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.