User:Maximilian Janisch/latexlist/Ablowitz–Kaup–Newell–Segur hierarchy

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An infinite tower of non-linear evolution equations that derives its name from the simplest non-trivial system of equations contained in it, the AKNS-equations

\begin{equation} \tag{a1} \left\{ \begin{array} { l } { i \frac { \partial } { \partial t } q ( x , t ) = i q _t = - \frac { 1 } { 2 } q _{x x} + q ^ { 2 } r, } \\ { i \frac { \partial } { \partial t } r ( x , t ) = i r _t = \frac { 1 } { 2 } r _{xx} - q r ^ { 2 }. } \end{array} \right. \end{equation}

It were M.J. Ablowitz, D.J. Kaup, A.C. Newell, and H. Segur who showed that the initial value problem of this system of equations (cf. also Differential equation, partial, discontinuous initial (boundary) conditions) could be solved with the inverse scattering transform (cf. also Korteweg–de Vries equation). To get a natural embedding of the AKNS-equations in a larger system, one rewrites (a1) in zero-curvature form as

\begin{equation} \tag{a2} \frac { \partial } { \partial t } P _ { 1 } - \frac { \partial } { \partial x } Q _ { 2 } + [ P _ { 1 } , Q _ { 2 } ] = 0, \end{equation}

where

\begin{equation*} P _ { 1 } = \left( \begin{array} { c c c } { 0 } & { \square } & { q } \\ { r } & { \square } & { 0 } \end{array} \right) , Q _ { 2 } = \left( \begin{array} { c c } { - \frac { i } { 2 } q r } & { \frac { i } { 2 } q_x } \\ { - \frac { i } { 2 } r _ { x } } & { \frac { i } { 2 } q r } \end{array} \right). \end{equation*}

Consider now the following polynomial expressions in the parameter $z$:

\begin{equation} \tag{a3} P = P _ { 0 } z + P _ { 1 } : = \left( \begin{array} { c c } { - i } & { 0 } \\ { 0 } & { i } \end{array} \right) z + \left( \begin{array} { l l } { 0 } & { q } \\ { r } & { 0 } \end{array} \right), \end{equation}

\begin{equation*} Q ^ { ( n ) } : = Q _ { 0 } z ^ { n } + Q _ { 1 } z ^ { n - 1 } \ldots Q _ { n }, \end{equation*}

where the $Q_i$ are $\operatorname{sl} _ { 2 }$-valued functions depending on the variables $x$, $t = ( t _ { n } )$, and $Q _ { 0 } = P _ { 0 }$ and $Q _ { 1 } = P _ { 1 }$. For these data the zero-curvature equations read

\begin{equation} \tag{a4} \frac { \partial } { \partial t _ { n } } P - \frac { \partial } { \partial x } Q ^ { ( n ) } + [ P , Q ^ { ( n ) } ] = 0 \Leftrightarrow \end{equation}

\begin{equation*} \Leftrightarrow \left[ \frac { \partial } { \partial x } - P , \frac { \partial } { \partial t _ { n } } - Q ^ { ( n ) } \right] = 0, \end{equation*}

which is an infinite tower of equations extending the system (a1). The system (a4) generalizes from $\operatorname{SL} _ { 2 } ( {\bf C })$ to a general simple complex Lie algebra $\frak g$, a regular element $P_0$ in a Cartan subalgebra $\mathfrak h $ of $\frak g$ and an element $Q_0$ in $\mathfrak h $, see [a4] and [a10]. Solutions of the equations (a4) can be obtained by the Zakharov–Shabat dressing method (cf. also Soliton). Namely, consider the function

\begin{equation} \tag{a5} \phi ( x , t , z ) = \end{equation}

\begin{equation*} = \operatorname { exp } \left( x P _ { 0 } z + \sum _ { r = 1 } ^ { \infty } Q _ { 0 } z ^ { r } \right) g ( z ) . . \operatorname { exp } \left( - x P _ { 0 } z - \sum _ { r = 1 } ^ { \infty } Q _ { 0 } z ^ { r } \right), \end{equation*}

with $g ( z )$ belonging to the loop group $C ^ { \infty } ( S ^ { 1 } , \operatorname{SL}_ { 2 } ( {\bf C} ) )$. If this $\phi$ factorizes as $\phi = \phi _ { - } \phi _ { + }$, with

\begin{equation} \tag{a6} \phi _ { - } ( x , t , z ) = \operatorname { exp } \left( \sum _ { i = 1 } ^ { \infty } \chi _ { i } ( x , t ) z ^ { - i } \right), \end{equation}

\begin{equation*} \phi _ { + } = \operatorname { exp } \left( \sum _ { j = 1 } ^ { \infty } \phi _ { j } ( x , t ) z ^ { j } \right), \end{equation*}

then conjugating the trivial connections (cf. also Connections on a manifold) $( \partial / \partial x ) - P _ { 0 } z$ and $( \partial / \partial t _ { n } ) - Q _ { 0 } z ^ { n }$ with $\phi_{-}$ gives connections of the required form:

\begin{equation} \tag{a7} \phi_{-} ^ { -1 } \left( \frac { \partial } { \partial x } - P _ { 0 }z \right) \phi _ { - } = \frac { \partial } { \partial x } - P, \end{equation}

\begin{equation*} \phi _ { - } ^ { - 1 } \frac { \partial } { \partial t _ { n } } - Q _ { 0 } z ^ { n } \phi _ { - } = \frac { \partial } { \partial t _ { n } } - Q ^ { ( n ) }. \end{equation*}

Since flatness is preserved by this procedure, this leads to solutions of (a4). If $Q _ { 0 } = P _ { 0 }$, as in the AKNS-case, one can take just as well $\partial / \partial x = \partial / \partial t _ { 1 }$.

It was observed by H. Flaschka, Newell and T. Ratiu [a5] that the equations (a4) for the $\operatorname{SL} _ { 2 } ( {\bf C })$-case could be captured in the system

\begin{equation} \tag{a8} \frac { \partial } { \partial t _ { n } } Q = [ Q ^ { ( n ) } , Q ] , n \geq 1, \end{equation}

for the single series

\begin{equation} \tag{a9} Q = \sum _ { j = 0 } ^ { \infty } Q _ { j } z ^ { - j } , Q _ { j } = \left( \begin{array} { c c } { h _ { j } } & { e _ { j } } \\ { f _ { j } } & { - h _ { j } } \end{array} \right), \end{equation}

\begin{equation*} Q ^ { ( n ) } = \sum _ { j = 0 } ^ { n } Q _ { j } z ^ { n - j }. \end{equation*}

They showed that these equations are commuting Hamiltonian flows (cf. also Hamiltonian system) on the Lie algebra $\sum _ { i = 0 } ^ { \infty } X _ { i } z ^ { - i }$, $X _ { i } \in \operatorname { sl } _ { 2 } ( {\bf C} )$, with respect to natural Poisson brackets. Further, they introduced the flux tensor $F _ { j k }$ by

\begin{equation} \tag{a10} F _ { j k } = \end{equation}

\begin{equation*} = \frac { 1 } { 2 } \operatorname { Tr } \left( \sum _ { r = 0 } ^ { j } ( j - r ) Q _ { r } Q _ { k + j - r } + \frac { 1 } { 2 } \sum _ { r = 0 } ^ { j } ( r - k ) Q _ { r } Q _ { k + j - r } \right) \end{equation*}

and proved the local conservation laws of the system, namely

\begin{equation} \tag{a11} \frac { \partial } { \partial t _ { k } } F _ { i j } = \frac { \partial } { \partial t _ { i } } F _ { j k }. \end{equation}

The left-hand side of (a11) is, in fact, even symmetric under permutations of the indices $i$, $j$, $k$ and this property made them introduce a potential $\tau$ by

\begin{equation} \tag{a12} F _ { j k } = \frac { \partial } { \partial t _ { j } } \frac { \partial } { \partial t _ { k } } \operatorname { log } ( \tau ). \end{equation}

The equations (a8) are called the Lax equations of the AKNS-hierarchy. As such, the AKNS-hierarchy is a natural reduction of the two-component KP-hierarchy (cf. also KP-equation; [a7]), a fact that enables a description in the Grassmannian of that hierarchy.

It was shown by M.J. Bergvelt and A.P.E. ten Kroode [a1] that it is natural to consider a system of zero-curvature relations (a4) labelled by the root lattice of the Lie algebra, where the operators at different sites of the lattice are linked by Toda-type differential-difference equations. For example, for nearest neighbour sites there holds

\begin{equation} \tag{a13} q ^ { ( l + 1 ) } = - ( q ^ { ( l ) } ) ^ { 2 } r ^ { ( l ) } + q ^ { ( l ) } \operatorname { log } ( q ^ { ( l ) } ) , r ^ { ( l + 1 ) } = \frac { 1 } { q ^ { ( l ) } }, \end{equation}

where

\begin{equation*} P ^ { ( l ) } = \left( \begin{array} { c c } { - i } & { 0 } \\ { 0 } & { i } \end{array} \right) z + \left( \begin{array} { c c } { 0 } & { q ^ { ( l ) } } \\ { r ^ { ( l ) } } & { 0 } \end{array} \right). \end{equation*}

This phenomenon is due to the fact that there is a natural lattice group that commutes with the commuting flows corresponding to the parameters $t _ { n }$.

In the representation-theoretic approach to soliton equations (see [a3], [a8]), the soliton equations occur as the equations describing the group orbit of the highest weight vector. A similar description holds for these combined differential-difference equations. Let $L ( \Lambda _ { 0 } )$ be the basic representation of the Kac–Moody Lie algebra $A _ { 1 } ^ { ( 1 ) }$ (cf. also Kac–Moody algebra). In $A _ { 1 } ^ { ( 1 ) }$ one takes the homogeneous Heisenberg algebra

\begin{equation*} s = \sum _ { i > 0 } \mathbf{C} \lambda ^ { i } \left( \begin{array} { c c } { - 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) \bigoplus \sum _ { i > 0 } \mathbf{C} \lambda ^ { - i } \left( \begin{array} { c c } { - 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) \bigoplus \mathbf{C} _{c}, \end{equation*}

where $c$ is the central element of $A _ { 1 } ^ { ( 1 ) }$ that is in the kernel of the projection of $A _ { 1 } ^ { ( 1 ) }$ onto the loop algebra of $\operatorname{SL} _ { 2 } ( {\bf C })$. The $A _ { 1 } ^ { ( 1 ) }$-module $L ( \Lambda _ { 0 } )$ decomposes with respect to the action of the homogeneous Heisenberg algebra $s$ as a direct sum of irreducible $s$-modules $\mathbf{C} [ t ] = \mathbf{C} [ t _ { 1 } , t _ { 2 } , \ldots]$ labelled by the root lattice. Thus, one can write each element of $L ( \Lambda _ { 0 } )$ as $\tau ( t ) = ( \tau _ { l } ( t ) ) _ { l \in \mathbf Z }$. The group orbit of the highest weight vector can then be characterized by a set of so-called Hirota bilinear relations for the components $( \tau _ { l } )$. By using the representation theory one constructs a series of elements $( g _ l)$ in a suitable completion of the Kac–Moody group associated with $A _ { 1 } ^ { ( 1 ) }$ such that the vacuum expectation value of $g_{l}$ is exactly $\tau_l$. The Birkhoff decomposition of the $( g _ l)$ in that group then enables one to construct solutions of the lattice zero-curvature equations, [a2]. In particular, the operators $P ^ { ( l ) }$ from (a13) obtained in this way can be expressed in the components $( \tau _ { l } )$ by

\begin{equation} \tag{a14} q ^ { ( l ) } = 2 i \frac { \tau _ { l + 1 }} { \tau _ { l } } ,\, r ^ { ( l ) } = - 2 i \frac { \tau _ { l - 1} } { \tau _ { l } }. \end{equation}

By using the adjoint action of the Kac–Moody group, Bergvelt and ten Kroode also showed [a2] that the

\begin{equation} \tag{a15} F _ { j k } ^ { ( l ) } : = \frac { \partial } { \partial t _ { j } } \frac { \partial } { \partial t _ { k } } \operatorname { log } ( \tau _ { l } ) \end{equation}

give exactly the flux tensor from (a10), thus furnishing a representation-theoretic basis for the results in [a5].

A geometric way to look at $\tau$-functions, see e.g. [a9], is to consider a homogeneous space over the relevant loop group $L$, a holomorphic line bundle $\mathcal{L}$ over this space and its pull-back over the corresponding central extension $\hat{L}$ of $L$. If this last line bundle has a global holomorphic section, then $\tau$-functions measure the failure of equivariance of partial liftings from $L$ to $\hat{L}$ with respect to this section. With this point of view, one can also arrive at the formulas in (a14) by lifting the discrete group of transformations that commute with the flows from (a6) appropriately, see [a11].

An important class of equations associated with the AKNS-hierarchy are the so-called stationary AKNS-equations. These are the differential equations for the functions $q$ and $r$ from the first-order differential operator

\begin{equation*} L : = P _ { 0 } \frac { d } { d x } + P _ { 1 } = \left( \begin{array} { c c } { - i } & { 0 } \\ { 0 } & { i } \end{array} \right) \frac { d } { d x } + \left( \begin{array} { c c } { 0 } & { q } \\ { r } & { 0 } \end{array} \right), \end{equation*}

resulting from the existence of a $( 2 \times 2 )$-matrix-valued differential operator

\begin{equation*} \mathcal{P} _ { n + 1 } = \sum _ { i = 0 } ^ { n + 1 } u _ { i } \left( \frac { d } { d x } \right) ^ { i } \end{equation*}

that commutes with $L$. Such a pair is naturally associated with a hyper-elliptic curve of genus $n$ and that is why one calls $P _ { 1 }$ an algebro-geometric AKNS-potential. The elliptic algebro-geometric AKNS-potentials have been characterized in [a6]. They correspond exactly to the potentials for which the equation $L ( \psi ) = z \psi$ has a meromorphic fundamental system of solutions with respect to $x$ for all values of the spectral parameter $z \in \mathbf{C}$.

References

[a1]	M.J. Bergvelt, A.P.E. ten Kroode, "Differential-difference $A K N S$ equations and homogeneous Heisenberg algebras" J. Math. Phys. , 28 (1987) pp. 302–306
[a2]	M.J. Bergvelt, A.P.E. ten Kroode, "$\tau$-functions and zero curvature equations of Toda–AKNS type" J. Math. Phys. , 29 (1988) pp. 1308–1320
[a3]	E. Date, M. Jimbo, M. Kashiwara, T. Miwa, "Transformation groups for soliton equations" , Non-linear Integrable Systems; Classical Theory and Quantum Theory (Proc. RIMS Symp.) , World Sci. (1983) pp. 41–119
[a4]	V.G. Drinfel'd, V.V. Sokolov, "Lie algebras and equations of Korteweg de Vries type" Itogi Nauki i Tekhn. Ser. Sovrem. Probl. Mat. , 24 (1984) pp. 81–180 (In Russian)
[a5]	H. Flaschka, A.C. Newell, T. Ratiu, "Kac–Moody Lie algebras and soliton equations II. Lax equations associated with $A _ { 1 } ^ { ( 1 ) }$" Physica , 9D (1983) pp. 300–323
[a6]	F. Gesztesy, R. Weikard, "A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy" Acta Math. , 181 (1998) pp. 63–108
[a7]	G.F. Helminck, G.F. Post, "A convergent framework for the multicomponent $K P$-hierarchy" Trans. Amer. Math. Soc. , 324 : 1 (1991) pp. 271–292
[a8]	V.G. Kac, "Infinite dimensional Lie algebras" , Cambridge Univ. Press (1989) (Edition: Third)
[a9]	G. Segal, G. Wilson, "Loop groups and equations of KdV type" Publ. Math. IHES , 63 (1985) pp. 1–64
[a10]	G. Wilson, "The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras" Ergod. Th. Dynam. Syst. , 1 (1981) pp. 361–380
[a11]	G. Wilson, "The $\tau$-functions of the AKNS equations" , Integrable Systems: the Verdier Memorial Conf. (Actes Colloq. Internat. Luminy) , Progress in Math. , 115 (1993) pp. 131–145