Abstract:
In [1] and [2] I.M. Krichever and S.P. Novikov introduced a
remarkable class of exact solutions of soliton equations — solutions of
rank $l>1.$ In this article we study solutions of rank two of the
following system
$$
\tag{1}
V_{t}=\frac{1}{4}(6VV_x+6W_x+V_{xxx}),\ \ W_{t}=\frac{1}{2}(-3VW_x-W_{xxx}).
$$
This system is equivalent to the commutativity condition of the self-adjoint
operator $$L_{4}=(\partial_{x}^{2}+V(x,t))^{2}+W(x,t)$$ and the
skew-symmetric operator
$\partial_{t}-\partial_{x}^{3}-\frac{3}{2}V(x,t)\partial_{x}-\frac{3}{4}V_{x}(x,t).$
In this case “solutions of rank two” means that for every $t\in {\mathbb R}$ every operator commuting with $L_4$ has even order. It also means that the dimension of space of common eigenfunctions of commuting operators $L_{4}$ and
$L_{4g+2}$ is equal to two
$$
dim_{{\mathbb C}}\left\{\psi: L_{4}\psi=z\psi, L_{4g+2}\psi=w\psi\right\}=2
$$
for generic eigenvalues $(z,w).$
The set of eigenvalues $P=(z,w)$ forms hyperelliptic curve
$$w^2=F_g(z)=z^{2g+1}+c_{2g}z^{2g}+\ldots+c_{0}. $$ This curve is
called spectral.
There is a classification of commutative rings of ordinary
differential operators of arbitrary rank obtained by Krichever
[3] but in general case such operators are not found.
Krichever and Novikov [1] found operators of rank two corresponding to an elliptic spectral curve.
Mokhov found operators of rank three corresponding to an elliptic spectral curve.
In the case of spectral curves of genus $2,$$3$ and $4$ it is known only examples of operators of rank greater than one.
Operators $L_4$ and $L_{4g+2}$ of rank two corresponding to hyperelliptic
spectral curves were studied in [4]. Operators $L_4-z,$$L_{4g+2}-w$ have common right divisor
$L_2=\partial_{x}^{2}-\chi_1(x,P)\partial_{x}-\chi_0(x,P)$:
$$
L_4-z=\tilde{L}_2L_2,\ \ \ L_{4g+2}-w=\tilde{L}_{4g}L_2.
$$
Functions $\chi_0(x,P),$$\chi_1(x,P)$ are rational functions on $\Gamma,$
they satisfy the Krichever's equations. The operator $L_4$ is self-adjoint
if and only if $\chi_1(x,P)=\chi_1(x,\sigma(P))$[4]. If $g\geq
1$, then the following theorem holds [4].
$ $ Theorem 1[4].If$L_4$is self-adjoint operator, then $$\chi_{0}=-\frac{Q_{xx}}{2Q}+\frac{w}{Q}-V, \ \ \ \ \ \chi_{1}=\frac{Q_{x}}{Q},$$ where $Q=z^{g}+\alpha_{g-1}(x)z^{g-1}+\ldots+\alpha_{0}(x).$ Polynomial$Q$satisfies equation $$
\tag{4}
4F_{g}(z)=4(z-W)Q^{2}-4V(Q_{x})^{2}+(Q_{xx})^{2}-2Q_{x}Q_{xxx}
+2Q(2V_{x}Q_{x}+4VQ_{xx}+Q_{xxxx}).
$$
The main aim of this paper is as follows. We study dynamics of polynomial
$Q$ provided that $V$ and $W$ satisfy (1).
$ $ Theorem 2.Suppose that potentials$V$and$W$of operator$L_4=(\partial_{x}^{2}+V(x,t))^2+W(x,t)$commuting with operator$L_{4g+2}$satisfy the system (1). Then polynomial$Q$satisfies the following
equation$Q_{t}=\frac{1}{2}\left(-3VQ_{x}-Q_{xxx}\right).$ $ $ Remark 1.
Similarly one can obtain the evolution equation on $Q$ if in (2) one
substitutes operator $A$ by a skew-symmetric operator of order $2n+1.$ For
example, in case of $n=2, 3.$ $ $ The following theorems are proved in [4] and [6].
$ $ Theorem 3.The operator $$
L_4^\sharp=(\partial_x^2+\alpha_3 x^3+\alpha_2 x^2+\alpha_1
x+\alpha_0)^2+\alpha_3 g(g+1)x
$$ commutes with an operator$L_{4g+2}^\sharp$of order$4g+2.$The spectral curve is given by the equation$w^2=F_{2g+1}(z),$where$F_{2g+1}$is a polynomial of degree$2g+1$.
$ $ Theorem 4.The operator $$
L_4^\natural=(\partial_x^2+\alpha_1\cosh(x)+\alpha_0)^2+\alpha_1g(g+1)\cosh(x), \ \ \alpha_1\neq0
$$ commutes with an operator$L_{4g+2}^\natural$of order$4g+2.$The spectral curve is given by the equation$w^2=F_{2g+1} (z),$ \emph{where\emph{ $F_{2g+1}$is a polynomial of degree$2g+1$.
$ $ The following theorems were proved in collaboration with E.I. Shamaev.
$ $ Theorem 5.The operator$L_4^\sharp$does not commute with any differential operator
of odd order. $ $ Theorem 6.The operator$L_4^\natural$does not commute with any differential operator of odd order. $ $ Theorems 5 and 6 rigorously prove that $L_4^\natural$ from [4] and $L_4^\sharp$ from [6] are differential operators of rank two.
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