KdV hierarchy

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

Let ${\displaystyle T}$ be translation operator defined on real valued functions as ${\displaystyle T(g)(x)=g(x+1)}$. Let ${\displaystyle {\mathcal {C}}}$ be set of all analytic functions that satisfy ${\displaystyle T(g)(x)=g(x)}$, i.e. periodic functions of period 1. For each ${\displaystyle g\in {\mathcal {C}}}$, define an operator ${\displaystyle L_{g}(\psi )(x)=\psi ''(x)+g(x)\psi (x)}$ on the space of smooth functions on ${\displaystyle \mathbb {R} }$. We define the Bloch spectrum ${\displaystyle {\mathcal {B}}_{g}}$ to be the set of ${\displaystyle (\lambda ,\alpha )\in \mathbb {C} \times \mathbb {C} ^{*}}$ such that there is a nonzero function ${\displaystyle \psi }$ with ${\displaystyle L_{g}(\psi )=\lambda \psi }$ and ${\displaystyle T(\psi )=\alpha \psi }$. The KdV hierarchy is a sequence of nonlinear differential operators ${\displaystyle D_{i}:{\mathcal {C}}\to {\mathcal {C}}}$ such that for any ${\displaystyle i}$ we have an analytic function ${\displaystyle g(x,t)}$ and we define ${\displaystyle g_{t}(x)}$ to be ${\displaystyle g(x,t)}$ and ${\displaystyle D_{i}(g_{t})={\frac {d}{dt}}g_{t}}$, then ${\displaystyle {\mathcal {B}}_{g}}$ is independent of ${\displaystyle t}$.

The KdV hierarchy arises naturally as a statement of Huygens' principle for the D'Alembertian.^[1][2]

The first three partial differential equations of the KdV hierarchy are $${\displaystyle {\begin{aligned}u_{t_{0}}&=u_{x}\\u_{t_{1}}&=6uu_{x}-u_{xxx}\\u_{t_{2}}&=10uu_{xxx}-20u_{x}u_{xx}-30u^{2}u_{x}-u_{xxxxx}.\end{aligned}}}$$ where each equation is considered as a PDE for ${\displaystyle u=u(x,t_{n})}$ for the respective ${\displaystyle n}$.^[3]

The first equation identifies ${\displaystyle t_{0}=x}$ and ${\displaystyle t_{1}=t}$ as in the original KdV equation. These equations arise as the equations of motion from the (countably) infinite set of independent constants of motion ${\displaystyle I_{n}[u]}$ by choosing them in turn to be the Hamiltonian for the system. For ${\displaystyle n>1}$, the equations are called higher KdV equations and the variables ${\displaystyle t_{n}}$ higher times.

One can consider the higher KdVs as a system of overdetermined PDEs for $${\displaystyle u=u(t_{0}=x,t_{1}=t,t_{2},t_{3},\cdots ).}$$ Then solutions which are independent of higher times above some fixed ${\displaystyle n}$ and with periodic boundary conditions are called finite-gap solutions. Such solutions turn out to correspond to compact Riemann surfaces, which are classified by their genus ${\displaystyle g}$. For example, ${\displaystyle g=0}$ gives the constant solution, while ${\displaystyle g=1}$ corresponds to cnoidal wave solutions.

For ${\displaystyle g>1}$, the Riemann surface is a hyperelliptic curve and the solution is given in terms of the theta function.^[4] In fact all solutions to the KdV equation with periodic initial data arise from this construction (Manakov, Novikov & Pitaevskii et al. 1984).

• Witten's conjecture
• Huygens' principle

• Gesztesy, Fritz; Holden, Helge (2003), Soliton equations and their algebro-geometric solutions. Vol. I, Cambridge Studies in Advanced Mathematics, vol. 79, Cambridge University Press, ISBN 978-0-521-75307-4, MR 1992536

• KdV hierarchy at the Dispersive PDE Wiki.