please help improve this section by adding citations to reliable sources. The energy-critical nonlinear Schroedinger equation, global solutions to the defocusing problem, and scattering are the focus of the second part.

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The third part describes wave and Schroedinger maps. Read more Read less. Product details Paperback: pages Publisher: Springer Basel; ed. From the Back Cover The first part of the book provides an introduction to key tools and techniques in dispersive equations: Strichartz estimates, bilinear estimates, modulation and adapted function spaces, with an application to the generalized Korteweg-de Vries equation and the Kadomtsev-Petviashvili equation.

## AMS :: Proceedings of the American Mathematical Society

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## Exact Multisoliton Solutions of General Nonlinear Schrödinger Equation with Derivative

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