AKNS system

Inmathematics,theAKNS systemis anintegrable systemofpartial differential equations,introduced by and named afterMark J. Ablowitz,David J. Kaup,Alan C. Newell,and Harvey Segur from their publication inStudies in Applied Mathematics:Ablowitz, Kaup, and Newell et al. (1974).

Definition

The AKNS system is a pair of two partialdifferential equationsfor twocomplex-valued functionspandqof 2 variablestandx:

$p_{t}=+ip^{2}q-{\frac {i}{2}}p_{xx}$

$q_{t}=-iq^{2}p+{\frac {i}{2}}q_{xx}$

Ifpandqarecomplex conjugatesthis reduces to thenonlinear Schrödinger equation.

Huygens' principleapplied to theDirac operatorgives rise to the AKNS hierarchy.

Applications to General Relativity

In 2021, the dynamics ofthree-dimensional (extremal) black holes on General Relativity with negative cosmological constantwere shown equivalent to two independent copies of the AKNS system.This duality was addressed through the imposition of suitable boundary conditions to theChern-Simons action.In this scheme, the involution of conserved charges of the AKNS system yields an infinite-dimensional commuting asymptotic symmetry algebra of gravitational charges.

See also

• Huygens principle

References

• Ablowitz, Mark J.; Kaup, David J.; Newell, Alan C.; Segur, Harvey (1974), "The inverse scattering transform-Fourier analysis for nonlinear problems",Studies in Appl. Math.,53(4): 249–315,doi:10.1002/sapm1974534249,MR0450815