KdV hierarchy

In mathematics, the KdV hierarchy is an infinite sequence of partial differential equations which contains the Korteweg–de Vries equation.

Details

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

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

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).

See also

• Witten's conjecture
• Huygens' principle

References

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.

Sources

• Gesztesy, Fritz; Holden, Helge (2003), Soliton equations and their algebro-geometric solutions. Vol. I, Cambridge Studies in Advanced Mathematics, vol. 79, Cambridge University Press, ISBN 978-0-521-75307-4, MR 1992536

External links

• KdV hierarchy at the Dispersive PDE Wiki.