KdV hierarchy

Let $T$ be translation operator defined on real valued functions as $T(g)(x)=g(x+1)$. Let $\backslash mathcal\{C\}$ be set of all analytic functions that satisfy $T(g)(x)=g(x)$, i.e. periodic functions of period 1. For each $g\; \backslash in\; \backslash mathcal\{C\}$, define an operator

$L\_g(\backslash psi)(x)\; =\; \backslash psi\text{'}\text{'}(x)\; +\; g(x)\; \backslash psi(x)$

on the space of smooth functions on $\backslash mathbb\{R\}$. We define the Bloch spectrum $\backslash mathcal\{B\}\_g$ to be the set of $(\backslash lambda,\backslash alpha)\; \backslash in\; \backslash mathbb\{C\}\backslash times\backslash mathbb\{C\}^*$ such that there is a nonzero function $\backslash psi$ with $L\_g(\backslash psi)=\backslash lambda\backslash psi$ and $T(\backslash psi)=\backslash alpha\backslash psi$. The KdV hierarchy is a sequence of nonlinear differential operators $D\_i:\; \backslash mathcal\{C\}\; \backslash to\; \backslash mathcal\{C\}$ such that for any $i$ we have an analytic function $g(x,t)$ and we define $g\_t(x)$ to be $g(x,t)$ and

$D\_i(g\_t)=\; \backslash frac\{d\}\{dt\}\; g\_t$,

then $\backslash mathcal\{B\}\_g$ is independent of $t$.

The KdV hierarchy arises naturally as a statement of Huygens' principle for the D'Alembertian.{{cite journal |first1=Fabio A. C. C. |last1=Chalub |first2=Jorge P. |last2=Zubelli |title=Huygens' Principle for Hyperbolic Operators and Integrable Hierarchies |journal=Physica D: Nonlinear Phenomena |volume=213 |issue=2 |year=2006 |pages=231–245 |doi=10.1016/j.physd.2005.11.008 |bibcode=2006PhyD..213..231C }}{{cite journal |first1=Yuri Yu. |last1=Berest |first2=Igor M. |last2=Loutsenko |date=1997 |title=Huygens' Principle in Minkowski Spaces and Soliton Solutions of the Korteweg-de Vries Equation |journal=Communications in Mathematical Physics |volume=190 |issue=1 |pages=113–132 |arxiv=solv-int/9704012 |doi=10.1007/s002200050235 |bibcode=1997CMaPh.190..113B |s2cid=14271642 }}

The first three partial differential equations of the KdV hierarchy are

where each equation is considered as a PDE for $u\; =\; u(x,\; t\_n)$ for the respective $n$.{{cite book |last1=Dunajski |first1=Maciej |title=Solitons, instantons, and twistors |date=2010 |publisher=Oxford University Press |location=Oxford |isbn=9780198570639 |pages=56–57}}

The first equation identifies $t\_0\; =\; x$ and $t\_1\; =\; t$ as in the original KdV equation. These equations arise as the equations of motion from the (countably) infinite set of independent constants of motion $I\_n[u]$ by choosing them in turn to be the Hamiltonian for the system. For $n\; >\; 1$, the equations are called higher KdV equations and the variables $t\_n$ higher times.

File:Cnoidal wave m=0.9.svg solution to the Korteweg–De Vries equation, in terms of the square of the Jacobi elliptic function cn (and with value of the parameter {{nowrap|1=m {{=}} 0.9}}).]]

One can consider the higher KdVs as a system of overdetermined PDEs for

$$u\; =\; u(t\_0\; =\; x,\; t\_1\; =\; t,\; t\_2,\; t\_3,\; \backslash cdots).$$

Then solutions which are independent of higher times above some fixed $n$ 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 $g$. For example, $g\; =\; 0$ gives the constant solution, while $g\; =\; 1$ corresponds to cnoidal wave solutions.

For $g\; >\; 1$, the Riemann surface is a hyperelliptic curve and the solution is given in terms of the theta function.{{cite book |last1=Manakov |first1=S. |last2=Novikov |first2=S. |last3=Pitaevskii |first3=L. |last4=Zakharov |first4=V. E. |title=Theory of solitons : the inverse scattering method |date=1984 |location=New York |isbn=978-0-306-10977-5}} In fact all solutions to the KdV equation with periodic initial data arise from this construction {{harvs|last1=Manakov|last2=Novikov|last3=Pitaevskii|last4=Zakharov|year=1984}}.

• Witten's conjecture
• Huygens' principle

{{Reflist}}

• {{Citation | last1=Gesztesy | first1=Fritz | last2=Holden | first2=Helge | title=Soliton equations and their algebro-geometric solutions. Vol. I | publisher=Cambridge University Press | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-75307-4 | mr=1992536 | year=2003 | volume=79}}

• [https://web.archive.org/web/20160303220759/http://wiki.math.toronto.edu/DispersiveWiki/index.php/KdV_hierarchy KdV hierarchy] at the Dispersive PDE Wiki.

Category:Partial differential equations

Category:Solitons

Category:Exactly solvable models