# Four Symmetries of the KdV equation

@inproceedings{Rasin2021FourSO, title={Four Symmetries of the KdV equation}, author={Alexander G. Rasin and Jeremy Schiff}, year={2021} }

We identify 4 nonlocal symmetries of KdV depending on a parameter. We explain that since these are nonlocal symmetries, their commutator algebra is not uniquely determined, and we present three possibilities for the algebra. In the first version, 3 of the 4 symmetries commute; this shows that it is possible to add further (nonlocal) commuting flows to the standard KdV hierarchy. The second version of the commutator algebra is consistent with Laurent expansions of the symmetries, giving rise to… Expand

#### References

SHOWING 1-10 OF 31 REFERENCES

Symmetries of the KdV equation and four hierarchies of the integrodifferential KdV equations

- Mathematics
- 1994

Using the inverse strong symmetry of the Korteweg–de Vries (KdV) equation on the trivial symmetry and τ0 symmetry, one gets four new sets of symmetries of the KdV equation. These symmetries are… Expand

Additional symmetries for integrable equations and conformal algebra representation

- Mathematics
- 1986

AbstractWe present a regular procedure for constructing an infinite set of additional (spacetime variables explicitly dependent) symmetries of integrable nonlinear evolution equations (INEEs). In our… Expand

A Lax representation for the vertex operator and the central extension

- Mathematics
- 1995

Integrable hierarchies, viewed as isospectral deformations of an operatorL may admit symmetries; they are time-dependent vector fields, transversal to and commuting with the hierarchy and forming an… Expand

Nonlocal symmetries of the KdV equation

- Physics
- 1993

A loop algebra of nonlocal isovectors of the Korteweg–de Vries (KdV) equation is introduced which is derived from the bi‐Hamiltonian structure of that equation by inverting the usual recursion… Expand

The Gardner method for symmetries

- Mathematics, Physics
- 2011

The Gardner method, traditionally used to generate conservation laws of integrable equations, is generalized to generate symmetries. The method is demonstrated for the KdV, Camassa-Holm and… Expand

Differential equations in the spectral parameter, Darboux transformations and a hierarchy of master symmetries for KdV

- Mathematics
- 1991

We study a certain family of Schrödinger operators whose eigenfunctions ϕ(χ, λ) satisfy a differential equation in the spectral parameter λ of the formB(λ,∂λ)ϕ=Θ(x)ϕ. We show that the flows of a… Expand

Nonlocal symmetries related to Bäcklund transformation and their applications

- Mathematics, Physics
- 2012

Starting from nonlocal symmetries related to Backlund transformation (BT), many interesting results can be obtained. Taking the well-known potential KdV (pKdV) equation as an example, a new type of… Expand

More non-local symmetries of the KdV equation

- Mathematics
- 1993

The inverse of the usual recursion operator for the Korteweg-de Vries equation is applied to the Galilean symmetry. A new family of non-local symmetries results. A symmetry algebra isomorphic to the… Expand

Soliton Equations and Hamiltonian Systems

- Mathematics
- 2003

Integrable Systems Generated by Linear Differential nth Order Operators Hamiltonian Structures Hamiltonian Structure of the GD Hierarchies Modified KdV and GD. The Kupershmidt-Wilson Theorem The KP… Expand

Nonlocal symmetries and the theory of coverings: An addendum to A. M. vinogradov's ‘local symmetries and conservation laws”

- Mathematics
- 1984

For a systemY of partial differential equations, the notion of a coveringŶ∞→Y∞ is introduced whereY∞ is infinite prolongation ofY. Then nonlocal symmetries ofY are defined as transformations ofŶ∞… Expand