1. Introduction
The R-matrix approach has two important applications. One is to systematically construct consistent Lax pairs
generate dispersionless integrable systems; the other is to systematically construct an infinite hierarchy of commuting symmetries for a given dispersionless system. First, we recall some basic facts on R-matrix formalism [
1,
2].
Let g be a Lie algebra (in general, infinite-dimensional). The Lie bracket defines the adjoint action of g on g:
Definition 1 ([
3])
. An R-structure is a Lie algebra g equipped with a linear map R: (called the R-matrix) such that the bracketis another Lie product on g. The skew symmetry of (1) is obvious. Lemma 1 ([
3])
. A sufficient condition for R to be an R-matrix iswhere α is some real number. Equation (2) is called the Yang–Baxter equation. It can be verified that a sufficient condition for the Jacobi identity to hold is the Yang–Baxter equation for R. How do we find such an R? Assume that the Lie algebra g can be split into a direct sum of Lie subalgebras and , that is,Denoting the projections onto these subalgebras by , it is easy to verify thatsolves Equation (2) when Hence, it defines an R-structure on g. Let
We consider the associated hierarchies of flows (Lax hierarchies):
Suppose that R commutes with all derivatives
, that is,
and obeys the classical modified Yang–Baxter equation in (2). One can verify that the following conditions are equivalent:
Lemma 2 ([
4])
. (i) The zero-curvature equations(ii) All commute in g: Consider an
and its associated Lax hierarchy, which extends the systems by adding an extra independent variable:
Suppose that
are such that Lemma 2 holds for all
and the R-matrix satisfies (4). Then, the flows in (7) commute. Via the so-called Lax Novikov equation
and noting
Equations (3), (5), and (7) take the following forms:
The usual approach for constructing a commutative subalgebra spanned by , whose existence ensures commutativity of the flows in (3) and (7), is as follows:
The commutative subalgebra is generated by rational powers of a given element
when the Lie algebra in question is a Poisson algebra that obeys the Leibniz rule:
However, in the (3+1)-dimensional setting, this construction does not work anymore when the Leibniz rule is no long required to hold. It is the case of (3+1)-dimensional dispersionless systems when the Lie algebra is a Jacobi algebra. Hence, instead of explicit construction of a commuting
, as in [
4], the zero-curvature constraints in (5) are imposed on chosen elements
For the (3+1)-dimensional case, we consider a commutative and associative algebra
A of formal series in
p:
with ordinary dot multiplication:
The coefficients
of these series are assumed to be smooth functions of
and time
t. The Jacobi structure on
A is induced by the contact bracket:
where
We call the algebra
the Jacobi algebra.
The flow in (8) can be bearded as the compatibility condition of the Lax pair:
Similarly, the flows in (9) and (10) can be regarded as having the following Lax pairs, respectively:
The Lax pairs (13) and (15) can be abstracted as follows:
where
p can be taken as some functions in
. For example, if
(16) and (17) become (3) and (4) in [
4,
5], and
are polynomials in
Setting
and taking
(17) turns into the following form:
which can be used to generate (3+1)-dimensional integrable systems. Therefore, by applying the compatibility conditions of the Lax pairs in (16)–(18), some (1+1), (2+1), and (3+1) dimensional integrable hierarchies can be produced. In addition, the Lax pairs (16) and (17) can be expressed by the Poisson brackets. For a
-dimensional symplectic manifold
M and any
, an associative algebra of smooth functions on
M, the Poisson bracket is defined as
where
are local coordinates of
M, called the Darboux coordinates. Thus, the Lax pairs (13) and (15) can be written as
where
where
Using the results from contact geometry, the two kinds of linear nonisospectral Lax pairs in (3+1)-dimensions that generalize (20) are as follows: The first one replaces the Poisson bracket
with the contact bracket in (12) and gives us the Lax pair of the following form:
where, now,
The second one replaces the Hamiltonian vector field
with their contact counterparts
, and we have
where
The Lax pair of the form (22) is called a linear contact Lax pair, where
It is easy to verify that the compatibility condition of (22) presents
where
which gives rise to the following zero-curvature type equation due to
being arbitrary:
In the paper, we apply the Lax pair in (16), taking and the R-matrix method to generate a new (2+1)-dimensional integrable system from which a generalized system of the DS equation is obtained based on a matrix associative algebra; here, we call it a generalized DS hierarchy. By reducing the integrable hierarchy, we obtain linear and nonlinear scalar (2+1)-dimensional equations. With the help of the nonisospectral zero-curvature-type equation in (23), a kind of integrable hierarchy is obtained, which can be reduced to several (3+1)-dimensional integrable systems, the recursion operators, linearizations of some of them generated by using a Lie-group analysis.
2. A Generalized DS Hierarchy and Its Reductions
In the section, we mainly focus on deriving a new generalized DS integrable hierarchy by choosing a new Lax pair with matrix function coefficients.
Consider a linear Lax pair
where
and
D are all matrices to be determined;
is a real constant independent of
and
t; and
u and
v are functions in
and
Actually, the Lax pair in (24) reads as in the following form (16):
the compatibility of which is just right with the Lax equation:
It is easy to calculate that (26) admits the following equation system:
Taking
the first equation above leads to
Noting
then the last equation becomes
From the second equation in (27), we have
The third equation in (27) gives that
The fourth equation in (27) admits, by using (29), that
The last two equations in (27) can be written as
which is equivalent to
Hence, the matrix
In terms of (32)–(35) and by the use of (37), one infers that
Let
; then, (38) becomes
Set
and
; then, (39) reduces to
which is a rational DS-type equation. Setting the matrix
, similar to the discussion in Ref. [
6], we can obtain the DS equation:
where
S satisfies
where
and
k are constants.
Remark 1. In the section, given an explicit expression of the first equation in (24), we can determine the second linear spectral expression in (24) by virtue of the Lax equation. Obviously, choosing a different linear spectral problem, (24), we can generate various integrable systems. For example, we consider the following modified linear spectral problem, which is simpler than (24) (not matrices, but scalar functions) by setting By letting one obtains The compatibility condition of the Lax pair (42) and (43) leads to Taking (44) reduces to Setting (45) again reduces towhich is the Pavlov equation. Hence, Equation (45) is known as a generalized Pavlov equation. In what follows, we want to deduce the recursion operator of the generalized Pavlov equation in the setting of Ref. [5]. First, we recall some preliminaries. Consider a system of m PDEsin d independent variables for an unknown N-component vector function A total derivative with respect to readswhere and For a local
N-component vector function
U, it is a symmetry for the system (46) if and only if that
U satisfies the linearized version of this system, namely,
where
Denoting
where
are
matrices and
are
matrices. In Ref. [
7], three propositions are presented as follows:
Proposition 1. For the system (24), suppose that
- (i)
- (ii)
- (iii)
- (iv)
There are such that we can express and from the relations and then, (47) defines a recursion operator for (24), i.e., whenever U is a symmetry for (24), so is defined by (47).
Proposition 2. For the system (24), suppose that
- (i)
- (ii)
- (iii)
- (iv)
There exist such that we can express and from the relation and then, (48) defines an adjoint recursion operator for (24), i.e., whenever γ is a cosymmetry for (24), then so is defined by (48).
Remark 2. A so-called cosymmetry γ means that it is a quantity that is the dual to a symmetry that satisfies the system
Proposition 3. Under the assumptions of Propositions 1 and 2, the operators where λ is a spectral parameter, satisfy which constitute a Lax pair for (24).
According to the above known basic facts, it is easy to find the Lax pair of (45) as follows:
For a nonlocal symmetry for (45) with the form
where
we require that there exist operators
that are linear in
and such that
Then, one should extract
and
based on Proposition 1 and
How do we seek such operators? For Equation (
45), starting with its Lax pair in (49), we have
Applying the Garteax derivative
the linearized equation of the gPe (45) presents that
Hence, the recursion operator for Equation (
45) is obtained by using Proposition 1 and Proposition 3:
which maps a (possible nonlocal) symmetry
to a new symmetry
This is the recursion operator found in [
7], rewritten as a Bäcklund auto-transformation for (50).
In the following, we consider some solutions of (50). It is easy to see that the conjugate equation of (50) is given by
It can be verified that if
are conserved densities of Equation (
50), then the variation
is the solution of Equation (
52). Therefore, assuming that
is a solution of (52), we can verify that
is a solution of Equation (
50), where
In fact, supposing that
is a solution to Equation (
50), we put it into Equation (
45) and calculate that
which implies that (53) holds true.
3. A Linear Nonisospectral Lax Pair and Applications
In the section, we apply the linear nonisospectral Lax pair (22) in contact geometry to consider the generation of (3+1)-dimensional integrable systems, which are known as nonisospectral integrable systems because operator has a derivative , which indicates that function H is dependent on parameter p.
According to the discussion on the R-matrix method in [
4], for the following general Lax functions,
A special case of the above Lax functions is chosen as
The corresponding nonisospectral Lax pair exhibits that
The compatibility condition of (55) and (56) leads to the following equations with (3+1) dimensions:
In order to recognize what the system of equations in (57)–(63) is, we now consider their special cases. Taking
, we obtain a (3+1)-dimensional integrable system:
When
(64) reduces to
Denoting
(65) can be transformed to a (3+1)-dimensional nonlinear equation:
which can be written as
This is a (3+1)-dimensional rational equation that looks beautiful! When
(64) becomes the following (3+1)-dimensional integrable system:
Setting
the above equation can be reduced to
which is called a Boussinesq-type equation; the reason why it has this name will be explained later.
4. The Recursion Operators and Linearizations
In the section, we apply a Lie-group analysis to discuss the recursion operators and the linearizations of Equation (
68). Such a method is not only suitable for Equation (
68) but also suitable for other associated integrable equations or integrable systems.
For convenience, we rewrite (68) as follows:
Firstly, we need to recall some basic facts for transforming nonlinear PDEs into linear PDEs using Lie groups (see [
8]).
Lemma 1 (necessary conditions for the existence of an invertible mapping): If there exists an invertible transformation , which maps a given nonlinear system of PDEs to a linear system of PDEs , then
- (i)
The mapping must be a point transformation of the form
- (ii)
must admit an infinite-parameter Lie group of point transformations having infinitesimal generator
with
characterized by
where
are some functions of
and
is an arbitrary solution of some linear system of PDEs
with
representing a linear differential operator depending on independent variables
Lemma 3 ([
8])
. (Sufficient conditions for the existence of an invertible mapping): Let a given nonlinear system of PDEs admit an infinitesimal generator (71) for which the coefficients are of the forms in (72) and (73), with F being an arbitrary solution of a linear system (74) with specific independent variables:If the linear homogeneous system of m first-order PDEs for scalar Φ
has n functionally independent solutionsand the linear system of first-order PDEs iswhere is the Kronecker symbol, has a solutionthen the invertible mapping μ given bytransforms into a linear system of PDEs for some nonhomogeneous term Proposition 4. The Boussinesq-type equation in (69) has the following linearizations: Proof. Assume that Equation (
69) has the Lie–Bäcklund symmetry
then its prolongation
given by
acts on Equation (
69) and leads to
which means that
If
, then (82) reduces to a linear partial differential equation:
which is the standard liner wave equation.
If
then (82) becomes a linear equation:
which is just Equation (
80).
The proof is completed. □
In addition, we can also obtain other linearizations of (69) from (82). For example, if
then (82) has the following reduction:
In terms of (81) and Theorem 5.2.4.-4 (see [
8]), the characteristic function
leads to
For Equation (
83),
From Lemma 1, we have
Equation (
76) can be written as
Thus, we obtain an invertible mapping between Equation (
69) and Equation (
83) as follows:
The resulting linearized equation reads that
Therefore, as long as some exact solutions of (87) are obtained, the solutions of (69) could also be known.
Next, we consider an invertible mapping between Equation (
69) and its linearized Equation (
84). According to (86), we find that
Solving Equation (
75) yields that
Equation (
76) becomes
which has a solution
Hence, an integrable mapping can be given by
The resulting linearization of Equation (
69) is shown to be
under the constraint
For Equation (
85), we can similarly discuss the invertible mapping among (69) and (85); here, we omit it.
In what follows, we consider a possible invertible mapping of the nonlinear Equation (
69) and its linearizations by means of contact transformation. Noting
(69) can be written as
Assuming that (88) has a Lie–Bäcklund transformation
where
, we introduce two lemmas.
Lemma 4 ([
8])
. If there exists an invertible transformation μ that maps a given nonlinear scalar PDE to a linear scalar PDE ; then, - (i)
The mapping μ has the form - (ii)
admits the infinitesimal generator with given by where is an arbitrary solution of some linear PDE
Lemma 5 ([
8])
. Let a given nonlinear scalar PDE(m=1) admit a generator (89) with coefficients of the form in (90). Suppose that the following four conditions hold: - (i)
as n functionally independent solutions;
- (ii)
have a solution ;
- (iii)
have n functionally independent solutions - (iv)
define a contact transformation.
Then, the invertible mapping μ given bytransforms into a linear PDE :for some nonhomogenous term In the following, we consider some linearizations of Equation (
69) by using the above Lemmas 3 and 4 via the contact transformations. For later convenience, we copy (88) as follows:
Assume that (94) admits a Lie–Bäcklund symmetry
then, one infers that
Thus, the linearization of (94) presents that
Proposition 5. Assume then, (95) reduces to a linearized equation as follows: An invertible mapping between Equation (69) i.e., Equations (94) and (96), can be established. In fact, in terms of (86), one has that, if , According to Lemma 4, we have
Again applying Lemmas 3 and 4, the function
satisfies
which has a solution
In addition, (93) admits that
Hence, we obtain the invertible mapping
between (69) and the following linearization equation:
where
can be regarded as a free variable and
can regarded as a parameter function. When
u is independent of
, Equation (
97) reduces to
5. Bäcklund Transformations and Invariant Solutions of Equation (69)
In the section, we investigate the Bäcklund transformation of Equation (
69) via a undetermined method such as (100) (see, below). Given seed solutions, we can apply the Bäcklund transformation to deduce other exact solutions of integrable equations.
We all know that the Boussinesq equation reads as
while the Boussinesq-type Equation (
69) can be written as
Compared with (98), we find the nonlinear term
in (99) different from the linear term
in (98). Defining
we see that
which indicates that (99) has many of the same properties as (98) in some aspects. Because the Boussinesq Equation (
98) has some Bäcklund transformation and conservation laws (see [
9]), we guess that the Boussinesq-type Equation (
99) may also have a Bäcklund transformation. In what follows, we want to follow the approach to look for such a property.
Setting
Equation (
99) becomes
Let a Bäcklund transformation of (100) be as follows:
Noting
that is
w and
satisfy Equation (
100), i.e.,
which implies that
Via calculation, one infers from (102) that
Comparing (103) with (104) yields that
Taking
we find that
with a constraint condition
where
and
are constants. Thus, we obtain the Bäcklund transformation of Equation (
100) with parameters
and
:
along with Constraint (105).
Let
then
Hence, (106) becomes that when
:
Moreover, from (105), we obtain
Therefore, (107) becomes that when
:
which indicates that Equation (
100) only has a constant solution.
Remark 3. When , for example, we can compute thatHence, Inserting (109) into (106) and (105), we can obtain a new solution w by using the above similar calculations; here, we omit them. In what follows, we discuss the invariant solution of (100). Firstly, we write Equation (
100) as a form of the conservation laws:
Next, we introduce a variable
v that satisfies
Using Maple, the infinitesimal generators of (110) are given by
For the vector field , an invariance is presented as by solving the equation
The characteristic equation corresponding to
reads as
which has invariant functions that satisfy
where
are arbitrary smooth functions that meet the following ODEs with variable coefficients:
Differentiating the first equation in (111) with respect to
gives that
Again integrating (112) with respect to
leads to
where the integral constant is taken to be zero. Similarly, we differentiate the second equation in (111) for
and yield the ODE:
Instituting (112) and (113) into (114), one infers that
As long as some solution of (115) is obtained, the resulting solution of Equation (
100) can be also presented.
Remark 4. How do we solve Equation (115) with variable coefficients? A feasible way may be seeking its series solutions. Concerning this problem, we would like to discuss it in another paper.