Journal ArticleDOI
Soliton equations and pseudospherical surfaces
TLDR
In this paper, all the soliton equations in 1 + 1 dimensions that can be solved by the AKNS 2 × 2 inverse scattering method are shown to describe pseudospherical surfaces, i.e., surfaces of constant negative Gaussian curvature.About:
This article is published in Nuclear Physics.The article was published on 1979-07-23. It has received 197 citations till now. The article focuses on the topics: sine-Gordon equation & Dissipative soliton.read more
Citations
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MonographDOI
Bäcklund and Darboux transformations : geometry and modern applications in soliton theory
Colin Rogers,Wolfgang K. Schief +1 more
TL;DR: Backlund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory have been explored in this article, where the authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Backlund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged.
Journal ArticleDOI
Integrable equations arising from motions of plane curves
Kai-Seng Chou,Changzheng Qu +1 more
TL;DR: The motion of plane curves in Klein geometry is studied in this paper, where it is shown that the KdV, Harry-Dym, Sawada-Kotera, Burgers, the defocusing mKdV hierarchies, the Camassa-Holm and the Kaup-Kupershmidt equation naturally arise from the motions of planes in SL(2)-, Sim(2), SA(2) and A(2-) geometries.
Journal ArticleDOI
An elementary geometric characterization of the integrable motions of a curve
Adam Doliwa,Paolo Maria Santini +1 more
TL;DR: In this article, it was shown that the motion of a curve selects hierarchies of integrable dynamics, such as the Korteweg-de Vries hierarchy, the Schrodinger hierarchy, and the Schroff hierarchy.
Journal ArticleDOI
Pseudospherical surfaces and evolution equations
TL;DR: In this paper, a methode systematique for determiner un probleme lineaire pour lequel une equation donnee est la condition d'integrabilite is defined.
References
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Journal ArticleDOI
Method for solving the Korteweg-deVries equation
TL;DR: In this paper, a method for solving the initial value problem of the Korteweg-deVries equation is presented which is applicable to initial data that approach a constant sufficiently rapidly as
Book
Inverse Problems in Quantum Scattering Theory
TL;DR: The physical importance of inverse problems in quantum scattering theory is clear since all the information we can obtain on nuclear, particle, and subparticle physics is gathered from scattering experiments as discussed by the authors.
Journal ArticleDOI
Nonlinear-evolution equations of physical significance
TL;DR: In this article, the inverse scattering method was used to solve the initial value problem for a broad class of nonlinear evolution equations, including sine-Gordon, sinh-Gordon and Benney-Newell.
Journal ArticleDOI
Korteweg‐de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion
TL;DR: In this article, a variety of conservation laws and constants of motion for the Kortewegde Vries and related equations are derived for the Sturm-Liouville eigenvalue problem.
Journal ArticleDOI
Korteweg-de Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation
TL;DR: In this article, an explicit nonlinear transformation relating solutions of the Kortewegde Vries equation and a similar nonlinear equation is presented, which is generalized to solutions of a one-parameter family of similar non-linear equations.