KdV hierarchy
In mathematics, the KdV hierarchy is an infinite sequence of partial differential equations which contains the Korteweg–de Vries equation.
Details
Let [math]\displaystyle{ T }[/math] be translation operator defined on real valued functions as [math]\displaystyle{ T(g)(x)=g(x+1) }[/math]. Let [math]\displaystyle{ \mathcal{C} }[/math] be set of all analytic functions that satisfy [math]\displaystyle{ T(g)(x)=g(x) }[/math], i.e. periodic functions of period 1. For each [math]\displaystyle{ g \in \mathcal{C} }[/math], define an operator [math]\displaystyle{ L_g(\psi)(x) = \psi''(x) + g(x) \psi(x) }[/math] on the space of smooth functions on [math]\displaystyle{ \mathbb{R} }[/math]. We define the Bloch spectrum [math]\displaystyle{ \mathcal{B}_g }[/math] to be the set of [math]\displaystyle{ (\lambda,\alpha) \in \mathbb{C}\times\mathbb{C}^* }[/math] such that there is a nonzero function [math]\displaystyle{ \psi }[/math] with [math]\displaystyle{ L_g(\psi)=\lambda\psi }[/math] and [math]\displaystyle{ T(\psi)=\alpha\psi }[/math]. The KdV hierarchy is a sequence of nonlinear differential operators [math]\displaystyle{ D_i: \mathcal{C} \to \mathcal{C} }[/math] such that for any [math]\displaystyle{ i }[/math] we have an analytic function [math]\displaystyle{ g(x,t) }[/math] and we define [math]\displaystyle{ g_t(x) }[/math] to be [math]\displaystyle{ g(x,t) }[/math] and [math]\displaystyle{ D_i(g_t)= \frac{d}{dt} g_t }[/math], then [math]\displaystyle{ \mathcal{B}_g }[/math] is independent of [math]\displaystyle{ t }[/math].
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 are [math]\displaystyle{ \begin{align}u_{t_0} &= u_x \\ u_{t_1} &= 6uu_x - u_{xxx} \\ u_{t_2} &= 10u u_{xxx} - 20u_x u_{xx} - 30u^2 u_x - u_{xxxxx}.\end{align} }[/math] where each equation is considered as a PDE for [math]\displaystyle{ u = u(x, t_n) }[/math] for the respective [math]\displaystyle{ n }[/math].[3]
The first equation identifies [math]\displaystyle{ t_0 = x }[/math] and [math]\displaystyle{ t_1 = t }[/math] as in the original KdV equation. These equations arise as the equations of motion from the (countably) infinite set of independent constants of motion [math]\displaystyle{ I_n[u] }[/math] by choosing them in turn to be the Hamiltonian for the system. For [math]\displaystyle{ n \gt 1 }[/math], the equations are called higher KdV equations and the variables [math]\displaystyle{ t_n }[/math] higher times.
Application to periodic solutions of KdV
One can consider the higher KdVs as a system of overdetermined PDEs for [math]\displaystyle{ u = u(t_0 = x, t_1 = t, t_2, t_3, \cdots). }[/math] Then solutions which are independent of higher times above some fixed [math]\displaystyle{ n }[/math] 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 [math]\displaystyle{ g }[/math]. For example, [math]\displaystyle{ g = 0 }[/math] gives the constant solution, while [math]\displaystyle{ g = 1 }[/math] corresponds to cnoidal wave solutions.
For [math]\displaystyle{ g \gt 1 }[/math], 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).
See also
- Witten's conjecture
- Huygens' principle
References
- ↑ Chalub, Fabio A. C. C.; Zubelli, Jorge P. (2006). "Huygens' Principle for Hyperbolic Operators and Integrable Hierarchies". Physica D: Nonlinear Phenomena 213 (2): 231–245. doi:10.1016/j.physd.2005.11.008. Bibcode: 2006PhyD..213..231C.
- ↑ Berest, Yuri Yu.; Loutsenko, Igor M. (1997). "Huygens' Principle in Minkowski Spaces and Soliton Solutions of the Korteweg-de Vries Equation". Communications in Mathematical Physics 190 (1): 113–132. doi:10.1007/s002200050235. Bibcode: 1997CMaPh.190..113B.
- ↑ Dunajski, Maciej (2010). Solitons, instantons, and twistors. Oxford: Oxford University Press. pp. 56–57. ISBN 9780198570639.
- ↑ Manakov, S.; Novikov, S.; Pitaevskii, L.; Zakharov, V. E. (1984). Theory of solitons : the inverse scattering method. New York. ISBN 978-0-306-10977-5.
Sources
- Gesztesy, Fritz; Holden, Helge (2003), Soliton equations and their algebro-geometric solutions. Vol. I, Cambridge Studies in Advanced Mathematics, 79, Cambridge University Press, ISBN 978-0-521-75307-4
External links
- KdV hierarchy at the Dispersive PDE Wiki.
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