We investigate non-homogeneous Hamiltonian operators composed of a first order Dubrovin–Novikov operator and an ultralocal one. The study of such operators turns out to be fundamental for the inverted system of equations associated with a class of Hamiltonian scalar equations. Often, the involved operators are degenerate in the first order term. For this reason, a complete classification of the operators with a degenerate leading coefficient in systems with two and three components is presented.

The Hamiltonian formalism for Partial Differential Equations (PDEs) is one of the leading tools to study nonlinear systems,1,2 following the well-known developed theory for the finite-dimensional ones. As shown in Refs. 3 and 4, Hamiltonian operators link conserved quantities with symmetries of the system, mapping the former onto the latter and then leading to a deeper investigation of the structure of solutions. This formalism represents a strong theoretical and practical connection between geometry and mathematical physics.5–7 Dubrovin and Novikov introduced differential-geometric Poisson brackets as a natural extension of finite-dimensional symplectic structures in traditional Hamiltonian mechanics that turned out to arise in several examples in nonlinear PDEs. The characterization of these structures is related to the (pseudo-)Riemannian geometry and algebraic geometry, especially for systems in 1 + 1 dimensions (or in independent variables x, t).

More in general, geometrical methods are well-established tools widely used to find solutions to systems, such as the generalized hodograph method introduced by Tsarev,8 valid for strictly hyperbolic systems. The Hamiltonian formalism is also used to discuss the integrability. In particular, finding two compatible Hamiltonian structures is strictly related to the existence of infinitely many commuting symmetries and conservation laws, as proved by Magri.9 In the context of hydrodynamic-type systems, an approach to describe integrability is based on the analysis of geometrical elements in the so-called method of hydrodynamic reductions, developed for systems in 2 + 1 dimensions10,11 and then extended to systems in 1 + 1 dimensions with infinitely many components.12

First, we recall some basic notions concerning the Hamiltonian formalism.5 Let us consider a system described by n field variables,
$ui=ui(t,x),i=1,…,n,$
(1)
depending on the independent variables t, x, and let $uσi$ denote the x-derivatives of times. A Hamiltonian operator is a linear operator Aij = aijσDσ such that the associated bracket for functionals f, g,
${f,g}=∫δfδuiAijδgδujdx,$
(2)
is a Poisson bracket, i.e., it is bilinear and skew-symmetric and satisfies the Jacobi identity,
${f,g}=−{g,f},{f,{g,h}}+{g,{h,f}}+{h,{f,g}}=0.$
An evolutionary system
$uti=Fi(x,u,uσ),$
(3)
with $u={ui}i=1n$, is Hamiltonian if it admits the following representation:
$uti=Fi(x,u,uσ)=AijδHδuj,$
(4)
where δ is the variational derivative; H is the Hamiltonian functional H = ∫h(u)dx, written in terms of the Hamiltonian density h; and A is a Hamiltonian operator.
Dubrovin and Novikov6,13 presented a class of Hamiltonian operators, which are homogeneous in the order of derivation and are also known as homogeneous Hamiltonian operators. They proved13 that first order homogeneous operators of the form
$gij∂x+Γkijuxk$
(5)
with det g ≠ 0 are Hamiltonian if and only if
$gij=(gij)−1,Γkij=−gisΓskj,$
(6)
i.e., gij is a flat metric and the coefficients $Γjki$ are Christoffel symbols for the metric tensor g. Operators of this type naturally arise in homogeneous quasilinear systems of first order PDEs, also known as hydrodynamic type systems,
$uti=vji(u)uxj,i=1,…,n,$
(7)
where $v(u)=(vji)1≤i,j≤n$ is the coefficient matrix depending on field variables. Indeed, if (7) is Hamiltonian with a Dubrovin–Novikov operator (5), it can be expressed as
$uti=vji(u)uxj=Aij∂h∂uj=gij∂x+Γkijuxk∂h∂uj=∇i∇jhuxj,$
(8)
where h = h(u) is the hydrodynamic Hamiltonian density and ∇i is the covariant derivative.
In the case of higher order homogeneous operators of degree m, the Dubrovin–Novikov operator generalizes as
$Aij=gijDxm+bkijuxkDxm−1+Ckijuxxk+CklijuxkuxlDxm−2+⋯+dkijunxk+⋯+dk1⋯kmijuxk1⋯uxkm,$
(9)
where the coefficients $bkij,Ckij,…$ depend on field variables. Dubrovin and Novikov also presented an extension of homogeneous structures,6 introducing non-homogeneous Hamiltonian operators as a sum of two or more homogeneous ones. A leading example in this context is offered by the Korteweg–de Vries equation,
$ut=6uux+uxxx,$
(10)
which possesses a Hamiltonian structure through the operator
$A=∂x3+2u∂x+ux,$
(11)
given by the sum of the third order operator $∂x3$ and the first order operator 2u∂x + ux.

Following the notation used by Dubrovin and Novikov, if an operator is given by the sum of two homogeneous operators of orders k and m, respectively, we denote the order of the non-homogeneous operator via the sum k + m.

Let us consider the simplest case, with k = 1 and m = 0 for the so-called non-homogeneous hydrodynamic-type operators 1 + 0. They naturally arise in non-homogeneous quasilinear systems of first-order PDEs. Let Cij = Aij + ωij, where Aij is homogeneous of order 1 and ωij is a symplectic structure of order 0. One can easily generalize (8) to systems of this type,
$uti=gij∂x+Γkijuxk+ωij∂h∂uj=∇i∇jhuxj+∇̃ih,$
(12)
where $∇̃i=ωis∇s$ and ∇s is the standard gradient.
A remarkable example of a non-homogeneous quasilinear system possessing such a construction is given by the three-wave equation,5
$ut1=−c1ux1−2(c2−c3)u2u3,ut2=−c2ux2−2(c1−c3)u1u3,ut3=−c3ux3−2(c2−c1)u1u2,$
(13)
which is a Hamiltonian with operator
$Cij=1000−1000−1∂x+0−2u32u22u302u1−2u2−2u10.$
(14)
Finally, following the approach introduced by Tsarev,14 we will see how non-homogeneous hydrodynamic operators arise in a class of systems obtained by the inversion of an evolutionary Hamiltonian equation (see Theorem IV.1). Often, the inversion of such equations leads to a degeneration of the leading coefficient gij in the first order operator. This is a strong motivation for the investigation of degenerate 1 + 0 structures.
In this paper, we present a complete classification of Hamiltonian operators for systems in two and three components of the form
$Cij=gij∂x+bkijuxk+ωij,$
(15)
focusing on the case when the leading coefficient is degenerate (i.e., its rank is lower than the number of components of the system) and some related remarkable examples of systems in 1 + 1 dimensions exhibiting this feature. The importance of a deeper study of such operators has been remarked not only by Mokhov in Ref. 5, who finds Hamiltonian structures of this type in the study of the real reduction of two-wave interaction system, but also by Dubrovin and Novikon themselves.6

In Sec. II, we introduce conditions for non-homogeneous operators of hydrodynamic type to be Hamiltonian, with either non-degenerate or degenerate assumptions. We establish the connection between such operators and non-homogeneous systems of first order PDEs, introducing the corresponding inverted systems and their associated Hamiltonian structures. In Sec. III, we show a complete classification, up to diffeomorphisms of the manifold defined by field variables, of degenerate operators of type 1 + 0 for systems with two and three components. In Sec. IV, we provide several examples with Hamiltonian structures fitting the above-mentioned classification, with particular emphasis on inverted Hamiltonian systems.

In this section, we review non-homogeneous operators of hydrodynamic type, as originally introduced in Ref. 6 and further investigated in Refs. 15 and 16.

Non-homogeneous operators of hydrodynamic type are introduced as the natural generalization of homogeneous Hamiltonian operators (9),
$Cij=gij∂x+bkijuxk+ωij,$
(16)
where $gij,bkij$, and ωij depend on field variables u. Then, the underlying non-homogeneous local Poisson structure of hydrodynamic type is defined as
${ui(x),uj(y)}=gij(u(x))δx(x−y)+bkij(u(x))uxkδ(x−y)+ωij(u(x))δ(x−y).$
(17)
Note that operators of type 1 + 0 are composed of two homogeneous operators $Aij=gij∂x+bkijuxk$ of order 1 and ωij of order 0. Conditions for Cij to be Hamiltonian are given by the constraints for each of its homogeneous part to be Hamiltonian (Theorems II.2 and II.1, respectively) and an additional compatibility condition between the two (Theorem II.3).

We recall that operators of order zero, also known as ultralocal operators, are Hamiltonian if the following conditions are satisfied.

Theorem II.1
(Ref. 5). The operator ωij(u) is Hamiltonian if and only if it forms a finite-dimensional Poisson structure, i.e., it satisfies the following conditions:
$ωij(u)=−ωji(u),$
(18)
$ωis∂ωjk∂us+ωjs∂ωki∂us+ωks∂ωij∂us=0.$
(19)

We remark that in the non-degenerate case, i.e., det ωij ≠ 0, conditions (18) and (19) are, respectively, skew-symmetry and closedness of the two-form ω.

In the case of operators of first order, the following result holds.

Theorem II.2
(Ref. 5). The operator Aij is Hamiltonian if and only if
$gij=gji,$
(20)
$∂gij∂uk=bkij+bkji,$
(21)
$gisbsjk−gjsbsik=0,$
(22)
$gis∂bsjr∂uk−∂bkjr∂us+bsijbksr−bsirbksj=0,$
(23)
$gis∂bqjr∂us−bsijbqsr−bsirbqjs=gjs∂bqir∂us−bsjibqsr−bqisbsjr,$
(24)
and
$∑(q,k)∂∂uqgis∂bsjr∂uk−∂bkjr∂us+bsijbksr−bsirbksj+∑(i,j,k)bqsi∂bkjr∂us−∂bsjr∂uk=0,$
(25)
with the sum over (q, k) and (i, j, k) on cyclic permutations of the indices.

Let us remark that here there is no assumption about non-degeneracy properties of metric. The conditions for non-homogeneous operators of hydrodynamic type to be Hamiltonian are shown in the following theorem.

Theorem II.3
(Refs. 5 and 15). Operator (16) is Hamiltonian if and only if $gij∂x+bkijuxk$ is Hamiltonian, ωij is Hamiltonian, and the compatibility conditions are satisfied,
$Φijk=Φkij,$
(26)
$∂Φijk∂ur=∑(i,j,k)brsi∂ωjk∂us+∂brij∂us−∂bsij∂urωsk,$
(27)
where Φijk is the (3, 0)-tensor,
$Φijk=gis∂ωjk∂us−bsijωsk−bsikωjs.$
(28)

Savoldi17 presented a complete classification of degenerate first order homogeneous operators for systems with two and three components. Starting from these results, in this section, we provide a novel complete classification of degenerate operators of type 1 + 0. To obtain an explicit form of ωij by means of Theorem II.3, it is sufficient to solve conditions (26) and (27) with fixed tensors gij and $bkij$, giving rise to an overdetermined system of PDEs. In addition, we require the ultralocal operator ωij to be Hamiltonian imposing (18) and (19) via Theorem II.1. In the  Appendix, we report the details of computations.

The following computations are carried out with the support of computer algebra methods, implemented in Maple, Reduce, and Mathematica. The use of symbolic computation for integrable systems and Hamiltonian structures is itself an ongoing topic of research.4,18

Let us consider systems with two components, with field variables u, v. In general, given n, the number of components of the hydrodynamic system, in the degenerate case, the operator gij can be classified by its rank, with $rankgij=m. In the following, we explicit the number of components n for the operator $Cn,kij$, while the index k is used to distinguish between different operators.

For n = 2, rank(gij) is in {0, 1}. The only solution for the case rank(gij) = 0 is the trivial one; then, the operator reduces to a symplectic form. In the case rank(gij) = 1, we can construct two different operators,
$C2,1ij=∂x000+0f(v)−f(v)0,$
(29)
$C2,2ij=∂x000+0−vxuvxu0+0f(v)u−f(v)u0,$
(30)
where f(v) is an arbitrary function depending only on the variable v.

Theorem III.1.

Every degenerate operator of type 1 + 0 in two components can be mapped either onto an ultralocal Hamiltonian operator or onto one between $C2,1ij$ and $C2,2ij$.

Proof.

Considering Theorem II.3, we compute the symplectic structure satisfying (26) and (27) for each degenerate operator of the classification introduced by Savoldi in two components.□

Let us consider the case of systems with three components u, v, w, for which the degenerate metric has $rankgij$ in {0, 1, 2}. We denote with f, g, h, l arbitrary functions, specifying the explicit dependence on the variables, and with c arbitrary constants.

• $rankgij=0$:
$C3,1ij=0wx0−wx00000+0f(u,v,w)0−f(u,v,w)00000.$
(31)
• $rankgij=1$:
$C3,2ij=∂x00000000+0f(v,w)g(v,w)−f(v,w)0h(v,w)−g(v,w)−h(v,w)0.$
(32)
Here, the function f(v, w) is expressed in terms of the functions h(v, w) and g(v, w) as
$f(v,w)=h(v,w)l(w)+∫1vg(s,w)∂wh(s,w)−h(s,w)∂wg(s,w)h(s,w)2ds,$
(33)
$C3,3ij=∂x00000000+0wx0−wx00000+0f(v,w)0−f(v,w)00000,$
(34)
$C3,4ij=∂x00000000+00−wxu000wxu00+00f(v,w)u000−f(v,w)u00,$
(35)
$C3,5ij=∂x00000000+0−vxu−wxuvxu00wxu00+0f(v,w)ug(v,w)u−f(v,w)u0h(v,w)u−g(v,w)u−h(v,w)u0,$
(36)
with f(v, w) given in (33).
• $rankgij=2$:
$C3,6ij=∂x000∂x0000+0f(w)g(w)−f(w)0cg(w)−g(w)−cg(w)0,$
(37)
$C3,7ij=∂x000∂x0000+00000−wxv0wxv0+00cf(w)00(1−cu)f(w)v−cf(w)−(1−cu)f(w)v0,$
(38)
$C3,8ij=∂x000∂x0000+00−wwxuw−v00wxuw−vwwxuw−v−wxuw−v0+(1+w2)f(w)01(1+w2)w−cv1+w2uw−v−1(1+w2)0−1−cu1+w2uw−v−w−cv1+w2uw−v1−cu1+w2uw−v0,$
(39)
$C3,9ij=0∂x0∂x00000+0f(w)cg(w)−f(w)0g(w)−cg(w)−g(w)0,$
(40)
$C3,10ij=0∂x0∂x00000+00−wxv000wxv00+0f(w)h(w)−ug(w)v−f(w)0g(w)−h(w)−ug(w)v−g(w)0,$
(41)
$h(w)g′(w)−g(w)f(w)+h′(w)=0,$
(42)
$C3,11ij=0∂x0∂x00000+00wxuw−v00−wwxuw−v−wxuw−vwwxuw−v0+f(w)0cwuw−2cwuw−v−cw0−wv−2cwuw−v−uw−2cwuw−vwwv−2cwuw−v0.$
(43)

Remark III.1.

Condition (42) can be explicitly solved with respect to any function among f, g, and h.

Theorem III.2.

Every degenerate operator of type 1 + 0 in three components can be mapped either onto an ultralocal operator satisfying the closure relation or onto one among $C3,kij$ with k = 1, …, 11.

Proof.

Imposing the conditions on the operators to be Hamiltonian, we obtain the extension of the classification for degenerate first order operators presented by Savoldi in three components.17 See the  Appendix for more details.□

Remark III.2.

In the proposed classification, we have considered three arbitrary functions for the sake of generality and in view of possible relevance for applications. However, we emphasize that changes of variables can simplify the form of operators. To do so, one should look for those changes of variables that leave the order 1 operator invariant and then apply them to the order 0 one.

In this section, we present some examples of non-homogeneous quasilinear systems with a degenerate Hamiltonian structure of order 1 + 0 in two and three components.

Example IV.1
(two-wave interaction system). Mokhov5 studied the real reduction of the two-wave interaction system formulated in terms of the system of hydrodynamic equations in two field variables u = u(x, t) and v = v(x, t),
$ut=auv,vt=avx+u2,$
(44)
with a being a constant. The system admits a Hamiltonian formulation, with the operator
$Cij=000∂x+0−uu0,$
(45)
and the Hamiltonian functional
$H=12∫av2−u2dx.$
(46)
The 1 + 0 operator (45) is degenerate since the rank of the order 1 term is lower than the number of components of the system. Moreover, by applying the exchange uv, it is evident that the operator found by Mokhov is of type $C2,1ij$ in (29).

Example IV.2
(Sinh–Gordon equation). Let us consider the Sinh–Gordon equation,
$φτξ=sinhφ.$
(47)
Applying the change of variables φ = 2 logu, we have
$2uτuξ=12u2−1u2.$
(48)
Introducing v = 2uτ/u and considering the light-cone coordinates τ = t, ξ = tx,
$ut=12uv,vt=vx+12u2−1u2,$
(49)
we show that the system is Hamiltonian with the non-homogeneous hydrodynamic operator of shape (29) with the exchange of variables uv and the function f(u) = u/2,
$Cij=000∂x+120u−u0.$
(50)
The corresponding Hamiltonian density is
$h(u,v)=12v2−u2+1u2.$
(51)

In this section, we show the connection between degenerate operators of type 1 + 0 and scalar equations possessing a local Hamiltonian structure.

Let us briefly recall that the momentum of a Hamiltonian equation ut = AijδH/δuj is a functional defining a x-translation,
$uxi=AijδPδuj, with P=∫p(u,uσ)dx,$
for i = 1, …, n. Tsarev14 proved that under the inversion of the independent variables x and t, the Hamiltonian property is preserved by the system. It is well known that the momentum is a conserved quantity in a Hamiltonian system; hence, there exists q(u, uσ) such that pt = qx. Then, one can choose H′ = q(u, uσ)dt as the Hamiltonian functional of the inverted system.
Non-homogeneous operators of hydrodynamic type are related to the study of scalar evolutionary equations possessing a local Hamiltonian structure. Indeed, by introducing the new set of variables,
$u1=u,u2=ux,u3=uxx,…,$
(52)
it is in some cases possible to write an equivalent non-homogeneous quasilinear system that can be seen as evolutionary with respect to the independent variable x, obtaining the inverted system.

Remark IV.1.
Let us observe that every invertible system of order k has the form
$ut=F1(u,ux,…,u(k−1)x)+F2(u,ux,…,u(k−1)x)ukx,$
(53)
where F1, F2 are arbitrary functions. Note that this is the case of Korteweg–de Vries (KdV) and many other examples in nonlinear phenomena. Indeed, considering the lower derivatives as parameters, we need the system to be linear in ukx in order to conserve linearity in ut once inverted.

The following result offers an explicit connection between non-homogeneous hydrodynamic operators and inverted systems.

Proposition IV.1.

Let us consider the evolutionary equation ut = F(u, uσ) endowed with a local Hamiltonian structure and a momentum density p in Sec. IV A depending on u only. Then, if the inverted system in the set of variables (52) admits a local Hamiltonian structure, this is given in terms of a non-homogeneous operator of hydrodynamic type.

Proof.
We observe the following:
$qx=pt=pu(u)ut=pu(u)F(u,uσ),σ≤k,$
(54)
where pt is of order $≤k$, at most equal to the order of the equation, and so is qx. Hence, q(u, uσ) is of order at most k − 1. This implies that the Hamiltonian H′ = ∫q(u1, …, uk−1)dt is of hydrodynamic type for the inverted system in the new variables. In Refs. 14 and 19, it was proved that the Hamiltonian property is preserved after a change of dependent variables and an inversion of t and x. Then, the inverted system is quasilinear of first order and already Hamiltonian. The operator Bij in
$uxi=BijδH′δuj,$
(55)
being local, must be of type 1 + 0, i.e., a non-homogeneous operator of hydrodynamic type.□

Proposition IV.1 justifies a deeper investigation of such operators, for which KdV offers a leading example, as follows: We emphasize that the previous theorem does not guarantee that the operator is, in general, non-degenerate.

Example IV.3
(KdV equation-I). Let us consider the KdV equation,
$ut=6uux+uxxx,$
(56)
which is widely known to be Hamiltonian. Inverting the equation, we obtain the evolutionary system with respect to x in three components u1(x, t), u2(x, t), and u3(x, t) defined as u = u1, ux = u2, uxx = u3, yielding the following non-homogeneous system of hydrodynamic type:
$ux1=u2,ux2=u3,ux3=ut1+6u1u2.$
(57)
This system is Hamiltonian with the following non-homogeneous hydrodynamic-type operator:14
$Cij=000000001∂t+010−106u10−6u10,$
(58)
with the leading coefficient gij being degenerate. It is easy to see that applying the change of variables u1 = w, we again obtain operator (32), where
$g(v,w)=0,f(v,w)=h(v,w)l(w),l(w)=6w,h(v,w)=−1.$

Example IV.4
(KdV equation-II). Mokhov20 found a transformation of variables (also known as local quadratic unimodular change),
$u1=w1−w32,u2=w2,u3=w1+w32+w1−w32,$
(59)
such that the KdV equation reads as
$wx1=−12w1−w3t+w2w1−w3+12w2,wx2=w1−w32+12w1+w3,wx3=−12w1−w3t+w2w1−w3−12w2.$
(60)
After this local change, the KdV is a bi-Hamiltonian system with respect to two non-homogeneous operators 1 + 0 of hydrodynamic type, one of these being the operator
$Cij=12101000101∂t+0w1−w3+120w3−w1−120w3−w1+120w1−w3−120,$
(61)
which is degenerate, since rank(gij) = 1. The Hamiltonian given in terms of the new variables is
$H=∫(w1)2−(w2)2−(w3)2dx.$
(62)
To show that the obtained operator is indeed one of those classified above, we consider a new change of variables,
$w1=ū1−ū32,w2=ū2,w3=ū3−ū12.$
(63)
The degenerate first order operator is written with the leading coefficient $ḡ=dū1⊗dū1$ and the skew-symmetric bivector,
$ω̄=−2ū3dū1∧dū2−dū2∧dū3.$
Operator (61) is of type $C2,2ij$ in three components showed in (32). In particular,
$g(v,w)=0,f(v,w)=l(w)h(v,w),l(w)=−1,h(v,w)=2w.$

Example IV.5
(generalized KdV equation). Let us consider the generalized KdV equation,
$ut+3(n+1)unux+uxxx=0,$
(64)
where n is a positive integer. It is known that (64) is Hamiltonian with the operator x for any n. The case n = 2 corresponds to the modified KdV equation (mKdV); it is integrable, and it has a second Hamiltonian structure, with operator $∂x3+6∂xu∂x−1u∂x$. The Hamiltonians associated with mKdV are
$H1=∫34u4+12ux2dx,$
(65)
$H2=∫12u2dx.$
(66)
In (64), we introduce the variables u1 = u, u2 = ux, u3 = uxx so that the equation reads as a quasilinear system of first order PDEs,
$ux1=u2,ux2=u3,ux3=−ut1−3(n+1)(u1)nu2.$
(67)
The Hamiltonian structure is still conserved after the scalar equation is transformed into a system, i.e., (67) has a Hamiltonian structure with the operator
$Cij=000000001∂t+010−10−3(n+1)(u1)n−103(n+1)(u1)n−10$
(68)
and the Hamiltonian functional
$H=∫3(u1)n+1−u1u3+(u2)22dx.$
(69)
Operator (68) is $C3,2ij$ in (32) with the exchange u1u3 and
$g(u1,u2)=0,f(u1,u2)=h(u1,u2)l(u1),l(u1)=3(n+1)(u1)n−1,h(u1,u2)=−1.$

Remark IV.2.

Let us observe that for n > 2, the generalized KdV equation is not integrable, even if it is Hamiltonian as proved in the previous example. We emphasize that this feature is more general than the integrability property.

We finally present two examples violating the hypothesis of locality, either in terms of the momentum or in terms of the operator defined for the inverted Hamiltonian structure.

Example IV.6.
We consider the linearized KdV equation,
$ut=uxxx,$
(70)
for which the inverted system is easily given in the new variables by
$ux1=u2,ux2=u3,ux3=ut1.$
(71)
The associated momentum is given in terms of the density $p(u)=∂x−2u$. Here, again, it is not possible to write the resulting system with Hamiltonian operator of type 1 + 0.14

Example IV.7.
We consider the Harry–Dym equation,16,21
$ut=1uxxx=−158u−7/2ux3+94u−5/2uxuxx−12u−3/2uxxx,$
(72)
$ut=A1δH1δu=−12∂x3δH1δu,$
(73)
$ut=A2δH2δu=−2u∂x−uxδH2δu,$
(74)
where
$H1=−∫4udx,$
(75)
$H2=−∫1532u−7/2ux−116u−5/2uxxdx.$
(76)
Introducing the variables ux = u2, uxx = u3, the inverted system is
$ux1=u2,ux2=u3,ux3=−2(u1)3/2ut1−154(u1)−2(u2)3+92u1u2u3.$
(77)
The momentum P associated with the operator A2 is
$ux=−2u∂x−uxδPδu,P=∫p(u)dx=−∫udx,$
and following the above procedure, the Hamiltonian H′ as a functional in the new variables is
$H′=−∫34(u1)−5/2(u2)2−12(u1)−3/2u3dx.$
(78)
With this Hamiltonian, it is not possible to build a local operator of the form 1 + 0 for the inverted system; hence, this operator will be non-local.

The study of non-homogeneous quasilinear systems of first order PDEs is an ongoing research topic in integrable systems and Hamiltonian PDEs. To the authors’ knowledge, a general criterion to discuss the integrability for these kinds of systems is not currently known, unlike the homogeneous systems.8 This paper represents a first step toward the investigation of integrability of non-homogeneous systems, focusing on the Hamiltonian property. The study of possible bi-Hamiltonian structures associated with these types will be the subject of a future paper. Non-homogeneous operators of order k + m play an important role in nonlinear phenomena, and their investigation represents another interesting topic.16,22–25 Even in the simplest case where k = 1 and m = 0, we showed how the conditions for the operator to be Hamiltonian lead to a specific form, this being exactly solvable. Higher order operators require a more general study, especially for what concerns the degenerate case.

As future perspectives, the authors emphasize not only the necessity to further investigate the integrability of non-homogeneous quasilinear systems and the compatibility conditions between systems and operators in the sense of Ref. 26 but also their associated geometric structure, following the lead of the homogeneous case, where both operators and systems are linked to projective algebraic geometry27–30 and differential Riemannian geometry. Finally, the discrete analogs of non-homogeneous operators were introduced by Dubrovin in Ref. 31, letting the classification method described in this paper suitable for discrete operators as well.

The authors thank C. Benassi, F. Coppini, E. V. Ferapontov, A. Moro, M. Pavlov, and R. Vitolo for stimulating discussions and interesting remarks. P.V. also acknowledges the financial support of GNFM of the Istituto Nazionale di Alta Matematica, of PRIN 2017 “Multiscale phenomena in Continuum Mechanics: singular limits, off-equilibrium and transitions,” Project No. 2017YBKNCE, and the research project Mathematical Methods in Non-Linear Physics (MMNLP) by the Commissione Scientifica Nazionale—Gruppo 4—Fisica Teorica of the Istituto Nazionale di Fisica Nucleare (INFN). M.D. acknowledges the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the program “Dispersive hydrodynamics: mathematics, simulation, and experiments, with applications in nonlinear waves” (HDY2) where work on this paper was undertaken. This work was supported by EPSRC under Grant No. EP/R014604/1.

The authors have no conflicts to disclose.

Marta Dell’Atti: Conceptualization (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Pierandrea Vergallo: Conceptualization (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

We give the details of the procedure followed to compute the classifications of Sec. III. The computations are carried out with the support of computer algebra systems (Maple, Reduce, and Mathematica) and finally checked by hand.

For the sake of simplicity, we describe the first nontrivial operator obtained in three components, in the text identified as C3,2. We start by considering the degenerate operator of order 1 for a system in three components in Savoldi’s classification,17
$gij=∂x00000000,bkij=000000000∀k∈{1,2,3}.$
(A1)
We add to the operator an order 0 operator,
$∂x00000000+ω11ω12ω13ω21ω22ω23ω31ω32ω33.$
(A2)
Being ω an ultralocal tail, its elements ωij are functions at most depending on the three variables of the system u, v, w.
Operator (A2) as a whole is Hamiltonian if its parts are Hamiltonian and the compatibility conditions established in Theorem II.3 are satisfied. In particular, the ultralocal term ω is Hamiltonian if it fulfills Theorem II.1. We reduce the number of free functions in (A2) by using the skew-symmetry property,
$∂x00000000+0ω12ω13−ω120ω23−ω13−ω230,$
(A3)
so that the unknown functions are
$ω12(u,v,w),ω13(u,v,w),ω23(u,v,w).$
(A4)
To implement the constraints expressed in Theorem II.3, we evaluate the tensor Φijk for case (A1),
$Φijk=gis∂ωjk∂us−bsijωsk−bsikωjs=gis∂ωjk∂us,$
(A5)
where the sum over s is intended via repeated indices. For three components, s = 1, 2, 3 and we use the notation for the variables u1 = u, u2 = v, and u3 = w. The only nonzero element in g is g11 = 1; hence, constraint (26) on the non-zero elements of the tensor Φijk takes the form Φ1jk = Φk1j, with
$Φ1jk=0∂ω12∂u∂ω13∂u−∂ω12∂u0∂ω23∂u−∂ω13∂u−∂ω23∂u0,$
(A6)
$Φk1j=000∂ω12∂u00∂ω13∂u00.$
(A7)
The constraints on the field variables are
$∂ωij(u,v,w)∂u=0⇒ωij(u,v,w)=ωij(v,w),$
(A8)
i.e., they do not depend on the variable u. We introduce the notation
$ω12(v,w)=f(v,w),ω13(v,w)=g(v,w),ω23(v,w)=h(v,w).$
(A9)
Constraint (27) yields
$∂2ωij∂u2=0,∂2ωij∂u∂v=0,∂2ωij∂u∂w=0,$
(A10)
not producing any further restriction for the form of the functions, given (A8). Finally, the closure requirement (19) is
$ω12∂ω23∂v−ω23∂ω12∂v+ω13∂ω23∂w−ω23∂ω13∂w=0.$
(A11)
With notation (A9), this becomes
$f(v,w)∂h(v,w)∂v−h(v,w)∂f(v,w)∂v+g(v,w)∂h(v,w)∂w−h(v,w)∂g(v,w)∂w=0.$
(A12)
We solve the last constraint with respect to the field f(v, w). Observing that
$∂∂vfh=1h∂f∂v−fh2∂h∂v,$
(A13)
we obtain the expression for f(v, w) given in (33). The operator is then
$C3,2ij=∂xf(v,w)g(v,w)−f(v,w)0h(v,w)−g(v,w)−h(v,w)0.$
We remark that for fixed rank of gij, the resulting conditions strongly depend on the structure of $bkij$. For instance, for the operator $C3,3ij$, we have the same operator gij as in (A1), but different $bkij$,
$b1ij=b2ij=000000000,b3ij=010−100000.$
(A14)
We look for the corresponding operator ω after considering the skew-symmetry property; hence, we have three field variables ω12, ω13, ω23. The first conditions are imposed by comparing the tensors
$Φ1jk=0−ω13+∂ω12∂u∂ω13∂uω13−∂ω12∂u0∂ω23∂u−∂ω13∂u−∂ω23∂u0,$
(A15)
$Φk1j=000−ω13+∂ω12∂uω230∂ω13∂u00.$
(A16)
At this stage, we can already reduce the number of free functions since ω23 = 0. Solving the remaining equations, we find the dependence of the fields ω12, ω13 on the variables u, v, w,
$ω12(u,v,w)=f(v,w)+ug(v,w),ω13(u,v,w)=g(v,w).$
(A17)
The constraints from relation (19) further reduce the number of free functions; in particular, g(v, w) = 0. The corresponding operator is then
$C3,3ij=∂xwx+f(v,w)0−wx−f(v,w)00000.$
The same procedure has been carried out for all the possible forms of operators gij and $bkij$, obtaining the above-mentioned classification.
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