Compatible Hamiltonian Operators for the Krichever-Novikov Equation

Sylvain Carpentier

arXiv:1705.04834·math.AP·May 16, 2017

Compatible Hamiltonian Operators for the Krichever-Novikov Equation

Sylvain Carpentier

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TL;DR

This paper demonstrates that the Krichever-Novikov equation's hierarchy is Hamiltonian with a set of compatible operators, including a non-local Hamiltonian operator and two weakly non-local recursion operators, expanding understanding of its integrable structure.

Contribution

It introduces new compatible Hamiltonian operators for the Krichever-Novikov hierarchy, including compositions of existing operators, enhancing the framework for its integrability analysis.

Findings

01

H_0, L_4H_0, and L_6H_0 are compatible Hamiltonian operators.

02

The hierarchy remains Hamiltonian under these operators.

03

The operators extend the known Hamiltonian structure of the equation.

Abstract

It has been proved by V. Sokolov that the Krichever-Novikov equation's hierarchy is hamiltonian for the non-local Hamiltonian operator H_0=u_x D^{-1} u_x and possesses twi weakly non-local recursion operatos of degree 4 and 6, L_4 and L_6. We show here that H_0, L_4H_0 and L_6H_0 are compatible Hamiltonian operators for which the Krichever-Novikov equation's hierarchy is hamiltonian.

Equations36

\frac{d u}{d t} = u_{3} - \frac{3}{2} \frac{u _{2}^{2}}{u _{1}} + \frac{P ( u )}{u _{1}},

\frac{d u}{d t} = u_{3} - \frac{3}{2} \frac{u _{2}^{2}}{u _{1}} + \frac{P ( u )}{u _{1}},

\frac{d u}{d t _{i}} = G_{i} \in V, i \geq 0,

\frac{d u}{d t _{i}} = G_{i} \in V, i \geq 0,

{F, G} := X_{F} (G) - X_{G} (F) = 0,

{F, G} := X_{F} (G) - X_{G} (F) = 0,

X_{F} = n \geq 0 \sum F^{(n)} \frac{\partial}{\partial u _{n}} .

X_{F} = n \geq 0 \sum F^{(n)} \frac{\partial}{\partial u _{n}} .

G_{0} = G_{1} = G_{2} = u_{1}, u_{3} - \frac{3}{2} \frac{u _{2}^{2}}{u _{1}} + \frac{P ( u )}{u _{1}}, u^{(5)} - 5 \frac{u _{4} u _{2}}{u _{1}} - \frac{5}{2} \frac{u _{3}^{2}}{u _{1}} + \frac{25}{2} \frac{u _{3} u _{2}^{2}}{u _{1}^{2}} - \frac{45}{8} \frac{u _{2}^{4}}{u _{1}^{3}} - \frac{5}{3} P \frac{u _{3}}{u _{1}^{2}} + \frac{25}{6} P \frac{u _{2}^{2}}{u _{1}^{3}} - \frac{5}{3} P_{u} \frac{u _{2}}{u _{1}} - \frac{5}{18} \frac{P ^{2}}{u _{1}^{3}} + \frac{5}{9} u_{1} P_{uu},

G_{0} = G_{1} = G_{2} = u_{1}, u_{3} - \frac{3}{2} \frac{u _{2}^{2}}{u _{1}} + \frac{P ( u )}{u _{1}}, u^{(5)} - 5 \frac{u _{4} u _{2}}{u _{1}} - \frac{5}{2} \frac{u _{3}^{2}}{u _{1}} + \frac{25}{2} \frac{u _{3} u _{2}^{2}}{u _{1}^{2}} - \frac{45}{8} \frac{u _{2}^{4}}{u _{1}^{3}} - \frac{5}{3} P \frac{u _{3}}{u _{1}^{2}} + \frac{25}{6} P \frac{u _{2}^{2}}{u _{1}^{3}} - \frac{5}{3} P_{u} \frac{u _{2}}{u _{1}} - \frac{5}{18} \frac{P ^{2}}{u _{1}^{3}} + \frac{5}{9} u_{1} P_{uu},

G_{3} = u_{7} - 7 \frac{u _{2} u _{6}}{u _{1}} - \frac{7}{6} \frac{u _{5}}{u _{1}^{2}} (2 P + 12 u_{3} u_{1} - 27 u_{2}^{2}) - \frac{21}{2} \frac{u _{4}^{2}}{u _{1}} + \frac{21}{2} \frac{u _{4}}{u _{1}^{3}} (2 P - 11 u_{2}^{2}) - \frac{7}{3} \frac{u _{4}}{u _{1}^{2}} (2 P_{u} u_{1} - 51 u_{2} u_{3}) + \frac{49}{2} \frac{u _{3}^{3}}{u _{1}^{2}} + \frac{7}{12} \frac{u _{3}^{2}}{u _{1}^{3}} (22 P - 417 u_{2}^{2}) + \frac{2499}{8} \frac{u _{2}^{4}}{u _{1}^{4}} u_{3} \frac{91}{3} P_{u} \frac{u _{2}}{u _{1}^{2}} u_{3} - \frac{595}{6} P \frac{u _{2}^{2}}{u _{1}^{4}} u_{3} - \frac{35}{18} \frac{u _{3}}{u _{1}^{4}} (2 P_{uu} u_{1}^{4} - P^{2}) - \frac{1575}{16} \frac{u _{2}^{6}}{u _{1}^{5}} + \frac{1813}{24} \frac{u _{2}^{4}}{u _{1}^{5}} P - \frac{203}{6} \frac{u _{2}^{3}}{u _{1}^{3}} P_{u} + \frac{49}{36} \frac{u _{2}^{2}}{u _{1}^{5}} (6 P_{uu} u_{1}^{4} - 5 P^{2}) - \frac{7}{9} \frac{u _{2}}{u _{1}^{3}} (2 P_{uuu} u_{1}^{4} - 5 P P_{u}) + \frac{7}{54} \frac{P ^{3}}{u _{1}^{5}} - \frac{7}{9} P_{uu} \frac{P}{u _{1}} + \frac{7}{9} P_{uuuu} u_{1}^{3} - \frac{7}{18} \frac{P _{u}^{2}}{u _{1}} .

G_{3} = u_{7} - 7 \frac{u _{2} u _{6}}{u _{1}} - \frac{7}{6} \frac{u _{5}}{u _{1}^{2}} (2 P + 12 u_{3} u_{1} - 27 u_{2}^{2}) - \frac{21}{2} \frac{u _{4}^{2}}{u _{1}} + \frac{21}{2} \frac{u _{4}}{u _{1}^{3}} (2 P - 11 u_{2}^{2}) - \frac{7}{3} \frac{u _{4}}{u _{1}^{2}} (2 P_{u} u_{1} - 51 u_{2} u_{3}) + \frac{49}{2} \frac{u _{3}^{3}}{u _{1}^{2}} + \frac{7}{12} \frac{u _{3}^{2}}{u _{1}^{3}} (22 P - 417 u_{2}^{2}) + \frac{2499}{8} \frac{u _{2}^{4}}{u _{1}^{4}} u_{3} \frac{91}{3} P_{u} \frac{u _{2}}{u _{1}^{2}} u_{3} - \frac{595}{6} P \frac{u _{2}^{2}}{u _{1}^{4}} u_{3} - \frac{35}{18} \frac{u _{3}}{u _{1}^{4}} (2 P_{uu} u_{1}^{4} - P^{2}) - \frac{1575}{16} \frac{u _{2}^{6}}{u _{1}^{5}} + \frac{1813}{24} \frac{u _{2}^{4}}{u _{1}^{5}} P - \frac{203}{6} \frac{u _{2}^{3}}{u _{1}^{3}} P_{u} + \frac{49}{36} \frac{u _{2}^{2}}{u _{1}^{5}} (6 P_{uu} u_{1}^{4} - 5 P^{2}) - \frac{7}{9} \frac{u _{2}}{u _{1}^{3}} (2 P_{uuu} u_{1}^{4} - 5 P P_{u}) + \frac{7}{54} \frac{P ^{3}}{u _{1}^{5}} - \frac{7}{9} P_{uu} \frac{P}{u _{1}} + \frac{7}{9} P_{uuuu} u_{1}^{3} - \frac{7}{18} \frac{P _{u}^{2}}{u _{1}} .

X_{F} (L) = [D_{F}, L]

X_{F} (L) = [D_{F}, L]

D_{F} = n \sum \frac{\partial F}{\partial u _{n}} \partial^{n} \in V [\partial] .

D_{F} = n \sum \frac{\partial F}{\partial u _{n}} \partial^{n} \in V [\partial] .

L_{4} = H_{1} H_{0}^{- 1}, L_{6} = H_{2} H_{0}^{- 1},

L_{4} = H_{1} H_{0}^{- 1}, L_{6} = H_{2} H_{0}^{- 1},

H_{0} = H_{1} = H_{2} = u_{1} \partial^{- 1} u_{1}, \frac{1}{2} (u_{1}^{2} \partial^{3} + \partial^{3} u_{1}^{2}) + (2 u_{3} u_{1} - \frac{9}{2} u_{2}^{2} - \frac{2}{3} P) \partial + \partial (2 u_{3} u_{1} - \frac{9}{2} u_{2}^{2} - \frac{2}{3} P) + G_{1} \partial^{- 1} G_{1} + u_{1} \partial^{- 1} G_{2} + G_{2} \partial^{- 1} u_{1}, \frac{1}{2} (u_{1}^{2} \partial^{5} + \partial^{5} u_{1}^{2}) + (3 u_{3} u_{1} - \frac{19}{2} u_{2}^{2} - P) \partial^{3} + \partial^{3} (3 u_{3} u_{1} - \frac{19}{2} u_{2}^{2} - P) + (u_{5} u_{1} - 9 u_{3} u_{2} + \frac{19}{2} u_{3}^{2} - \frac{2}{3} \frac{u _{3}}{u _{1}} (5 P - 39 u_{2}^{2}) + \frac{u _{2}^{2}}{u _{1}^{2}} (5 P - 9 u_{2}^{2}) + \frac{2}{3} \frac{P ^{2}}{u _{1}^{2}} + u_{1}^{2} P_{uu}) \partial + \partial (u_{5} u_{1} - 9 u_{3} u_{2} + \frac{19}{2} u_{3}^{2} - \frac{2}{3} \frac{u _{3}}{u _{1}} (5 P - 39 u_{2}^{2}) + \frac{u _{2}^{2}}{u _{1}^{2}} (5 P - 9 u_{2}^{2}) + \frac{2}{3} \frac{P ^{2}}{u _{1}^{2}} + u_{1}^{2} P_{uu}) + G_{1} \partial^{- 1} G_{2} + G_{2} \partial^{- 1} G_{1} + u_{1} \partial^{- 1} G_{3} + G_{3} \partial^{- 1} u_{1} .

H_{0} = H_{1} = H_{2} = u_{1} \partial^{- 1} u_{1}, \frac{1}{2} (u_{1}^{2} \partial^{3} + \partial^{3} u_{1}^{2}) + (2 u_{3} u_{1} - \frac{9}{2} u_{2}^{2} - \frac{2}{3} P) \partial + \partial (2 u_{3} u_{1} - \frac{9}{2} u_{2}^{2} - \frac{2}{3} P) + G_{1} \partial^{- 1} G_{1} + u_{1} \partial^{- 1} G_{2} + G_{2} \partial^{- 1} u_{1}, \frac{1}{2} (u_{1}^{2} \partial^{5} + \partial^{5} u_{1}^{2}) + (3 u_{3} u_{1} - \frac{19}{2} u_{2}^{2} - P) \partial^{3} + \partial^{3} (3 u_{3} u_{1} - \frac{19}{2} u_{2}^{2} - P) + (u_{5} u_{1} - 9 u_{3} u_{2} + \frac{19}{2} u_{3}^{2} - \frac{2}{3} \frac{u _{3}}{u _{1}} (5 P - 39 u_{2}^{2}) + \frac{u _{2}^{2}}{u _{1}^{2}} (5 P - 9 u_{2}^{2}) + \frac{2}{3} \frac{P ^{2}}{u _{1}^{2}} + u_{1}^{2} P_{uu}) \partial + \partial (u_{5} u_{1} - 9 u_{3} u_{2} + \frac{19}{2} u_{3}^{2} - \frac{2}{3} \frac{u _{3}}{u _{1}} (5 P - 39 u_{2}^{2}) + \frac{u _{2}^{2}}{u _{1}^{2}} (5 P - 9 u_{2}^{2}) + \frac{2}{3} \frac{P ^{2}}{u _{1}^{2}} + u_{1}^{2} P_{uu}) + G_{1} \partial^{- 1} G_{2} + G_{2} \partial^{- 1} G_{1} + u_{1} \partial^{- 1} G_{3} + G_{3} \partial^{- 1} u_{1} .

E (\partial) \in V [\partial] + i \sum p_{i} \partial^{- 1} \frac{δ ρ _{i}}{δ u},

E (\partial) \in V [\partial] + i \sum p_{i} \partial^{- 1} \frac{δ ρ _{i}}{δ u},

\frac{δ F}{δ u} = D_{F}^{*} (1) = n \sum (- \partial)^{n} (\frac{\partial F}{\partial u _{n}}) .

\frac{δ F}{δ u} = D_{F}^{*} (1) = n \sum (- \partial)^{n} (\frac{\partial F}{\partial u _{n}}) .

G_{i} = H_{0} (\frac{δ ϕ _{i}}{δ u}) for all i \geq 0.

G_{i} = H_{0} (\frac{δ ϕ _{i}}{δ u}) for all i \geq 0.

A^{*} (D_{B (F)} (A (G)) + D_{A (G)}^{*} (B (F)) - D_{B (G)} (A (F)) + D_{B (G)}^{*} (A (F))) = B^{*} (D_{A (G)} (A (F)) - D_{A (F)} (A (G)))

A^{*} (D_{B (F)} (A (G)) + D_{A (G)}^{*} (B (F)) - D_{B (G)} (A (F)) + D_{B (G)}^{*} (A (F))) = B^{*} (D_{A (G)} (A (F)) - D_{A (F)} (A (G)))

X_{E} (L) = D_{E} L + L D_{E}^{*},

X_{E} (L) = D_{E} L + L D_{E}^{*},

(1.15) ⟺ X_{E} (A) - D_{E} A = A B^{- 1} (X_{E} (B) + D_{E}^{*} B) ⟺ X_{E} (A) - D_{E} A = - B^{*}^{- 1} A^{*} (X_{E} (B) + D_{E}^{*} B) ⟺ A^{*} (X_{E} (B) + D_{E}^{*} B) = B^{*} (D_{E} A - X_{E} (A)) ⟺ A^{*} (X_{E} + D_{E}^{*}) B = B^{*} (D_{E} - X_{E}) A . ⟺ A^{*} (D_{B (F)} (E) + D_{E}^{*} (B (F))) = B^{*} (D_{E} (A (F)) - D_{A (F)} (E)) \forall F \in V

(1.15) ⟺ X_{E} (A) - D_{E} A = A B^{- 1} (X_{E} (B) + D_{E}^{*} B) ⟺ X_{E} (A) - D_{E} A = - B^{*}^{- 1} A^{*} (X_{E} (B) + D_{E}^{*} B) ⟺ A^{*} (X_{E} (B) + D_{E}^{*} B) = B^{*} (D_{E} A - X_{E} (A)) ⟺ A^{*} (X_{E} + D_{E}^{*}) B = B^{*} (D_{E} - X_{E}) A . ⟺ A^{*} (D_{B (F)} (E) + D_{E}^{*} (B (F))) = B^{*} (D_{E} (A (F)) - D_{A (F)} (E)) \forall F \in V

A^{*} (X_{A (G)} (B) (F) - X_{A (F)} (B) (G) + (D_{A})_{G}^{*} (B (F)) + (D_{B})_{G}^{*} (A (F))) = B^{*} (X_{A (F)} (A) (G) - X_{A (G)} (A) (F)),

A^{*} (X_{A (G)} (B) (F) - X_{A (F)} (B) (G) + (D_{A})_{G}^{*} (B (F)) + (D_{B})_{G}^{*} (A (F))) = B^{*} (X_{A (F)} (A) (G) - X_{A (G)} (A) (F)),

(D_{P})_{F} (G) = X_{G} (P) (F) \forall G \in V .

(D_{P})_{F} (G) = X_{G} (P) (F) \forall G \in V .

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Full text

Compatible Hamiltonian Operators for the Krichever-Novikov Equation

Sylvain Carpentier*

It has been proved by Sokolov that Krichever-Novikov equation’s hierarchy is hamiltonian for the Hamiltonian operator $H_{0}=u_{x}\partial^{-1}u_{x}$ and possesses two weakly non-local recursion operators of degree $4$ and $6$ , $L_{4}$ and $L_{6}$ . We show here that $H_{0}$ , $L_{4}H_{0}$ and $L_{6}H_{0}$ are compatible Hamiltonians operators for which the Krichever-Novikov equation’s hierarchy is hamiltonian.

††footnotetext: * Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

In the study of finite gap solutions of KP, an integrable $(1+1)$ -dimensional PDE was discovered, the Krichever-Novikov equation. One of its forms (equivalent to the original one in [KN80]) is

[TABLE]

where $u=u(t,x)$ , $u_{n}=(\frac{d}{dx})^{n}(u)$ , and $P$ is a polynomial of degree at most $4$ . Let ${\mathcal{V}}=\mathbb{C}[u,u_{1}^{\pm},u_{2},...]$ and ${\mathcal{K}}$ be the fraction field of ${\mathcal{V}}$ . Let us denote $\frac{d}{dx}$ by $\partial$ . The differential order $d_{F}$ of a function $F\in{\mathcal{V}}$ is the highest integer $n$ such that $\frac{\partial F}{\partial u_{n}}\neq 0$ .

One of the attributes of equation $(1)$ is to be part of an infinite hierarchy of compatible evolution PDEs of odd differential orders

[TABLE]

where $G_{i}$ has differential order $(2i+1)$ . One says that $F,G\in{\mathcal{V}}$ are compatible, or symmetries of one another, if

[TABLE]

where $X_{F}$ denotes the derivation of ${\mathcal{V}}$ induced by the evolution equation $u_{t}=F$ , that is

[TABLE]

$(3)$ endows ${\mathcal{V}}$ with a Lie algebra bracket, and the $G_{i}$ ’s span an infinite-dimensional abelian subalgebra of $({\mathcal{V}},\{.,.\})$ , which we will denote by ${\mathcal{S}}$ . The first four equations in the hierarchy are:

[TABLE]

It is known ([IS80], [MS08]) that all integrable hierarchies admit a pseudodifferential operators $L\in{\mathcal{V}}((\partial^{-1}))$ satisfying

[TABLE]

for all $F$ in the hierarchy, where $D_{F}$ denotes the Fréchet derivative of $F$ :

[TABLE]

A pseudodifferential operator satisfying $(7)$ is called a recursion operator (for $F$ ). In [DS08] two rational recursions operators for $(1)$ were found, of order $4$ and $6$ :

[TABLE]

where

[TABLE]

Moreover, $L_{4}$ and $L_{6}$ are both weakly non-local, i.e. of the form

[TABLE]

where the $\rho_{i}$ ’s are conserved densities of $(1)$ . Recall that the variational derivative $\frac{\delta}{\delta u}$ is defined as follows:

[TABLE]

In [S84], Sokolov showed that the space of symmetries of $(1)$ , ${\mathcal{S}}$ , is preserved by $L_{4}$ . The same argument applies to $L_{6}$ , which was found later. He also establishes that the hierarchy of the Krichever-Novikov equation is hamiltonian for $H_{0}$ : there exists a sequence $\phi_{i}\in{\mathcal{V}}$ such that

[TABLE]

A Hamiltonian operator $H=AB^{-1}\in{\mathcal{V}}(\partial)$ with $A$ and $B$ right coprime is a skewadjoint rational differential operator inducing a non-local poisson lambda bracket, which is equivalent to the following identity (see equation $(6.13)$ in [DSK13])

[TABLE]

for all $F,G\in{\mathcal{V}}$ .

Lemma 1.

Let $L\in{\mathcal{V}}(\partial)$ be a skewadjoint rational operator. If there exists an infinite-dimensional (over $\mathbb{C}$ ) subspace ${\mathcal{W}}\subset{\mathcal{V}}$ such that $B(W)\subset\frac{\delta}{\delta u}{\mathcal{V}}$ and such that for all $G\in{\mathcal{W}}$ , $E=A(G)$ satisfies

[TABLE]

then $L$ is a Hamiltonian operator. Conversely, if $L$ is a hamiltonian operator and $G\in{\mathcal{V}}$ , then $D_{B(G)}=D_{B(G)}^{*}$ if and only if $A(G)$ satisfies equation $(15)$

Proof.

Let us first give an equivalent form of $(15)$ involving only differential operators.

[TABLE]

Comparing the last line of $(16)$ with $(14)$ , it is clear that if $H$ is Hamiltonian, then $E=A(G)$ satisfies equation $(15)$ is and only if $D_{B(G)}$ is self-adjoint. It is also clear that if $A(G)$ satisfies $(15)$ and $D_{B(G)}$ is self-adjoint, then $(F,G)$ satisfies $(14)$ for any $F\in{\mathcal{V}}$ . Therefore, if we consider ${\mathcal{W}}\subset{\mathcal{V}}$ infinite-dimensional subspace of ${\mathcal{V}}$ such that $A({\mathcal{W}})$ satisfies $(15)$ and $B({\mathcal{W}})\subset\frac{\delta}{\delta u}{\mathcal{V}}$ , we deduce that $(14)$ is satisfied for any $(F,G)\in{\mathcal{V}}\times{\mathcal{W}}$ . To conclude, we note that $(14)$ can be rewritten as an identity of bidifferential operator, i.e. it amounts to say that some expression of the form $\sum m_{ij}F^{(i)}G^{(j)}$ , where $m_{ij}\in{\mathcal{V}}$ is trivial, i.e. $m_{ij}=0$ for all $i,j$ . Namely, $(14)$ is equivalent to

[TABLE]

where given a differential operator $P$ , an element $F\in{\mathcal{V}}$ , the differential operator $(D_{P})_{F}$ is defined by

[TABLE]

If a bidifferential operator vanishes on ${\mathcal{V}}\times{\mathcal{W}}$ , it must be identically [math], since ${\mathcal{W}}$ is infinite dimensional. Hence, $L$ is an Hamiltonian operator. ∎

Lemma 2.

Let $L=CD^{-1}$ be a rational operator and $(F_{n})_{n\geq 0}$ a sequence spanning an infinite-dimensional subspace of ${\mathcal{K}}$ satisfying $C(F_{n})=D(F_{n+1})\in{\mathcal{V}}$ for all $n\geq 0$ . Assume that $L$ is recursion for all the $D(F_{n})$ ’s and that the $D(F_{n})$ ’s are hamiltonian for some Hamiltonian operator $H\in{\mathcal{V}}(\partial)$ . Then, provided $LH$ is skew-adjoint, $LH$ is a Hamiltonian operator for which all the $D(F_{n})$ ’s are hamiltonian ( $n\geq 1$ ).

Proof.

By Lemma $1$ , $H$ satisfies equation $(15)$ for all $D(F_{n}),n\geq 0$ , hence so does $LH$ ( $L$ is recursion for $D(F_{n})$ for all $n\geq 0$ ). To conclude using Lemma $1$ , one needs to check that $D(F_{n})=LH(\frac{\delta\rho_{n}}{\delta u})$ for some $\rho_{n}\in{\mathcal{V}}$ for all $n\geq 1$ . Let $P,Q\in{\mathcal{V}}[\partial]$ be right coprime differential operators such that $LH=PQ^{-1}$ . Let $A,B$ be right coprime differential operators such that $H=AB^{-1}$ . $D(F_{n})$ is hamiltonian for $H$ for all $n\geq 0$ , meaning that there exist two sequences in ${\mathcal{V}}$ , $(\phi_{n})_{n\geq 0}$ and $(\rho_{n})_{n\geq 0}$ , such that $\frac{\delta\rho_{n}}{\delta u}=B(\phi_{n})$ and $D(F_{n})=A(\phi_{n})$ for all $n\geq 0$ . In the language of [CDSK15], $\frac{\delta\rho_{n}}{\delta u}$ and $C(F_{n})$ are $CD^{-1}AB^{-1}$ associated, hence (quote result) there exists $\psi_{n}$ such that $C(F_{n})=P(\psi_{n})$ and $Q(\psi_{n})=\frac{\delta\rho_{n}}{\delta u}$ for all $n\geq 0$ . Therefore, by Lemma $1.1$ , $LH$ is a Hamiltonian operator for which $(C(F_{n}))_{n\geq 0}$ are hamiltonian. ∎

Theorem .

$H_{0}$ , $H_{1}$ and $H_{2}$ are compatible Hamiltonian operators.

Proof.

Let $\alpha,\beta,\gamma\in\mathbb{C}$ and let $L_{\alpha,\beta,\gamma}=(\alpha H_{0}+\beta H_{1}+\gamma H_{2})H_{0}^{-1}$ . $L_{\alpha,\beta,\gamma}$ is a recursion operator for the whole Krichever-Novikov hierarchy ${\mathcal{S}}$ . Moreover, it maps ${\mathcal{S}}$ to itself as was proved in [S84], meaning that if $L_{\alpha,\beta,\gamma}=AB^{-1}$ with $A,B$ right coprime and $G\in{\mathcal{S}}$ , then $G=B(F)$ for some $F\in{\mathcal{K}}$ and $A(F)\in{\mathcal{S}}$ . The theorem follows from Lemma $2$ . ∎

Remark 3.

It follows from Lemma $1$ that $H=H_{2}H_{1}^{-1}H_{0}$ is a Hamiltonian operator of degree $1$ . However, it is not weakly non-local. More generally all the $(H_{2}H_{1}^{-1})^{n}H_{0}$ , for $n\in\mathbb{Z}$ are pairwise compatible Hamiltonian operators.

Remark 4.

Every Hamiltonian operator $K=AB^{-1}$ over ${\mathcal{V}}$ , where $A$ and $B$ are right coprime induces a Lie algebra bracket on the space of functionals ${\mathcal{F}}(K):=\{\int f\in{\mathcal{V}}/\partial{\mathcal{V}}|\frac{\delta f}{\delta u}\in ImB\}$ , (well-)defined by $\{\int f,\int g\}=\int\frac{\delta f}{\delta u}AB^{-1}(\frac{\delta g}{\delta u})$ (see section $7.2$ in [DSK13]). Note that ${\mathcal{F}}(H_{0})={\mathcal{V}}/\partial{\mathcal{V}}$ but that ${\mathcal{F}}(H_{1})$ and ${\mathcal{F}}(H_{2})$ consist only of the conserved densities of the Krichever-Novikov equation.

We recall that if a rational differential operator $L=AB^{-1}$ , with $A,B\in{\mathcal{V}}[\partial]$ right coprime generates an infinite dimensional abelian subspace of $({\mathcal{V}},\{.,.\})$ , in the sense that there exist $(F_{n})_{n\geq 0}\in{\mathcal{K}}$ such that $A(F_{n})=B(F_{n+1})$ for all $n\geq 0$ and such that the $B(F_{n})$ ’s span an infinite-dimensional abelian subspace of $({\mathcal{V}},\{.,.\})$ , then for all $\lambda\in\mathbb{C}$ , $Im(A+\lambda B)$ must be a sub Lie algebra of $({\mathcal{V}},\{.,.\})$ (see [C17]). The recursion operators $L_{\alpha,\beta,\gamma}$ satisfy this condition.

The author thanks Vladimir Sokolov for useful discussions, and Victor Kac for his interest in this work.

Bibliography8

The reference list from the paper itself. Each links out to its DOI / PubMed record.

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Lemma 1**.**

Proof.

Lemma 2**.**

Proof.

**Theorem **.

Proof.

Remark 3**.**

Remark 4**.**

Lemma 1.

Lemma 2.

Theorem .

Remark 3.

Remark 4.