Abstract
The hierarchy considered in the present chapter appeared in the early days of the soli-ton theory as a spatial discretization of one of the most famous hierarchies of soliton equations in partial derivatives, namely of the AKNS (Ablowitz-Kaup-Newell-Segur) hierarchy. The latter is most conveniently described as a hierarchy of equations attached to the Zakharov-Shabat spectral problem:
Here q = q(x, t), r = r(x, t) are unknown fields, in terms of which the hierarchy is formulated and ζ is the spectral parameter. Each equation of the hierarchy may be presented as a compatibility condition of (18.1.1) with a linear differential equation describing the evolution of the function Ψ in time:
where the matrix Q polynomially depends on the spectral parameter. Explicitly the abovementioned compatibility condition takes the form of the zero curvature equation
which is, for suitable Q, a non-linear evolution equation for q, r. In particular (see Section 18.2), we can get on this way the famous non-linear Schrödinger equation (NLS):
and the not less famous modified Korteweg-de Vries equation (MKdV):
To get NLS, one has to change the independent variable \( t \mapsto it\) and to perform the reduction
while for MKdV the following reduction is relevant:
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© 2003 Springer Basel AG
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Suris, Y.B. (2003). Ablowitz-Ladik Hierarchy. In: The Problem of Integrable Discretization: Hamiltonian Approach. Progress in Mathematics, vol 219. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8016-9_18
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DOI: https://doi.org/10.1007/978-3-0348-8016-9_18
Publisher Name: Birkhäuser, Basel
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Online ISBN: 978-3-0348-8016-9
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