Ablowitz-Ladik Hierarchy

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:

$$ {\Psi _x}=P\Psi,P=\left( {\begin{array}{*{20}{c}} \zeta &q \\r&{ - \zeta }\end{array}} \right).$$

(18.1.1)

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:

$$ {\Psi _t} = Q\Psi,$$

(18.1.2)

where the matrix Q polynomially depends on the spectral parameter. Explicitly the abovementioned compatibility condition takes the form of the zero curvature equation

$$ {P_t} - {Q_x} + \left[ {P,Q} \right] = 0,$$

(18.1.3)

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):

$$ - i{q_t} = {q_{xx}} \mp 2{\left| q \right|^2}q,$$

(18.1.4)

and the not less famous modified Korteweg-de Vries equation (MKdV):

$$ {q_t} = {q_{xxx}} \mp 6{q_x}{q^2}.$$

(18.1.5)

To get NLS, one has to change the independent variable $t \mapsto it$ and to perform the reduction

$$r = \pm {{q}^{*}}, $$

(18.1.6)

while for MKdV the following reduction is relevant:

$$ r = \pm q. $$

(18.1.7)