1. Introduction: Dubrovin’s Integrability Scheme
We begin by recalling some very interesting works by B. Dubrovin and collaborators [
1,
2,
3], in which the following classification problem was posed:
Consider a general evolution equation:
with graded homogeneous polynomials of the jet-variables
with
where
for arbitrary
. Introduce now the set
of smooth functions
with fixed
, such that Equation (
1) reduces by means of the following transformation:
to the Riemann-type equation
where numbers
are finite and
is a formal parameter. The transformation (
2) is often called a quasi-Miura transformation and naturally acts as
an automorphism of the ring
of formal functional series with respect to the parameter
. It is worth mentioning that this ring is a topological ring
with respect to the natural metric, within which adding and multiplication of series is continuous. Moreover, the related group of Miura-type automorphisms, which is the semidirect product of the local diffeomorphism group
of the real axis
and the quasi-identical authomorphism subgroup of self-mappings,
with finite
, naturally generates the Lie subalgbra
of the natural derivations of the ring
, whose representatives coincide with the above Equation (
1).
Dubrovin formulated the following integrability criterion:
Definition 1. The evolution Equation (1) is defined to be formally integrable, iff the corresponding inverse to (2) transformationwith finite orders upon application to an arbitrary Riemann type symmetry flowwith respect to an evolution parameter reduces to the formwith uniform homogeneous differential polynomials of order In their works, B. Dubrovin and his collaborators applied this scheme to the equation
and presented a list of 9 (!) equations [
3]:
The first two equations are the KdV and mKdV, the third equation is equivalent via the Miura transformation
to the KdV Equation (
1). The last ones (4)–(9) are reduced by means of suitable reciprocity transformations
parametrized by a smooth function
, to the Equations (1)–(3) from this listing.
Keeping in mind this result, we have decided to reanalyze the integrability of the evolution Equation (
3), having rewritten it in the following generalized form:
where
are constants. As the right-hand side of the flow (
10) defines a vector field
on a suitably chosen smooth functional manifold
(being here locally diffeomorphic to the jet-manifold
, we checked the existence of a suitable infinite hierarchy of conservation laws for the flow (
10) and related Hamiltonian structures on
M via the gradient-holonomic integrability scheme [
4,
5].
In particular, it means that this hierarchy is suitably ordered and satisfies the well known Noether-Lax equation on where —the functional gradient of a functional on M, depending on the constant parameter as , and chosen to be a generating function of conservation laws to the vector field .
As a result of calculations, we obtained the following two cases:
The first case gives rise to Equation (
3) from the (
9), and the second one gives rise to the new evolution equation
which possesses infinite hierarchy of suitably ordered conservation laws:
where we put, for brevity,
.
The new evolution Equation (
12) can be represented in the following Hamiltonian form
where the Poisson operators
are given by the expressions
and
and are compatible on the functional manifold
M, that is for any
the operator
is Hamiltonian.
Proposition 1. The above result simply means that the dynamical systemis a new completely integrable bi-Hamiltonian system on the functional manifold Remark 1. Concerning the Dubrovin-Zhang Equation (3) from the listing (9), we have stated, as a by-product, that it is also a bi-Hamiltonian system and representable in the formwhere the compatible Poisson operators are given, respectively, by the expressionsandjointly with the Hamiltonian operators 2. Reduction Integrability Properties
Now, we proceed to the following reduction of the new Equation (
12) putting
as
:
called here KN-3/4 and
a priory integrable and possessing an infinite hierarchy of conservation laws, which can be easily written down from the hierarchy (
13) via the limiting procedure
Remark 2. Here, we would like to remark that Equation (22) above is very similar to the well known Krichever-Novikov (KN-3/2) equationwhich differs from (22) only by the coefficient instead of the rational number and was studied before by V. Sokolov [6] by means of the well known Mikhailov-Shabat recursion symmetry analysis technique and by G. Wilson [7], using differential-algebraic Galois group solvability reasonings. We have reanalyzed these Novikov-Krichever type Equations (
22) and (
23), having performed the following manipulation:
where
are now arbitrary coefficients, and checked the latter equation for the existence of an infinite hierarchy of suitably ordered conservation laws. The corresponding calculations immediately revealed that the coefficients
should satisfy two related algebraic relationships:
whose solutions are the following two cases:
The first case
gives rise to the new integrable bi-Hamiltonian system on the functional manifold
M in the form
where
is an arbitrary parameter, yet
For the case
when
Equation (
24) reduces to the modified integrable Krichever-Novikov type system
possessing an infinite hierarchy of conservation laws and giving rise to the well known Krichever-Novikov bi-Hamiltonian system (
23) at
:
Remark 3. We would like to remark here that originally Krichever and Novikov [8] and Sokolov [6,9] analyzed integrability of the generalized equationwhere is a third order polynomial with constant coefficients, and reducing to the KN-3/2 (29) upon the change of dependent variable and We have reanalyzed the Equation (30) within the gradient-holonomic integrability scheme [10] for the case of an arbitrary N-s order polynomial and proved that it conserves the integrability for the polynomial order , that slightly generalizes the former result presented in References [6,8,9]. The derived above modified Krichever-Novikov type Equation (
28) is also an integrable bi-Hamiltonian flow on the functional manifold
M for arbitrary
:
where the Poisson operator
forms a compatible pair to the operator
and the corresponding Hamiltonian function equals
Moreover, one can also easily check that the next slightly modified Krichever-Novikov-type equation
for arbitrary
is also an integrable bi-Hamiltonian flow, possesses an infinite hierarchy of functionally independent conservation laws, which can be generated recursively:
via the Magri gradient recursion scheme:
for arbitrary
using the above mentioned compatible Poisson
-
pair (
32) and (
33).
The same can be stated about the new integrable Krichever-Novikov type equation
which is also a bi-Hamiltonian flow with respect to a compatible pair of the Poisson operators
Remark 4. It appears interesting to observe that the generalized Krichever-Novikov type Equation (24)transforms via the change of variables to the following modified Korteweg de Vries-type equation:which is, obviously, also integrable for two cases (26), mentioned before:The case reduces to the well known integrable modified Korteweg de Vries equation Respectively, at or equivalently at the modified Korteweg-de Vries Equation (42) reduces to the classical modified Korteweg de Vries equationwhich, evidently, is also integrable and bi-Hamiltonian on the functional manifold The second case of Equation (41) also reduces to the classical integrable modified Korteweg-de Vries equation The special case , which is equivalent to the choice , corresponds at exactly to the strictly linear equation, whose exact integrability is trivial.
3. The Integrability of the Riemann-Type Hydrodynamical Systems via the Gradient-Holonomic Integrability Scheme
In this section, we will dwell on the integrability theory aspects of a new Riemann type hierarchy
where
are arbitrary natural numbers, in the frame of the gradient-holonomic integrability scheme, devised and applied jointly with Maxim Pavlov and collaborators. This hierarchy was proposed before in Reference [
11] as a nontrivial generalization of the infinite hierarchy of the Riemann type flows, suggested recently by M. Pavlov and D. Holm [
12,
13] in the form of dynamical systems on a
-periodic functional manifold
where the vector
, and the differentiations
satisfy the Lie-algebraic commutator relationship
and
is an evolution parameter.
The mentioned above dynamical systems
at
,
and
,
, respectively, were extensively studied by many researchers. They appeared to be related to nontrivial generalizations of the Camassa-Holm and Degasperis-Procesi systems. The case
and
is a generalization of the known Gurevich-Zybin dynamical system in cosmology, whose integrability was analyzed by M. Pavlov in Reference [
12] and later in the works [
4,
10,
14] within the gradient-holonomic scheme. There was shown that this system, namely:
is a smooth integrable bi-Hamiltonian flow on the
-periodic functional maniifold
. This flow has the following Lax type representation
where
is an arbitrary spectral parameter and
.
Dynamical system (
49) for the case
and
is equivalent to the following evolution flow on a
-periodic functional manifold
for a point
3.1. Poissonian Structure on
Let us rewrite the dynamical system (
52) in the following component-wise form
where
is the corresponding vector field on
, and construct the Poissonian structures on
. To do that, we need to obtain additional solutions to the basic Noether-Lax gradient equation on the functional manifold
Here, the matrix operator
is an endomorphism of the cotangent space
adjoint to the corresponding Fréchet derivative
at
with respect to the bi-linear form
. Equation (
55) in the component-wise form can be rewritten as
where
. The following system of linear differential relationships follows from (
56)
where
. Here, the next operator relation
holds for any
and the function
, for which
. The relationship (
58) follows from the commutator relationship
.
Let us now construct a differential ring
which is invariant with respect to two differentiations
,
and generated by a fixed functional variable
. These differentiations satisfy the Lie-algebraic commutator relationship (
48) together with the constraint (
50). Taking into account that, for any
, any additive set
is an ideal in the functional ring
, we can solve the first equation of the linear system (
57) above and next recursively solve the remaining two equations. We can obtain that the three vector elements
are solutions to the linear system (
57). The first two elements of (
59) lead to the trivial conservation laws
. For the
we obtain the Volterra asymmetric vector
, which give rise to the following co-symplectic expression:
Then, the Poisson operator
can be obtained as
and the Hamiltonian representation with respect to the Poisson operator (
61) is
where the Hamiltonian function
is given by the polynomial functional
The same way makes it possible to derive the second Poisson operator
on the manifold
, which is compatible with the Poisson operator
(
61):
3.2. Lax-Type Integrability Analysis
Next, we return to the Lax type integrability analysis of the dynamical system (
53) on the functional manifold
As the Poissonian operators (
64) and (
61) on the manifold
are compatible [
4,
10,
15,
16], then for arbitrary
the operator pencil
is also Poissonian, and then all operators of the form
for arbitrary
are Poissonian too on the functional manifold
. Now, it is easy to reconstruct the related infinite hierarchy of the mutually commuting conservation laws
for the dynamical system (
65), using the recursion property of the Poissonian pair (
61), (
64) and the homotopy formula [
4,
10,
17]. Here,
and the corresponding recursion operator
satisfies the associated Lax-type commutator relationship
Remark 5. It is evident that the trace-functionalswhere Tr is the usual Adler type trace operation on the algebra of periodic pseudo-differential operators are conservation laws for our dynamical system (65) for any . In particular, this property was put into the background of the Shabat-Mikhailov integrability classification scheme. The following result is based on the analysis and observations above.
Proposition 2. The Riemann type hydrodynamic system (65) on the functional manifold is a bi-Hamiltonian dynamical system with respect to the compatible Poissonian structures (64) and (61) and possesses an infinite hierarchy of mutually commuting conservation laws (67). 4. Differential-Algebraic Integrability Analysis:
Let us consider the previously introduced differential ring
, which is generated by a fixed functional variable
and is invariant with respect to differentiations
,
. These differentiations satisfy the Lie-algebraic commutator relationship (
48) together with the constraint (
50), which is expressed in the following differential-algebraic functional form
Now, we take into account the fact that the Lax-type representation for (
65) can be interpreted [
10,
11] as the existence of a finite-dimensional invariant ideal
realizing the finite-dimensional representation of relationship (
48). The ideal can be constructed as
where
,
,
. The finite-dimensional representations of the
- and
-differentiations can be constructed if we find [
11] the
-invariant kernel
and then check its invariance with respect to the
-differentiation. The kernel can be found as follows:
where
End
is given as
The
-differentiation representation in the space
can be constructed if we find a matrix
, which satisfies the linear relationship for
when the corresponding ideal
is
-invariant with respect to the matrix differentiation representation (
73). The matrix
can be found by straightforward calculations. The following proposition is stated.
Proposition 3. The generalized Riemann-type dynamical system (65) for is a bi-Hamiltonian integrable flow with a non-autonomous Lax-type representation for with the arbitrary spectral parameter . Equation (
76) depends explicitly on the temporal evolution parameter
, which is not usual. Nonetheless, the matrices (
72) and (
75) satisfy the Zakharov–Shabat compatibility condition
for all
. This follows from the linear Lax type relationships (
71), (
73) and the commutator condition (
48). We can assume that the dynamical system (
65) possesses a usual autonomous Lax type representation, taking into account that it has a compatible Poissonian pair (
61) and (
64), which depends only on the variables
. This representation can be found by means of a corresponding gauge transformation of the linear relationships (
71) and (
73).