KdV hierarchy

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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 $L g (ψ)(x) = ψ''(x) + g (x)ψ(x)$ on the space of smooth functions on $\mathbb{R}$ . We define the Bloch spectrum $\mathcal{B}_g$ to be the set of $(\lambda,\alpha) \in \mathbb{C}\times\mathbb{C}^*$ so that there is a nonzero function $ψ$ with $L g (ψ) = λψ$ and $T (ψ) = αψ$ . The KdV hierarchy is a sequence of nonlinear differential operators $D_i: \mathcal{C} \to \mathcal{C}$ 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 $t$ .

KdV hierarchy

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