Ablowitz-Ladik Hierarchy

摘要: The hierarchy considered in the present chapter appeared early days of soli-ton theory as a spatial discretization one most famous hierarchies soliton equations partial derivatives, namely AKNS (Ablowitz-Kaup-Newell-Segur) hierarchy. latter is conveniently described attached to Zakharov-Shabat spectral problem: $$ {\Psi _x}=P\Psi,P=\left( {\begin{array}{*{20}{c}} \zeta &q \\r&{ - }\end{array}} \right).$$ (18.1.1) Here q = q(x, t), r r(x, t) are unknown fields, terms which formulated and ζ parameter. Each equation may be presented compatibility condition (18.1.1) with linear differential describing evolution function Ψ time: $$ _t} Q\Psi,$$ (18.1.2) where matrix Q polynomially depends on Explicitly abovementioned takes form zero curvature {P_t} {Q_x} + \left[ {P,Q} \right] 0,$$ (18.1.3) which is, for suitable Q, non-linear q, r. In particular (see Section 18.2), we can get this way Schrodinger (NLS): i{q_t} {q_{xx}} \mp 2{\left| \right|^2}q,$$ (18.1.4) and not less modified Korteweg-de Vries (MKdV): {q_t} {q_{xxx}} 6{q_x}{q^2}.$$ (18.1.5) To NLS, has change independent variable \( t \mapsto it\) perform reduction $$r \pm {{q}^{*}}, $$ (18.1.6) while MKdV following relevant: q. $$ (18.1.7)