# The symmetries of solitons

@article{Palais1997TheSO, title={The symmetries of solitons}, author={Richard Palais}, journal={Bulletin of the American Mathematical Society}, year={1997}, volume={34}, pages={339-403} }

In this article we will retrace one of the great mathematical adventures of this century—the discovery of the soliton and the gradual explanation of its remarkable properties in terms of hidden symmetries. We will take an historical approach, starting with a famous numerical experiment carried out by Fermi, Pasta, and Ulam on one of the first electronic computers, and with Zabusky and Kruskal’s insightful explanation of the surprising results of that experiment (and of a follow-up experiment of…

## 97 Citations

Glimpses of Soliton Theory: The Algebra and Geometry of Nonlinear Pdes

- Physics
- 2010

Solitons are explicit solutions to nonlinear partial differential equations exhibiting particle-like behavior. This is quite surprising, both mathematically and physically. Waves with these…

Geometry of Solitons

- Mathematics, Physics
- 2001

parameterized by θ ∈ R. But we do not expect that the “sum” of two such solutions will again be a solution. However, the special class of soliton equations, the subject of this article, does have a…

Geometric transformations and soliton equations

- Mathematics
- 2010

We give a survey of the following six closely related topics: (i) a general method for constructing a soliton hierarchy from a splitting of a loop algebra into positive and negative subalgebras,…

Geometry of Solitons Chuu-Lian Terng

- Mathematics, Physics
- 1999

parameterized by θ ∈ R . But we do not expect that the “sum” of two such solutions will again be a solution. However, the special class of soliton equations, the subject of this article, does have a…

Feedback Stabilization of Solitons and Phonons Using the Controlled Lax Form

- Mathematics
- 2009

We consider the problem of asymptotically stabilizing a desired family of soliton solutions of a completely integrable system. We proceed from the Lax form of the system, to which we add a suitable…

A Brief History of Solitons and the KDV Equation

- PhysicsCurrent Science
- 2018

The historical context necessary to appreciate these spectacular developments inoliton theory is provided, a brief overview of the early history of the field, and a list of references to consult for additional information.

A brief historical introduction to solitons and the inverse scattering transform–a vision of Scott Russell

- Mathematics
- 2007

‘...the great primary waves of translation cross each other without change of any kind in the same manner as the small oscillations produced on the surface of a pool by a falling stone.’ SCOTT…

1 Periodic Schroödinger equations in Hamiltonian form

- Mathematics
- 2012

In these lectures I will summarize some old and recent results concerning different aspects of periodic Schrödinger equations viewed as infinite dimension Hamiltonian systems. In the first lecture I…

Geometry and dynamics in Hamiltonian lattices

- Mathematics, Physics
- 2003

E. Fermi, J. Pasta and S. Ulam introduced the Fermi-Pasta-Ulam lattice in the 1950s as a classical mechanical model for a mono-atomic crystal or a one-dimensional continuum. The model consisted of a…

Fay's identity in the theory of integrable systems

- Mathematics
- 2011

Fay's identity on Riemann surfaces is a powerful tool in the context of algebro-geometric solutions to integrable equations. This relation generalizes a well-known identity for the cross-ratio…

## References

SHOWING 1-10 OF 139 REFERENCES

Soliton equations and differential geometry

- Mathematics
- 1997

In this paper we study certain symplectic, Lie theoretic, and differential geometric properties of soliton equations. The equation for harmonic maps from the Lorentz space R1,1 to a symmetric space,…

Bäcklund transformations and loop group actions

- Mathematics
- 1998

We construct a local action of the group of rational maps from 2 to GL(n, ℂ), on local solutions of flows of the ZS-AKNS sl(n, ℂ)-hierarchy. We show that the actions of simple elements (linear…

Adventures of a Mathematician

- Art
- 1976

This autobiography of mathematician Stanislaw Ulam, one of the great scientific minds of the twentieth century, tells a story rich with amazingly prophetic speculations and peppered with lively…

Poisson actions and scattering theory for integrable systems

- Mathematics
- 1997

Conservation laws, heirarchies, scattering theory and Backlund transformations are known to be the building blocks of integrable partial differential equations. We identify these as facets of a…

Integrals of Nonlinear Equations of Evolution and Solitary Waves

- Mathematics
- 1968

In Section 1 we present a general principle for associating nonlinear equations evolutions with linear operators so that the eigenvalues of the linear operator integrals of the nonlinear equation. A…

Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method

- Mathematics, Physics
- 1978

A method is proposed for deriving and classifying relativistically invariant integrable systems that are sufficiently general to encompass all presently known two-dimensional solvable models, and for…

Solitons in mathematics and physics

- Physics, Mathematics
- 1987

The History of the Soliton Derivation of the Korteweg-de Vries, Nonlinear Schrodinger and Other Important and Canonical Equations of Mathematical Physics Soliton Equation Families and Solution…

Nonlinear Science Today

- Computer Science
- 1992

This article is an example of the hands-on, open-ended quality of articles that include software supplements: it is easy to reproduce, modify and extend the results given in the article by using Mathematica with Sullivan's software 4-D Polytopes.