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KdV hierarchy - Wikipedia

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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]

Explicit equations for first three terms of hierarchy

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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.

Application to periodic solutions of KdV

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Cnoidal wave solution to the Korteweg–De Vries equation, in terms of the square of the Jacobi elliptic function cn (and with value of the parameter m = 0.9).

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).

  1. ^ 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. Bibcode:2006PhyD..213..231C. doi:10.1016/j.physd.2005.11.008.
  2. ^ 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. arXiv:solv-int/9704012. Bibcode:1997CMaPh.190..113B. doi:10.1007/s002200050235. S2CID 14271642.
  3. ^ Dunajski, Maciej (2010). Solitons, instantons, and twistors. Oxford: Oxford University Press. pp. 56–57. ISBN 9780198570639.
  4. ^ 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.{{cite book}}: CS1 maint: location missing publisher (link)