KdV hierarchy explained
In mathematics, the KdV hierarchy is an infinite sequence of partial differential equations which contains the Korteweg–de Vries equation.
Details
Let
be translation operator defined on real valued
functions as
. Let
be set of all
analytic functions that satisfy
, i.e.
periodic functions of period 1. For each
, define an operator
Lg(\psi)(x)=\psi''(x)+g(x)\psi(x)
on the space of
smooth functions on
. We define the
Bloch spectrum
to be the set of
such that there is a nonzero function
with
and
. The KdV hierarchy is a sequence of nonlinear differential operators
such that for any
we have an analytic function
and we define
to be
and
,then
is independent of
.
The KdV hierarchy arises naturally as a statement of Huygens' principle for the D'Alembertian.[1] [2]
Explicit equations for first three terms of hierarchy
The first three partial differential equations of the KdV hierarchy arewhere each equation is considered as a PDE for
for the respective
.
[3] The first equation identifies
and
as in the original KdV equation. These equations arise as the equations of motion from the (countably) infinite set of independent
constants of motion
by choosing them in turn to be the Hamiltonian for the system. For
, the equations are called
higher KdV equations and the variables
higher times.
Application to periodic solutions of KdV
One can consider the higher KdVs as a system of overdetermined PDEs forThen solutions which are independent of higher times above some fixed
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
. For example,
gives the constant solution, while
corresponds to
cnoidal wave solutions.
For
, 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 .
See also
External links
Notes and References
- Fabio A. C. C. . Chalub . Jorge P. . Zubelli . Huygens' Principle for Hyperbolic Operators and Integrable Hierarchies . Physica D: Nonlinear Phenomena . 213 . 2 . 2006 . 231–245 . 10.1016/j.physd.2005.11.008 . 2006PhyD..213..231C .
- Yuri Yu. . Berest . Igor M. . Loutsenko . 1997 . Huygens' Principle in Minkowski Spaces and Soliton Solutions of the Korteweg-de Vries Equation . Communications in Mathematical Physics . 190 . 1 . 113–132 . solv-int/9704012 . 10.1007/s002200050235 . 1997CMaPh.190..113B . 14271642 .
- Book: Dunajski . Maciej . Solitons, instantons, and twistors . 2010 . Oxford University Press . Oxford . 9780198570639 . 56–57.
- Book: Manakov . S. . Novikov . S. . Pitaevskii . L. . Zakharov . V. E. . Theory of solitons : the inverse scattering method . 1984 . New York . 978-0-306-10977-5.