AKNS system

In mathematics, the AKNS system is an integrable system of partial differential equations, introduced by and named after Mark J. Ablowitz, David J. Kaup, and Alan C. Newell et al. (1974).

Definition

The AKNS system is a pair of two partial differential equations for two complex-valued functions p and q of 2 variables t and x:

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

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

If p and q are complex conjugates this reduces to the nonlinear Schrödinger equation.

Huygens' principle applied to the Dirac operator gives rise to the AKNS hierarchy.

See also

• Huygen's 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, MR 0450815