Dispersionless (3+1)-dimensional integrable hierarchies

Abstract

In this paper, we introduce a multi-dimensional version of the R-matrix approach to the construction of integrable hierarchies. Applying this method to the case of the Lie algebra of functions with respect to the contact bracket, we construct integrable hierarchies of (3+1)-dimensional dispersionless systems of the type recently introduced in Sergyeyev (2014 (http://arxiv.org/abs/1401.2122)).

1. Introduction

Integrable systems are well known to play a prominent role in modern theoretical and mathematical physics, including quantum field theory and string theory; cf., for example, [1–21]. The R-matrix approach (see, for example, [2,3,13,14] and references therein) is one of the most general and best known constructions of such systems. In this approach, integrable systems result from the Lax equations on suitably chosen Lie algebras. The key advantage of this method is the possibility of systematic construction of infinite hierarchies of symmetries, conserved quantities and respective Hamiltonian, or rather multi-Hamiltonian, structures; see, for example, the recent surveys [3,14].

More than three decades of experience show that this approach, as well as other methods, works perfectly in (1+1) dimensions and admits an extension to (2+1) dimensions (e.g. [2,22–26]). However, to the best of our knowledge, all earlier attempts at extending the R-matrix approach to higher dimensions failed, even though a number of examples of integrable partial differential systems in (3+1) and higher dimensions were known (see, for example, [8,21,27–29] and references therein).

A significant advance was recently made in [30], where a novel systematic construction of (3+1)-dimensional integrable dispersionless systems was found. To put things into context, recall that zero-curvature equations involving the Poisson bracket with one degree of freedom give rise to (2+1)-dimensional dispersionless systems (see, for example, [20,23,31]). Roughly speaking, the key insight of the new construction in question is to replace the Poisson bracket by the contact bracket in the zero-curvature equations under study. Then these equations yield (3+1)- rather than (2+1)-dimensional systems. This approach gives rise to broad new classes of (3+1)-dimensional dispersionless integrable systems along with their Lax pairs.

The key message of this paper is that many of the systems of the type introduced in [30] admit a modified version of the R-matrix approach. Namely, inspired by the results of the work in question, we present below a multi-dimensional version of the R-matrix approach on appropriately chosen Lie algebras. In contrast with the standard version of the R-matrix method, we drop the requirement that the Lie algebras under study admit, in addition to the Lie bracket, an associative multiplication such that the adjoint action associated with the Lie bracket is a derivation (that is, this action obeys the Leibniz rule) with respect to the said multiplication. Unfortunately, in this case there appears to be no natural Hamiltonian structure on the dual Lie algebra, and thus no systematic method for constructing Hamiltonian representations for the systems under study is available.

In the particular setting introduced in [30] and considered in §§3 and 4, the Lie algebras belong to the class of Jacobi algebras which represent a natural generalization of the Poisson algebras. Even though the Jacobi algebras by definition admit an associative multiplication in addition to the Lie bracket, the adjoint action associated with the Lie bracket is not a derivation; instead it obeys a certain generalization of the Leibniz rule presented in §3. The systems in question are integrable in the sense of the existence of infinite hierarchies of commuting symmetries, and the construction of these hierarchies is given below. Note also that infinite hierarchies of non-local conservation laws for the systems under study could be obtained using the construction of non-isospectral Lax pairs from Sergyeyev [30] applied to our systems (cf. also [32]).

Using the R-matrix approach with suitably relaxed assumptions presented in §2, in §4 we construct infinite hierarchies of integrable dispersionless (3+1)-dimensional systems with infinitely many dependent variables associated with the contact bracket which is discussed in §3. Finally, some natural finite-component reductions of our systems are presented in §5. Section 6 contains conclusions and discussion.

2. The general R-matrix construction of integrable hierarchies

Let $\mathfrak{g}$ be an (infinite-dimensional) Lie algebra. The Lie bracket [⋅,⋅] defines the adjoint action of $\mathfrak{g}$ on $\mathfrak{g}$: ad_ab=[a,b].

Recall (e.g. [3,14,33] and references therein) that an $R\in \mathrm{End}(\mathfrak{g})$ is called a (classical) R-matrix if the R-bracket

$${[a,b]}_R:=[Ra,b]+[a,Rb]$$ 2.1

is a new Lie bracket on $\mathfrak{g}$. The skew symmetry of (2.1) is obvious. As for the Jacobi identity for (2.1), a sufficient condition for it to hold is the classical modified Yang–Baxter equation for R,

$$[Ra,Rb]-R{[a,b]}_R-\alpha [a,b]=0,\alpha \in \mathbb{R}.$$ 2.2

Let $L_i\in \mathfrak{g}$, $i\in \mathbb{N}$. Consider the associated hierarchies of flows (Lax hierarchies)

$${(L_n)}_{t_r}=[R{L}_r,L_n],r,n\in \mathbb{N}.$$ 2.3

We have the following result.

Theorem 2.1 —

Suppose that R is an R-matrix on $\mathfrak{g}$ which commutes with all derivatives ∂_tn, i.e.

$$\mathrm{\forall }L\in \mathfrak{g}{(RL)}_{t_n}=R{L}_{t_n},n\in \mathbb{N},$$ 2.4

and obeys the classical modified Yang–Baxter equation (2.2) for α≠0. Let $L_i\in \mathfrak{g}$, $i\in \mathbb{N}$ satisfy (2.3).

Then the following conditions are equivalent:

• (i) L_i satisfy the zero-curvature equations

  $${(R{L}_r)}_{t_s}-{(R{L}_s)}_{t_r}+[R{L}_r,R{L}_s]=0,r,s\in \mathbb{N};$$ 2.5
• (ii) all L_i commute in $\mathfrak{g}$

  $$[L_i,L_j]=0,i,j\in \mathbb{N}.$$ 2.6

Moreover, if one (and hence both) of the above equivalent conditions holds, then the flows (2.3) commute:

$${({(L_n)}_{t_r})}_{t_s}-{({(L_n)}_{t_s})}_{t_r}=0,n,r,s\in \mathbb{N}.$$ 2.7

Proof. —

Using (2.3) and the assumption (2.4), we see that the left-hand side of (2.5) takes the form

$$\begin{array}{rl}& \hspace{0.17em}{(R{L}_r)}_{t_s}-{(R{L}_s)}_{t_r}+[R{L}_r,R{L}_s]=R[R{L}_s,L_r]-R[R{L}_r,L_s]+[R{L}_r,R{L}_s]\\ & \hspace{0.17em}=[R{L}_r,R{L}_s]-R{[L_r,L_s]}_R\stackrel{(2.2)}{=}-\alpha [L_r,L_s],\end{array}$$

which establishes the equivalence of (2.6) and (2.5). To complete the proof it suffices to observe that the left-hand side of (2.7) can be written as

$$\begin{array}{rl}{({(L_n)}_{t_r})}_{t_s}-{({(L_n)}_{t_s})}_{t_r}& \hspace{0.17em}={[R{L}_r,L_n]}_{t_s}-{[R{L}_s,L_n]}_{t_r}\\ & \hspace{0.17em}=[{(R{L}_r)}_{t_s}-{(R{L}_s)}_{t_r},L_n]+[R{L}_r,[R{L}_s,L_n]]-[R{L}_s,[R{L}_r,L_n]]\\ & \hspace{0.17em}=[{(R{L}_r)}_{t_s}-{(R{L}_s)}_{t_r}+[R{L}_r,R{L}_s],L_n]\\ & \hspace{0.17em}=0,\end{array}$$

where the last equality follows from (2.5). □

Now we present a procedure of extending the systems under study by adding an extra independent variable. This procedure bears some resemblance to that of central extension (e.g. [3,14,22] and references therein).

Namely, we assume that all elements of $\mathfrak{g}$ depend on an additional independent variable y not involved in the Lie bracket, so all of the above results remain valid. Consider an $\mathcal{L}\in \mathfrak{g}$ and the associated Lax hierarchies defined by

$${\mathcal{L}}_{t_r}=[R{L}_r,\mathcal{L}]+{(R{L}_r)}_y,r\in \mathbb{N}.$$ 2.8

Theorem 2.2 —

Suppose that the R-matrix R on $\mathfrak{g}$ satisfies (2.4), and $\mathcal{L}\in \mathfrak{g}$ and $L_i\in \mathfrak{g}$, $i\in \mathbb{N}$ are such that the zero-curvature equations (2.5) hold for all $r,s\in \mathbb{N}$ and equations (2.8) hold for all $r\in \mathbb{N}$.

Then the flows (2.8) commute, i.e.

$${({\mathcal{L}}_{t_r})}_{t_s}-{({\mathcal{L}}_{t_s})}_{t_r}=0,r,s\in \mathbb{N}.$$ 2.9

Proof. —

Using equations (2.8) and the Jacobi identity for the Lie bracket, we obtain

$$\begin{array}{rl}{({\mathcal{L}}_{t_r})}_{t_s}-{({\mathcal{L}}_{t_s})}_{t_r}& \hspace{0.17em}=[{(R{L}_r)}_{t_s}-{(R{L}_s)}_{t_r}+[R{L}_r,R{L}_s],\mathcal{L}]\\ & \hspace{0.17em}+({(R{L}_r)}_{t_s}-{{(R{L}_s)}_{t_r}+[R{L}_r,R{L}_s])}_y\\ & \hspace{0.17em}=0.\end{array}$$

The right-hand side of the above equation vanishes by virtue of (2.5). □

It is well known (see, for example, [2,3,14,33]) that whenever $\mathfrak{g}$ admits a decomposition into two Lie subalgebras ${\mathfrak{g}}_{+}$ and ${\mathfrak{g}}_{-}$ such that

$$\mathfrak{g}={\mathfrak{g}}_{+}\oplus {\mathfrak{g}}_{-},[{\mathfrak{g}}_{\pm },{\mathfrak{g}}_{\pm }]\subset {\mathfrak{g}}_{\pm },{\mathfrak{g}}_{+}\cap {\mathfrak{g}}_{-}=\mathrm{\varnothing },$$

the operator

$$R=\frac{1}{2}(P_{+}-P_{-})=P_{+}-\frac{1}{2},$$ 2.10

where P_± are projectors onto ${\mathfrak{g}}_{\pm }$, satisfies the classical modified Yang–Baxter equation (2.2) with $\alpha =\frac{1}{4}$, hence R defined by (2.10) is a classical R-matrix.

Next, let us specify the dependence of L_j on y via the so-called Lax–Novikov equations (see [23] and references therein)

$$[L_j,\mathcal{L}]+{(L_j)}_y=0,j\in \mathbb{N}.$$ 2.11

Then, upon applying (2.6), (2.10) and (2.11), and putting B_i=P₊L_i, equations (2.3), (2.5) and (2.8) are readily seen to take the following form:

$$\begin{array}{rl}& \hspace{0.17em}{(L_s)}_{t_r}=[B_r,L_s],r,s\in \mathbb{N},\end{array}$$ 2.12

$$\begin{array}{rl}& \hspace{0.17em}{(B_r)}_{t_s}-{(B_s)}_{t_r}+[B_r,B_s]=0,r,s\in \mathbb{N},\end{array}$$ 2.13

$$\begin{array}{rl}& {\mathcal{L}}_{t_r}=[B_r,\mathcal{L}]+{(B_r)}_y,n,r\in \mathbb{N}.\end{array}$$ 2.14

Obviously, if upon the reduction to the case when all quantities are independent of y we put $\mathcal{L}=L_n$ for some $n\in \mathbb{N}$, then the hierarchies (2.8) boil down to hierarchies (2.3) and the Lax–Novikov equations (2.11) reduce to (a part of) the commutativity conditions (2.6). In particular, if the bracket [⋅,⋅] is such that equations (2.8) give rise to integrable systems in d independent variables, then equations (2.3) yield integrable systems in d−1 independent variables.

A standard construction of a commutative subalgebra spanned by L_i whose existence by theorem 2.1 ensures commutativity of the flows (2.8) is, in the case of Lie algebras which admit an additional associative multiplication ° which obeys the Leibniz rule

$${\mathrm{ad}}_a(b\circ c)={\mathrm{ad}}_a(b)\circ c+b\circ {\mathrm{ad}}_a(c)\hspace{0.17em}\iff \hspace{0.17em}[a,b\circ c]=[a,b]\circ c+b\circ [a,c],$$ 2.15

as follows: the commutative subalgebra in question is generated by fractional powers of a given element $L\in \mathfrak{g}$ (see, for example, [3,14] and references therein).

However, in our setting, when we no longer assume the existence of an associative multiplication on $\mathfrak{g}$ which obeys (2.15), the construction from the preceding paragraph does not work anymore. In order to circumvent this difficulty, instead of an explicit construction of commuting L_i we will impose the zero-curvature constraints (2.5) on chosen elements $L_i\in \mathfrak{g}$, $i\in \mathbb{N}$; it is readily seen that, in the setting of §§3, 4 and 5 we are interested in, this can be done in a consistent fashion. By theorem 2.1 this guarantees the commutativity of L_i for any R-matrix which obeys the classical modified Yang–Baxter equation (2.2) with α≠0.

3. The contact bracket

Consider a commutative and associative algebra A of formal series in p

$$A\ni f=\sum _i{u}_i{p}^i$$ 3.1

with the standard multiplication

$$f_1\cdot f_2\equiv f_1{f}_2,f_1,f_2\in A.$$ 3.2

The coefficients u_i of these series are assumed to be smooth functions of x,y,z and infinitely many times t₁,t₂,… .

Following [30], we define the contact bracket {⋅,⋅}_C on A as

$${\{f_1,f_2\}}_C=\frac{\mathrm{\partial }{f}_1}{\mathrm{\partial }p}\frac{\mathrm{\partial }{f}_2}{\mathrm{\partial }x}-p\frac{\mathrm{\partial }{f}_1}{\mathrm{\partial }p}\frac{\mathrm{\partial }{f}_2}{\mathrm{\partial }z}+f_1\frac{\mathrm{\partial }{f}_2}{\mathrm{\partial }z}-(f_1\leftrightarrow f_2).$$ 3.3

Note that the variable y is not involved in this bracket.

If we drop the dependence on z then this bracket reduces to the canonical Poisson bracket in one degree of freedom,

$${\{f_1,f_2\}}_{P,1}=\frac{\mathrm{\partial }{f}_1}{\mathrm{\partial }p}\frac{\mathrm{\partial }{f}_2}{\mathrm{\partial }x}-\frac{\mathrm{\partial }{f}_2}{\mathrm{\partial }p}\frac{\mathrm{\partial }{f}_1}{\mathrm{\partial }x},$$ 3.4

where the variable x is canonically conjugated to p.

Note that A is not a Poisson algebra as the contact bracket (3.3) does not obey the Leibniz rule. However, it belongs to a more general class of the so-called Jacobi algebras (see, for example, [34] and references therein for further details on these) that obey the following generalization of the Leibniz rule:

$${\{f_1{f}_2,f_3\}}_C={\{f_1,f_3\}}_C{f}_2+f_1{\{f_2,f_3\}}_C-f_1{f}_2{\{1,f_3\}}_C.$$ 3.5

More precisely, a Jacobi algebra is an associative commutative algebra (i.e. a vector space endowed with an associative commutative multiplication which is distributive with respect to addition and compatible with multiplication by elements of the ground field) which is further endowed with the Lie algebra structure that obeys the generalized Leibniz rule (3.5). If the unity 1 belongs to the centre of the Lie algebra in question, then (3.5) becomes the usual Leibniz rule and the algebra under study is then just a Poisson algebra.

Now let $\mathcal{A}$ be a Lie algebra of formal series in p_x and p_z whose coefficients again depend on x,y,z,t₁,t₂,… with respect to the standard Poisson bracket in two degrees of freedom,

$${\{h_1,h_2\}}_P=\frac{\mathrm{\partial }{h}_1}{\mathrm{\partial }{p}_x}\frac{\mathrm{\partial }{h}_2}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{h}_1}{\mathrm{\partial }{p}_z}\frac{\mathrm{\partial }{h}_2}{\mathrm{\partial }z}-(h_1\leftrightarrow h_2).$$ 3.6

It is readily checked that we have [30] a Lie algebra homomorphism from A to $\mathcal{A}$

$$f(p,x,y,z,t_1,t_2,\dots )\mapsto \overline{f}=p_{z}f(\frac{p_x}{p_z},x,y,z,t_1,t_2,\dots ).$$ 3.7

Note, however, that lifting this homomorphism to the Jacobi algebra homomorphism yields

$$\overline{f_1\hspace{0.17em}{f}_2}=\frac{1}{p_z}{\overline{f}}_1\hspace{0.17em}{\overline{f}}_2.$$

It is now readily seen that in fact we have the Jacobi algebra isomorphism, given by (3.7), that goes from the Jacobi algebra (A,{,}_C,⋅), defined via (3.1), (3.2) and (3.3), to the Jacobi algebra $(\overline{A},{\{,\}}_P,\circ )$ of formal series of the form

$$h=\sum _i{u}_i{p}_x^i{p}_z^{-i+1},$$ 3.8

which is a subalgebra of $(\mathcal{A},{\{,\}}_P,\circ )$, where

$$h_1\circ h_2=\frac{1}{p_z}{h}_1{h}_2.$$ 3.9

Note that the bracket (3.6) is not a Poisson bracket on the algebra $(\overline{A},{\{,\}}_P,\circ )$ as it does not obey the Leibniz rule with respect to the multiplication (3.9).

To make contact with the R-matrix approach of §2, we identify $\mathfrak{g}$ with A and the bracket [⋅,⋅] in $\mathfrak{g}$ with the contact bracket (3.3). As for the choice of the splitting of $\mathfrak{g}$ into Lie subalgebras ${\mathfrak{g}}_{\pm }$ with P_± being projections onto the respective subalgebras, so ${\mathfrak{g}}_{\pm }=P_{\pm }(\mathfrak{g})$, it is readily checked that we have two natural choices when the R’s defined by (2.10) satisfy the classical modified Yang–Baxter equation (2.2) and thus are R-matrices.

These two choices are P₊=P_≥k, where k=0 or k=1, and by definition

$$P_{\ge k}\left(\sum _{j=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_j{p}^j\right)=\sum _{j=k}^{\mathrm{\infty }}{a}_j{p}^j.$$

Note that, in contrast with the (1+1)-dimensional systems associated with the Poisson bracket (3.4) with one degree of freedom [35], the choice of k=2, i.e. taking P_≥2 for P₊, does not yield an R-matrix on A via (2.10), that is, in this case R defined via (2.10) does not satisfy (2.2).

4. Integrable (3+1)-dimensional infinite-component hierarchies and their reductions

Consider first the case of k=0 and the nth order Lax function from A of the form

$$\mathcal{L}=u_n{p}^n+u_{n-1}{p}^{n-1}+\cdots +u_0+u_{-1}{p}^{-1}+\cdots ,n>0,$$ 4.1

and let

$$B_m\equiv P_{+}{L}_m=v_{m,m}{p}^m+v_{m,m-1}{p}^{m-1}+\cdots +v_{m,0},m>0,$$ 4.2

where u_i=u_i(t,x,y,z), v_m,j=v_m,j(t,x,y,z) and t=(t₁,t₂,…).

Substituting $\mathcal{L}$ and B_m into the zero-curvature Lax equations

$${\mathcal{L}}_{t_m}={\{B_m,\mathcal{L}\}}_C+{(B_m)}_y,$$ 4.3

we obtain a hierarchy of infinite-component systems of the form

$$\begin{array}{rl}{(u_r)}_{t_m}& \hspace{0.17em}=X_r^m[u,v_m],r\le n+m,r\ne 0,\dots ,m,\\ {(u_r)}_{t_m}& \hspace{0.17em}=X_r^m[u,v_m]+{(v_{m,r})}_y,r=0,\dots ,m,\end{array}\}$$ 4.4

wherein (4.4) we put u_r≡0 for r>n and

$$\begin{array}{rl}{X}_r^m[u,v_m]& \hspace{0.17em}=\sum _{s=0}^m[s{v}_{m,s}{(u_{r-s+1})}_x-(r-s+1)u_{r-s+1}{(v_{m,s})}_x\\ & \hspace{0.17em}-(s-1)v_{m,s}{(u_{r-s})}_z+(r-s-1)u_{r-s}{(v_{m,s})}_z]\end{array}$$ 4.5

for r≤m+n, u=(u_n,u_n−1,…) and v_m=(v_m,0,…,v_m,m). The fields u_r for r≤n are dynamical variables while equations for n+m≥r>n can be seen as non-local constraints on u_r which define the variables v_m,s. The reader has to bear in mind that the additional dependent variables v_m,s are by construction related to each other for different m through the zero-curvature equations (2.13).

Upon using the homomorphism (3.7) we see that the hierarchy (4.4) can also be generated by

$$\begin{array}{rl}\overline{\mathcal{L}}& \hspace{0.17em}=u_n{p}_x^n{p}_z^{-n+1}+u_{n-1}{p}_x^{n-1}{p}_z^{-n+2}+\cdots +u_0{p}_z+u_{-1}{p}_x^{-1}{p}_z^2+\cdots ,\\ {\overline{B}}_m& \hspace{0.17em}=v_{m,m}{p}_x^m{p}_z^{-m+1}+v_{m,m-1}{p}_x^{m-1}{p}_z^{-m+2}+\cdots +v_{m,0}{p}_z,\end{array}$$

and the Lax equations

$${\overline{\mathcal{L}}}_{t_m}={\{{\overline{B}}_m,\overline{\mathcal{L}}\}}_P+{({\overline{B}}_m)}_y$$

with the Lie bracket (3.6). The same procedure can be applied to the other examples given below, but in what follows we shall continue to use the contact bracket formalism for the sake of simplicity. Let us also point out that using the contact bracket {⋅,⋅}_C and the algebra A instead of $\overline{A}$ and the Poisson bracket {⋅,⋅}_P naturally leads to non-isospectral Lax representations for systems written in the form of zero-curvature equations like (2.8) or (2.14) with [⋅,⋅] being the contact bracket (see theorem 1 of [30] for details).

The first equation from the system (4.4), i.e. the one for r=n+m, takes the form

$$(n-1)u_n{(v_{m,m})}_z-(m-1)v_{m,m}{(u_n)}_z=0$$

and hence, for n>1,m>1, admits the constraint

$$v_{m,m}={(u_n)}^{(m-1)/(n-1)}.$$ 4.6

For n=1, the constraint in question takes the form u₁=const.

The system (4.1)–(4.5) admits a natural constraint: u_n=c_n, v_m,m=c_m,m, where $c_n,c_{m,m}\in \mathbb{R}$. Then, if we put c_n=c_m,m=1, we have

$$\begin{array}{rl}\mathcal{L}& \hspace{0.17em}=p^n+u_{n-1}{p}^{n-1}+\cdots +u_0+u_{-1}{p}^{-1}+\cdots ,n>0,\end{array}$$ 4.7

$$\begin{array}{rl}{B}_m& \hspace{0.17em}\equiv P_{+}{L}_m=p^m+v_{m,m-1}{p}^{m-1}+\cdots +v_{m,0},m>0\end{array}$$ 4.8

and equations (4.3) take the form (4.4), where now r<n+m and

$$\begin{array}{rl}{X}_r^m[u,v_m]& \hspace{0.17em}=m{(u_{r-m+1})}_x-(m-1){(u_{r-m})}_z+\sum _{s=0}^{m-1}[s{v}_{m,s}{(u_{r-s+1})}_x\\ & \hspace{0.17em}-(r-s+1)u_{r-s+1}{(v_{m,s})}_x-(s-1)v_{m,s}{(u_{r-s})}_z+(r-s-1)u_{r-s}{(v_{m,s})}_z].\end{array}$$ 4.9

Again, the first equation from system (4.4), i.e. the one for r=n+m−1, takes the form

$$(n-1){(v_{m,m-1})}_z-(m-1){(u_{n-1})}_z=0,$$

so the system under study for n>1 admits a further constraint

$$v_{m,m-1}=\frac{(m-1)}{(n-1)}{u}_{n-1}.$$ 4.10

It is readily seen that for n=1 the constraint (4.10) should be replaced by u₀=const. Consider this case in more detail.

Upon taking u₀=0, the Lax equation (4.3) for

$$\mathcal{L}=p+u_{-1}{p}^{-1}+u_{-2}{p}^{-2}+\cdots$$ 4.11

and for m=2, with

$$B_2=p^2+v_{1}p+v_0,$$

generates the following infinite-component system:

$$\begin{array}{rl}{(v_1)}_y& \hspace{0.17em}={(v_1)}_x+{(u_{-1})}_z,\\ {(v_0)}_y& \hspace{0.17em}={(v_0)}_x+{(u_{-2})}_z-2{(u_{-1})}_x+2u_{-1}{(v_1)}_z,\\ {(u_r)}_{t_2}& \hspace{0.17em}=2{(u_{r-1})}_x-{(u_{r-2})}_z-(r+1)u_{r+1}{(v_0)}_x+v_0{(u_r)}_z\\ & \hspace{0.17em}+(r-1)u_r{(v_0)}_z+v_1{(u_r)}_x-r{u}_r{(v_1)}_x+(r-2)u_{r-1}{(v_1)}_z,\end{array}\}$$ 4.12

where r<0 and v_2,r≡v_r.

We have a natural (2+1)-dimensional reduction of (4.12) when u_j,v₀ and v₁ are independent of y,

$$\begin{array}{rl}0& \hspace{0.17em}={(v_1)}_x+{(u_{-1})}_z,\\ 0& \hspace{0.17em}={(v_0)}_x+{(u_{-2})}_z-2{(u_{-1})}_x+2u_{-1}{(v_1)}_z,\\ {(u_r)}_{t_2}& \hspace{0.17em}=2{(u_{r-1})}_x-{(u_{r-2})}_z-(r+1)u_{r+1}{(v_0)}_x+v_0{(u_r)}_z\\ & \hspace{0.17em}+(r-1)u_r{(v_0)}_z+v_1{(u_r)}_x-r{u}_r{(v_1)}_x+(r-2)u_{r-1}{(v_1)}_z,\end{array}\}$$ 4.13

another (2+1)-dimensional reduction,

$$\begin{array}{rl}{(v_1)}_y& \hspace{0.17em}={(u_{-1})}_z,\\ {(v_0)}_y& \hspace{0.17em}={(u_{-2})}_z+2u_{-1}{(v_1)}_z,\\ {(u_r)}_{t_2}& \hspace{0.17em}=-{(u_{r-2})}_z+v_0{(u_r)}_z+(r-1)u_r{(v_0)}_z+(r-2)u_{r-1}{(v_1)}_z,\end{array}\}$$ 4.14

when u_j,v₀ and v₁ are independent of x, and yet another (2+1)-dimensional reduction,

$$\begin{array}{rl}{(v_1)}_y& \hspace{0.17em}={(v_1)}_x,\\ {(v_0)}_y& \hspace{0.17em}={(v_0)}_x-2{(u_{-1})}_x,\\ {(u_r)}_{t_2}& \hspace{0.17em}=2{(u_{r-1})}_x-(r+1)u_{r+1}{(v_0)}_x+v_1{(u_r)}_x-r{u}_r{(v_1)}_x,\end{array}\}$$ 4.15

when u_j,v₀ and v₁ are independent of z.

Moreover, system (4.15) admits a further reduction v₁=0 to the form

$$\begin{array}{rl}{(v_0)}_y& \hspace{0.17em}={(v_0)}_x-2{(u_{-1})}_x,\\ {(u_r)}_{t_2}& \hspace{0.17em}=2{(u_{r-1})}_x-(r+1)u_{r+1}{(v_0)}_x+v_1{(u_r)}_x.\end{array}\}$$ 4.16

The system (4.16) reduces to the (1+1)-dimensional Benney system (cf., for example, [36,35])

$${(u_r)}_{t_2}=2{(u_{r-1})}_x-2(r+1)u_{r+1}{(u_{-1})}_x,r<0,$$ 4.17

when u_i are independent of both y and z, and we put v₀=2u₋₁.

On the other hand, system (4.14) admits no reductions to (1+1)-dimensional systems. Note that for systems (4.12)–(4.17) there are no obvious finite-component reductions.

For systems with the Lax functions (4.1), (4.2) and (4.7), (4.8) we have (2+1)-dimensional and (1+1)-dimensional reductions of the same types as above.

Now pass to the case of k=1, when P₊=P_≥1, and consider the general case when

$$\begin{array}{rl}\mathcal{L}& \hspace{0.17em}=u_n{p}^n+u_{n-1}{p}^{n-1}+\cdots +u_0+u_{-1}{p}^{-1}+\cdots ,n>0,\\ B_m& \hspace{0.17em}=v_{m,m}{p}^m+v_{m,m-1}{p}^{m-1}+\cdots +v_{m,1}p,m>0,\end{array}\}$$ 4.18

from which we again obtain the hierarchies of infinite-component systems

$$\begin{array}{rl}{(u_r)}_{t_m}& \hspace{0.17em}=X_r^m[u,v_m],r\le n+m,r\ne 1,\dots ,m,\\ {(u_r)}_{t_m}& \hspace{0.17em}=X_r^m[u,v_m]+{(v_{m,r})}_y,r=1,\dots ,m,\end{array}\}$$ 4.19

where in (4.19) we put u_r≡0 for r>n and

$$\begin{array}{rl}{X}_r^m[u,v_m]& \hspace{0.17em}=\sum _{s=1}^m[s{v}_{m,s}{(u_{r-s+1})}_x-(r-s+1)u_{r-s+1}{(v_{m,s})}_x\\ & \hspace{0.17em}-(s-1)v_{m,s}{(u_{r-s})}_z+(r-s-1)u_{r-s}{(v_{m,s})}_z],\end{array}$$ 4.20

for r≤m+n, u=(u_n,u_n−1,…) and v_m=(v_m,1,…,v_m,m).

For n>1,m>1, we again obtain the constraint (4.6), and for n=1 the constraint in question is replaced by u₁=const. Consider in more detail the simplest case when

$$\mathcal{L}=p+u_0+u_{-1}{p}^{-1}+\cdots$$ 4.21

and

$$B_m\equiv P_{+}{L}_m=v_{m,m-1}{p}^m+v_{m,m-2}{p}^{m-1}+\cdots +v_{m,1}p,m>1.$$ 4.22

The first flow for m=2, where we put v_2,r≡v_r to simplify writing, takes the form

$$\begin{array}{rl}{(v_2)}_y& \hspace{0.17em}={(v_2)}_x+u_0{(v_2)}_z+v_2{(u_0)}_z,\\ {(v_1)}_y& \hspace{0.17em}={(v_1)}_x+u_0{(v_1)}_z+v_2{(u_{-1})}_z+2u_{-1}{(v_2)}_z-2v_2{(u_0)}_x,\\ {(u_r)}_{t_2}& \hspace{0.17em}=v_1{(u_r)}_x-r{u}_r{(v_1)}_x+(r-2)u_{r-1}{(v_1)}_z+2v_2{(u_{r-1})}_x\\ & \hspace{0.17em}-(r-1)u_{r-1}{(v_2)}_x-v_2{(u_{r-2})}_z+(r-3)u_{r-2}{(v_2)}_z.\end{array}\}$$ 4.23

We have a natural (2+1)-dimensional reduction of (4.23) with u_j,v₁ and v₂ independent of y,

$$\begin{array}{rl}0& \hspace{0.17em}={(v_2)}_x+u_0{(v_2)}_z+v_2{(u_0)}_z,\\ 0& \hspace{0.17em}={(v_1)}_x+u_0{(v_1)}_z+v_2{(u_{-1})}_z+2u_{-1}{(v_2)}_z-2v_2{(u_0)}_x,\\ {(u_r)}_{t_2}& \hspace{0.17em}=v_1{(u_r)}_x-r{u}_r{(v_1)}_x+(r-2)u_{r-1}{(v_1)}_z+2v_2{(u_{r-1})}_x\\ & \hspace{0.17em}-(r-1)u_{r-1}{(v_2)}_x-v_2{(u_{r-2})}_z+(r-3)u_{r-2}{(v_2)}_z.\end{array}\}$$ 4.24

On the other hand, if u_j,v₁ and v₂ are independent of x, we obtain from (4.23) a (2+1)-dimensional system

$$\begin{array}{rl}{(v_2)}_y& \hspace{0.17em}=u_0{(v_2)}_z+v_2{(u_0)}_z,\\ {(v_1)}_y& \hspace{0.17em}=u_0{(v_1)}_z+v_2{(u_{-1})}_z+2u_{-1}{(v_2)}_z,\\ {(u_r)}_{t_2}& \hspace{0.17em}=(r-2)u_{r-1}{(v_1)}_z-v_2{(u_{r-2})}_z+(r-3)u_{r-2}{(v_2)}_z.\end{array}\}$$ 4.25

Finally, if u_j,v₁ and v₂ in (4.23) are independent of z, we arrive at a (2+1)-dimensional system

$$\begin{array}{rl}{(v_1)}_y& \hspace{0.17em}={(v_1)}_x,\\ {(v_0)}_y& \hspace{0.17em}={(v_0)}_x-2v_1{(u_0)}_x,\\ {(u_r)}_{t_2}& \hspace{0.17em}=v_0{(u_r)}_x-r{u}_r{(v_0)}_x+2v_1{(u_{r-1})}_x-(r-1)u_{r-1}{(v_1)}_x,\end{array}\}$$ 4.26

where we made use of an admissible reduction v₂=const.=1, and if we make a further reduction v₁=const.=1 we obtain

$$\begin{array}{rl}{(v_0)}_y& \hspace{0.17em}={(v_0)}_x-2{(u_0)}_x,\\ {(u_r)}_{t_2}& \hspace{0.17em}=2{(u_{r-1})}_x+v_0{(u_r)}_x-r{u}_r{(v_0)}_x.\end{array}\}$$ 4.27

If u_j,v₁ and v₂ are independent of both y and z, we can put v₁=2u₀ and obtain

$${(u_r)}_{t_2}=2{(u_{r-1})}_x+2u_0{(u_r)}_x-2r{u}_r{(u_0)}_x.$$ 4.28

Finally, when u_j,v₁ and v₂ are independent of both y and x, we have

$${(u_r)}_{t_2}=(r-2)u_{r-1}{(v_1)}_z-v_2{(u_{r-2})}_z+(r-3)u_{r-2}{(v_2)}_z,$$ 4.29

where a reduction $v_2=a{u}_0^{-1},\hspace{0.17em}{v}_1=-a{u}_{-1}{u}_0^{-2}$, was performed, and $a\in \mathbb{R}$ is an arbitrary constant. Thus, in this case the system under study is rational (rather than polynomial) in u₀.

5. Finite-component reductions

For k=0, in contrast with the simplest case (4.11), we do have natural reductions to finite-component systems by putting u_r=0 for r<1 or r<0 in (4.1) and (4.7), so we consider two cases:

$$\begin{array}{rl}\mathcal{L}& \hspace{0.17em}=u_n{p}^n+u_{n-1}{p}^{n-1}+\cdots +u_r{p}^r,r=0,1,\\ B_m& \hspace{0.17em}={(u_n)}^{\frac{m-1}{n-1}}{p}^m+v_{m,m-1}{p}^{m-1}+\cdots +v_{m,0}\end{array}\}$$ 5.1

and

$$\begin{array}{rl}\mathcal{L}& \hspace{0.17em}=p^n+u_{n-1}{p}^{n-1}+\cdots +u_r{p}^r,r=0,1,\\ B_m& \hspace{0.17em}=p^m+\frac{(m-1)}{(n-1)}{u}_{n-1}{p}^{m-1}+\cdots +v_{m,0}.\end{array}\}$$ 5.2

The case (5.2) for r=0 was considered for the first time in [30]. Note that in (5.1) and (5.2) for r=0 we have $\mathcal{L}=B_n$, and hence the variable y can be identified with t_n. Then equations (4.3) coincide with the zero-curvature equations (2.13) and the Lax–Novikov equations (2.11) reduce to equation (2.12).

The structure of the said finite-component reductions is best revealed in the matrix form of system (4.4). For the reduction (5.2) and n≥m, we obtain

$$\begin{array}{rl}0& \hspace{0.17em}=A_1^m(u){(V_m)}_z+A_2^m(u){(V_m)}_x+A_3^m(v){(U_m)}_z+A_4^m(v){(U_m)}_x,\\ {(U_n)}_{t_m}& \hspace{0.17em}=A_1^n(u){(V_n)}_z+A_2^n(u){(V_n)}_x+A_3^n(v){(U_n)}_z+A_4^n(v){(U_n)}_x+{(V_n)}_y,\end{array}\}$$ 5.3

where

$$\begin{array}{rl}{U}_m& \hspace{0.17em}={(u_{n-m},\dots ,u_{n-1})}^{\mathrm{T}},V_m={(v_{m,0},\dots ,v_{m,m-1})}^{\mathrm{T}},\\ V_n& \hspace{0.17em}={\underset{n}{\underbrace{(v_{m,0},\dots ,v_{m,m-1},0,\dots ,0)}}}^{\mathrm{T}},U_n={(u_0,\dots ,u_{n-1})}^{\mathrm{T}},\end{array}$$

$A_i^m$ and $A_i^n$ are, respectively, m×m and n×n square matrices and, as usual, the superscript T indicates the transposed matrix. The entries of the matrices $A_i^s$ are linear in the fields u_i and v_m,s.

On the other hand, for n<m we have

$$\begin{array}{rl}0& \hspace{0.17em}=B_1^m(u){(V_m)}_z+B_2^m(u){(V_m)}_x+B_3^m(v){(U_m)}_z+B_4^m(v){(U_m)}_x+{(V_{m,n})}_y,\\ {(U_n)}_{t_m}& \hspace{0.17em}=B_1^n(u){(V_n)}_z+B_2^n(u){(V_n)}_x+B_3^n(v){(U_n)}_z+B_4^n(v){(U_n)}_x+{(V_n)}_y,\end{array}\}$$ 5.4

where

$$\begin{array}{rl}{V}_m& \hspace{0.17em}={(v_{m,0},\dots ,v_{m,m-1})}^{\mathrm{T}},V_n={(v_{m,0},\dots ,v_{m,n-1})}^{\mathrm{T}},U_m={\underset{m}{\underbrace{(u_0,\dots ,u_{n-1},0,\dots ,0)}}}^{\mathrm{T}},\\ V_{m,n}& \hspace{0.17em}={\underset{m}{\underbrace{(v_{m,n},\dots ,v_{m,m-1},0,\dots ,0)}}}^{\mathrm{T}},U_n={(u_0,\dots ,u_{n-1})}^{\mathrm{T}}.\end{array}$$

The structure of the matrices $B_i^{(j)}$ is essentially the same as that of the matrices $A_i^{(j)}$ above.

Another class of natural reductions to finite-component systems arises for k=1, if we put

$$\begin{array}{rl}\mathcal{L}& \hspace{0.17em}=u_n{p}^n+u_{n-1}{p}^{n-1}+\cdots +u_r{p}^r,r=1,0,-1,\dots ,\\ B_m& \hspace{0.17em}={(u_n)}^{(m-1)/(n-1)}{p}^m+v_{m,m-1}{p}^{m-1}+\cdots +v_{m,1}p\end{array}\}$$ 5.5

or

$$\begin{array}{rl}\mathcal{L}& \hspace{0.17em}=p+u_0+u_{-1}{p}^{-1}+\cdots +u_r{p}^r,r=0,1,-1,\dots ,\\ B_m& \hspace{0.17em}=v_{m,m}{p}^m+v_{m,m-1}{p}^{m-1}+\cdots +v_{m,1}p,m>1.\end{array}\}$$ 5.6

For instance, let

$$\mathcal{L}=p+u_0+u_{-1}{p}^{-1}$$ 5.7

and, with a slight variation of the earlier notation, put

$$B_2=v_2{p}^2+v_{1}p\mathrm{and}{B}_3=w_3{p}^3+w_2{p}^2+w_{1}p.$$

The member of the hierarchy associated with B₂ reads

$$\begin{array}{rl}{(u_{-1})}_{t_2}& \hspace{0.17em}=u_{-1}{(v_1)}_x+v_1{(u_{-1})}_x,\\ {(u_0)}_{t_2}& \hspace{0.17em}=-2u_{-1}{(v_1)}_z+v_1{(u_0)}_x+u_{-1}{(v_2)}_x+2v_2{(u_{-1})}_x,\\ {(v_1)}_y& \hspace{0.17em}={(v_1)}_x+2u_{-1}{(v_2)}_z+v_2{(u_{-1})}_z+u_0{(v_1)}_z-2v_2{(u_0)}_x,\\ {(v_2)}_y& \hspace{0.17em}={(v_2)}_x+u_0{(v_2)}_z+v_2{(u_0)}_z,\end{array}\}$$ 5.8

and the one associated with B₃ has the form

$$\begin{array}{rl}{(u_{-1})}_{t_3}& \hspace{0.17em}=u_{-1}{(w_1)}_x+w_1{(u_{-1})}_x,\\ {(u_0)}_{t_3}& \hspace{0.17em}=w_1{(u_0)}_x-2u_{-1}{(w_1)}_z+u_{-1}{(w_2)}_x+2w_2{(u_{-1})}_x,\\ {(w_1)}_y& \hspace{0.17em}={(w_1)}_x+w_2{(u_{-1})}_z-u_{-1}{(w_3)}_x-2w_2{(u_0)}_x+2u_{-1}{(w_2)}_z\\ & \hspace{0.17em}+u_0{(w_1)}_z-3w_3{(u_{-1})}_x,\\ {(w_2)}_y& \hspace{0.17em}={(w_2)}_x-3w_3{(u_0)}_x+2w_3{(u_{-1})}_z+w_2{(u_0)}_z+u_0{(w_2)}_z+2u_{-1}{(w_3)}_z,\\ {(w_3)}_y& \hspace{0.17em}={(w_3)}_x+u_0{(w_3)}_z+2w_3{(u_0)}_z.\end{array}\}$$ 5.9

Commutativity of the flows associated with t₂ and t₃, i.e.

$${({(u_i)}_{t_2})}_{t_3}={({(u_i)}_{t_3})}_{t_2},i=0,1,$$

can be readily checked using the set of relations

$$\begin{array}{rl}{(v_1)}_z& \hspace{0.17em}=-\frac{v_2}{w_3}{(w_3)}_x-\frac{v_2{w}_2}{4w_3^2}{(w_3)}_z+\frac{v_2}{2w_3}{(w_2)}_z+\frac{3}{2}{(v_2)}_x,\\ {(v_2)}_z& \hspace{0.17em}={\displaystyle \frac{v_2}{2w_3}{(w_3)}_z,}\\ {(w_1)}_{t_2}& \hspace{0.17em}=v_1{(w_1)}_x-w_1{(v_1)}_x+{(v_1)}_{t_3},\\ {(w_2)}_{t_2}& \hspace{0.17em}=v_1{(w_2)}_x-w_1{(v_2)}_x+2v_2{(w_1)}_x-2w_2{(v_1)}_x+{(v_2)}_{t_3},\\ {(w_3)}_{t_2}& \hspace{0.17em}=\frac{v_2{w}_2}{2w_3}{(w_2)}_z-\frac{w_2}{2}{(v_2)}_x-\frac{v_2{w}_2^2}{4w_3^2}{(w_3)}_z+\frac{(v_1{w}_3-v_2{w}_2)}{w_3}{(w_3)}_x\\ & \hspace{0.17em}-v_2{(w_1)}_z+2v_2{(w_2)}_x-3w_3{(v_1)}_x,\end{array}\}$$ 5.10

which is equivalent to the zero-curvature equation

$${(B_2)}_{t_3}-{(B_3)}_{t_2}+{\{B_2,B_3\}}_C=0.$$ 5.11

Note that the compatibility conditions ((v_i)_y)_z=((v_i)_z)_y, i=1,2, are also satisfied by virtue of (5.8) and (5.11).

When u_i and v_j are independent of z we obtain (2+1)-dimensional systems with additional constraints v₂=const.=1, w₃=const.=1, namely

$$\begin{array}{rl}{(u_{-1})}_{t_2}& \hspace{0.17em}=u_{-1}{(v_1)}_x+v_1{(u_{-1})}_x,\\ {(u_0)}_{t_2}& \hspace{0.17em}=v_1{(u_0)}_x+2{(u_{-1})}_x,\\ {(v_1)}_y& \hspace{0.17em}={(v_1)}_x-2{(u_0)}_x\end{array}\}$$ 5.12

and

$$\begin{array}{rl}{(u_{-1})}_{t_3}& \hspace{0.17em}=u_{-1}{(w_1)}_x+w_1{(u_{-1})}_x,\\ {(u_0)}_{t_3}& \hspace{0.17em}=w_1{(u_0)}_x+u_{-1}{(w_2)}_x+2w_2{(u_{-1})}_x,\\ {(w_1)}_y& \hspace{0.17em}={(w_1)}_x-3{(u_{-1})}_x-2w_2{(u_0)}_x,\\ {(w_2)}_y& \hspace{0.17em}={(w_2)}_x-3{(u_0)}_x.\end{array}\}$$ 5.13

When u_i and v_j are independent of x we obtain other (2+1)-dimensional systems making use of a naturally arising extra constraint u₋₁=1, namely

$$\begin{array}{rl}{(u_0)}_{t_2}& \hspace{0.17em}=-2{(v_1)}_z,\\ {(v_1)}_y& \hspace{0.17em}=2{(v_2)}_z+u_0{(v_1)}_z,\\ {(v_2)}_y& \hspace{0.17em}={(u_0{v}_2)}_z\end{array}\}$$ 5.14

and

$$\begin{array}{rl}{(u_0)}_{t_2}& \hspace{0.17em}=-2{(w_1)}_z,\\ {(w_1)}_y& \hspace{0.17em}=2{(w_2)}_z+u_0{(w_1)}_z,\\ {(w_2)}_y& \hspace{0.17em}=2{(w_3)}_z+{(u_0{w}_2)}_z,\\ {(w_3)}_y& \hspace{0.17em}=u_0{(w_3)}_z+2w_3{(u_0)}_z.\end{array}\}$$ 5.15

Further reduction of (5.12) and (5.13) by assuming that u_i, v_j and w_k are independent of y leads to (1+1)-dimensional systems of the form

$$\begin{array}{rl}{(u_{-1})}_{t_2}& \hspace{0.17em}=2{(u_{-1}{u}_0)}_x,\\ {(u_0)}_{t_2}& \hspace{0.17em}=2{(u_{-1}+u_0^2)}_x,\end{array}\}$$ 5.16

with the constraint v₁=2u₀, and

$$\begin{array}{rl}{(u_{-1})}_{t_3}& \hspace{0.17em}=3{(u_{-1}{u}_0^2+u_{-1}^2)}_x,\\ {(u_0)}_{t_3}& \hspace{0.17em}={(u_0^3+6u_0{u}_{-1})}_x,\end{array}\}$$ 5.17

with the constraints w₂=3u₀, $w_1=3u_0^2+3u_{-1}$.

Likewise, the reduction of (5.14) and (5.15) by assuming that u_i, v_j and w_k are independent of y leads to the (1+1)-dimensional systems of the form

$${(u_0)}_{t_2}=2{(u_0^{-2})}_z$$ 5.18

and

$${(u_0)}_{t_3}=-6{(u_0^{-4})}_z,$$ 5.19

as we have $v_2=u_0^{-1}$, $v_1=-u_0^{-2}$, $w_3=u_0^{-2}$, $w_2=-2u_0^{-3}$ and $w_1=3u_0^{-4}$.

The simplest non-trivial example of Lax pair (5.5) is given by

$$\begin{array}{rl}\mathcal{L}& \hspace{0.17em}=u_3{p}^3+u_2{p}^2+u_{1}p,\\ B_2& \hspace{0.17em}=v_2{p}^2+v_{1}p,\end{array}$$

and the associated system reads

$$\begin{array}{rl}0& \hspace{0.17em}=2u_3{(v_2)}_z-v_2{(u_3)}_z,\\ 0& \hspace{0.17em}=u_2{(v_2)}_z-v_2{(u_2)}_z+2u_3{(v_1)}_z+2v_2{(u_3)}_x-3u_3{(v_2)}_x,\\ {(u_3)}_{t_2}& \hspace{0.17em}=v_1{(u_3)}_x+2v_2{(u_2)}_x-2u_2{(v_2)}_x-3u_3{(v_1)}_x-v_2{(u_1)}_z+u_2{(v_1)}_z,\\ {(u_2)}_{t_2}& \hspace{0.17em}={(v_2)}_y+v_1{(u_2)}_x+2v_2{(u_1)}_x-2u_2{(v_1)}_x-u_1{(v_2)}_x,\\ {(u_1)}_{t_2}& \hspace{0.17em}={(v_1)}_y+v_1{(u_1)}_x-u_1{(v_1)}_x.\end{array}$$

Here we have not yet imposed the constraint (4.6).

The first two of the above equations impose constraints on the ‘non-dynamical’ fields v₁ and v₂. The first of these constraints is satisfied once we impose (4.6), i.e. v₂=(u₃)^1/2, and then the second one becomes

$${(v_1)}_z={\left[\frac{1}{2}{u}_2{(u_3)}^{-1/2}\right]}_z-{\left[\frac{1}{2}{(u_3)}^{1/2}\right]}_x.$$

Assuming that u_i and v_j no longer depend on z naturally leads to further constraints

$$v_2=\mathrm{const}.=1,u_3=\mathrm{const}.=1,v_1=\frac{2}{3}{u}_2,$$

and then we obtain an evolutionary system

$$\begin{array}{rl}{(u_2)}_{t_2}& \hspace{0.17em}=2{(u_1)}_x-\frac{2}{3}{u}_2{(u_2)}_x,\\ {(u_1)}_{t_2}& \hspace{0.17em}=\frac{2}{3}[{(u_2)}_y+u_2{(u_1)}_x-u_1{(u_2)}_x].\end{array}\}$$ 5.20

On the other hand, assuming that u_i and v_j no longer depend on x yields

$$\begin{array}{rl}{(u_3)}_{t_2}& \hspace{0.17em}=u_2{(v_1)}_z-v_2{(u_1)}_z,\\ {(u_2)}_{t_2}& \hspace{0.17em}={(v_2)}_y,\\ {(u_1)}_{t_2}& \hspace{0.17em}={(v_1)}_y,\end{array}\}$$ 5.21

where we have v₂=(u₃)^1/2 and $v_1=\frac{1}{2}{u}_2{(u_3)}^{-1/2}$.

The reduction of (5.20) and (5.15) by assuming that the dependent variables involved are independent of y leads to a (1+1)-dimensional system

$$\begin{array}{rl}{(u_2)}_{t_2}& \hspace{0.17em}=2{(u_1)}_x-\frac{2}{3}{u}_2{(u_2)}_x,\\ {(u_1)}_{t_2}& \hspace{0.17em}=\frac{2}{3}[u_2{(u_1)}_x-u_1{(u_2)}_x],\end{array}\}$$ 5.22

while for (5.21) we are naturally led to imposing the constraints u₁=const.=0, u₂=const.=1, and then we obtain the equation

$${(u_3)}_{t_2}=\frac{1}{2}({{(u_3)}^{-1/2})}_z.$$ 5.23

In closing note that it would be interesting to find out whether the above hierarchies could be reproduced using the recursion operators in the spirit of [2,37–42] and references therein.

6. Outlook

In this paper, we have presented an extension of the R-matrix approach to the construction of (3+1)-dimensional integrable dispersionless hierarchies related to the contact bracket. We stress that this extension is rather non-trivial in that we had to drop a number of commonly used assumptions on the underlying Lie algebra as seen in §§2, 3 and 4. This approach has a number of advantages; in particular, it makes the whole construction significantly more transparent. Let us also reiterate that our results extend those of [30] in that we construct at once infinite hierarchies of commuting (3+1)-dimensional integrable systems rather than single integrable systems.

Our results lead to a number of open problems which we would like to touch upon below.

First of all, it would be very interesting to find out how exactly the dispersionless version of the inverse scattering transform in the spirit of [43,44] and references therein could be applied to the systems considered in this paper. Even though the associated linear Lax pairs belong [30] to a broader class of non-isospectral Lax pairs written in terms of vector fields, the study of the inverse scattering for our Lax pairs is a highly non-trivial challenge going far beyond the scope of this article. The same applies to the twistor approach (see, for example, [8] and references therein).

Second, (2+1)-dimensional integrable dispersionless systems are known to be intimately related to integrable hydrodynamic chains (see, for example, [45,46] and references therein) and, while it would be quite interesting to find out whether a similar connection exists for (3+1)-dimensional integrable systems studied by us, at present we do not quite know how to approach this problem.

Next, there is the hydrodynamic reduction integrability test for translation-invariant dispersionless integrable systems in more than two independent variables (see, for example, [25,47] and references therein). It is believed that if a dispersionless system in more than two independent variables is integrable in the sense that it possesses a dispersionless Lax pair, i.e. a Lax pair written in terms of vector fields, with or without the derivative with respect to the spectral parameter, then it should pass this test and vice versa.

Unfortunately, to the best of our knowledge, so far there have been no general theorems proving this in either direction, although Ferapontov & Khusnutdinova [25], Doubrov & Ferapontov [47] and related papers have treated certain classes of integrable dispersionless systems using the hydrodynamic reductions test and have shown that systems which pass this test also possess Lax pairs. Moreover, the computations involved in this test increase dramatically with increasing dimension, and, to the best of our knowledge, Doubrov & Ferapontov [47], and Ferapontov & Kruglikov [26] and references [25,28] therein, are pretty much the only papers dealing with this test for the case of systems in four independent variables. It would be very interesting to apply the said test to the systems from the present paper and to those from Sergyeyev [30], but this would require a huge amount of work which would merit a separate publication. In this connection, we reiterate that all our examples possess Lax pairs a priori and thus, in a sense, are integrable by construction.

Closely related to these matters are the recent results of Ferapontov, Kruglikov, Doubrov, Calderbank and Dunajski (see [26,47–50] and references therein), which, for the systems in three and four independent variables whose characteristic variety is a quadric hypersurface, show that the existence of a dispersionless Lax pair for such a system is equivalent to the canonical conformal structure defined by the symbol of the system being Einstein–Weyl on any solution in three dimensions, and self-dual on any solution in four dimensions. Unfortunately, the characteristic varieties for our examples in general do not appear to be of the form rendering this result applicable to them. Finding out whether suitable examples within our class do exist is an interesting open problem, which is, however, beyond the scope of the present paper.

Authors' contributions

Both authors contributed at all stages of manuscript preparation and gave final approval for publication.

Competing interests

We declare we do not have competing interests.

Funding

The research of A.S. was supported, in part, by the Ministry of Education, Youth and Sports of the Czech Republic (MŠMT ČR) under RVO funding for IČ47813059, and by the Grant Agency of the Czech Republic (GA ČR) under grant P201/12/G028.