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Vector Nonlinear Schrödinger Hierarchies as Approximate Kadomtsev-Petviashvili Hierarchies

P. D. Miller
Australian Photonics Cooperative Research Centre
Optical Sciences Centre
Research School of Physical Sciences and Engineering
The Australian National University, Canberra ACT 0200 Australia


The Kadomtsev-Petviashvili (KP) hierarchy, a collection of compatible nonlinear equations, each in 2+1 independent variables, can be consistently constrained in many different ways to yield hierarchies of equations in 1+1 independent variables. In particular, the N-component vector nonlinear Schrödinger (VNLS) hierarchies are contained within the KP hierarchy in this way. These hierarchies approximate the KP hierarchy in the limit of large N, and this permits the equations of the KP hierarchy to be approximated by nonlinear equations in 1+1 dimensions.