KdV hierarchy

In mathematics, the KdV hierarchy is an infinite sequence of partial differential equations which starts with the Korteweg–de Vries equation.

Let $T$ be translation operator defined on real valued functions as $T(g)(x)=g(x+1)$. Let $mathcal\{C\}$ be set of all analytic functions that satisfy $T(g)(x)=g(x)$, i.e. periodic functions of period 1. For each $g\; in\; mathcal\{C\}$, define an operator$L\_g(psi)(x)\; =\; psi"(x)\; +\; g(x)\; psi(x)$on the space of smooth functions on $mathbb\{R\}$. We define the Bloch spectrum $mathcal\{B\}\_g$ to be the set of $(lambda,alpha)\; in\; mathbb\{C\}\; imesmathbb\{C\}^*$ so that there is a nonzero function $psi$ with $L\_g(psi)=lambdapsi$ and $T(psi)=alphapsi$. The KdV hierarchy is a sequence of nonlinear differential operators $D\_i:\; mathcal\{C\}\; o\; mathcal\{C\}$ so that for any $i$ we have an analytic function $g(x,t)$ and we define $g\_t(x)$ to be $g(x,t)$ and$D\_i(g\_t)=\; frac\{d\}\{dt\}\; g\_t$,then $mathcal\{B\}\_g$ is independent of $t$.

External links

* [http://tosio.math.toronto.edu/wiki/index.php/KdV_hierarchy KdV hierarchy] at the Dispersive PDE Wiki.