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Korteweg-de Vries hierarchy (KdV hierarchy)

Korteweg-de Vries hierarchy
SolvableExactly solvable
The Cauchy problemSolvable by inverse scattering transform
First introducedLax, 1968

Definition

The Korteweg-de Vries equation is the first equation of the so-called Korteweg-de Vries hierarchy defined as

${u}_{t}+\frac{\partial }{\partial x}{L}_{n+1}\left[u\right]=0,$

where $L$ is Lenard operator defined by

$\frac{\partial }{\partial x}{L}_{n+1}\left[u\right]=\left(\frac{{\partial }^{3}}{\partial {x}^{3}}+4u\frac{\partial }{\partial x}+2\frac{\partial u}{\partial x}\right){L}_{n}\left[u\right],\text{ }{L}_{0}\left[u\right]=\frac{1}{2}.$

There are four first operators

${L}_{1}\left[u\right]=u,$

${L}_{2}\left[u\right]={u}_{xx}+3{u}^{2},$

${L}_{3}\left[u\right]={u}_{xxxx}+10u{u}_{xx}+5{u}_{x}^{2}+10{u}^{3},$

${L}_{4}\left[u\right]={u}_{xxxxxx}+14u{u}_{xxxx}+28{u}_{x}{u}_{xxx}+21{u}_{xx}^{2}+70{u}^{2}{u}_{xx}+70u{u}_{x}^{2}+35{u}^{4}.$

Corresponding equations of the Korteweg-de Vries hierarchy are

${u}_{t}+{u}_{x}=0,$

${u}_{t}+6u{u}_{x}+{u}_{xxx}=0,$

${u}_{t}+10u{u}_{xxx}+30{u}^{2}{u}_{x}+20{u}_{x}{u}_{xx}+{u}_{5x}=0,$

$\begin{array}{c}{u}_{t}+14u{u}_{5x}+70{u}^{2}{u}_{xxx}+42{u}_{x}{u}_{4x}+\\ +70{u}_{xx}{u}_{xxx}+280u{u}_{x}{u}_{xx}+70{u}_{x}^{3}+140{u}^{3}{u}_{x}+{u}_{7x}=0.\end{array}$

All equations of the Korteweg-de Vries hierarchy pass the Painlevé test, possess soliton solutions and are exactly solvable. Also all equations has odd degrees that corresponds to the lack of dissipative terms.

Derivation

In 1968 P.D. Lax solved problem of finding all partial differential equations for which the Cauchy problem can be solved by the inverse scattering transform with Schrödinger steady-state equation. He suggested to look for equation class representable as a system of two linear equations

${\psi }_{xx}+\left(\lambda +u\right)\psi =0$

${\psi }_{t}=B\left(u,{u}_{x},\dots ,\lambda \right)\psi -2E\left(u,{u}_{x},\dots ,\lambda \right){\psi }_{x},$

where $B$ and $E$ are functions of $\lambda$, $u$ and derivatives form $u$ by $x$ which are to be found.

Lax pair

Lax pair for the nth equation of the Korteweg-de Vries hierarchy can be written as the following system of linear partial differential equations

${\psi }_{xx}+\left(\lambda +u\right)\psi =0,$

${\psi }_{t}=\left(c\left(t\right)+\sum _{k=0}^{n}{\left(-4\lambda \right)}^{n-k}{L}_{x}^{k}\left[u\right]\right)\psi -2\left(\sum _{k=0}^{n}{\left(-4\lambda \right)}^{n-k}{L}^{k}\left[u\right]\right){\psi }_{x}.$

Korteweg-de Vries equation

Modified Korteweg-de Vries hierarchy

References

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3. Joshi N. The second Painlevé hierarchy and the stationary KdV hierarchy // Publ. Res. Inst. Math. Sci., 2004. 40:3. Pp.1039-1061. DOI:10.2977/prims/1145475502
4. Kudryashov N.A. Analytical theory of nonlinear differential equations (in Russian) // 2nd ed., Institute of Computer Investigation, Moscow-Izhevsk, 2004. — 360 p.
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7. Lax P.D. Integrals of nonlinear equations of evolution and solitary waves // Comm. Pure Appl. Math., 1968. 21:5. Pp.467-490. DOI:10.1002/cpa.3160210503
8. Lax P.D. Weak solutions of nonlinear hyperbolic equations and their numerical computation // Commun. Pure Appl. Math., 1954. 7:1. Pp.159–193. DOI:10.1002/cpa.3160070112
9. Olver P.J. Hamiltonian and non-Hamiltonian models for water waves // Lecture Notes in Physics, 1984. 195. Pp.273-290. DOI:10.1007/3-540-12916-2_62