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In mathematics, the KdV hierarchy is an infinite sequence of
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
s which starts with the Korteweg–de Vries equation.


Details

Let T be translation operator defined on real valued functions as T(g)(x)=g(x+1). Let \mathcal be set of all
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s that satisfy T(g)(x)=g(x), i.e.
periodic function A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to d ...
s of period 1. For each g \in \mathcal, define an operator L_g(\psi)(x) = \psi''(x) + g(x) \psi(x) on the space of
smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
s on \mathbb. We define the Bloch spectrum \mathcal_g to be the set of (\lambda,\alpha) \in \mathbb\times\mathbb^* such that there is a nonzero function \psi with L_g(\psi)=\lambda\psi and T(\psi)=\alpha\psi. The KdV hierarchy is a sequence of nonlinear differential operators D_i: \mathcal \to \mathcal 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)= \frac g_t , then \mathcal_g is independent of t. The KdV hierarchy arises naturally as a statement of Huygens' principle for the
D'Alembertian In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of M ...
.


See also

* Witten's conjecture * Huygens' principle


References


Sources

*{{Citation , last1=Gesztesy , first1=Fritz , last2=Holden , first2=Helge , title=Soliton equations and their algebro-geometric solutions. Vol. I , publisher=
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...
, series=Cambridge Studies in Advanced Mathematics , isbn=978-0-521-75307-4 , mr=1992536 , year=2003 , volume=79


External links


KdV hierarchy
at the Dispersive PDE Wiki. Partial differential equations Solitons Exactly solvable models