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.

## I. INTRODUCTION

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 dimensions^{10,11} and then extended to systems in 1 + 1 dimensions with infinitely many components.^{12}

^{5}Let us consider a system described by

*n*field variables,

*t*,

*x*, and let $u\sigma i$ denote the

*x*-derivatives of

*uσ*times. A Hamiltonian operator is a linear operator

*A*

^{ij}=

*a*

^{ijσ}

*D*

_{σ}such that the associated bracket for functionals

*f*,

*g*,

*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.

^{6,13}presented a class of Hamiltonian operators, which are homogeneous in the order of derivation and are also known as

*homogeneous Hamiltonian operators*. They proved

^{13}that first order homogeneous operators of the form

*g*≠ 0 are Hamiltonian if and only if

*g*

^{ij}is a flat metric and the coefficients $\Gamma jki$ 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*,

*h*=

*h*(

*u*) is the hydrodynamic Hamiltonian density and ∇

_{i}is the covariant derivative.

*m*, the Dubrovin–Novikov operator generalizes as

^{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,

*u∂*

_{x}+

*u*

_{x}.

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*.

*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

*C*

^{ij}=

*A*

^{ij}+

*ω*

^{ij}, where

*A*

^{ij}is homogeneous of order 1 and

*ω*

^{ij}is a symplectic structure of order 0. One can easily generalize (8) to systems of this type,

_{s}is the standard gradient.

^{5}

^{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

*g*

^{ij}in the first order operator. This is a strong motivation for the investigation of degenerate 1 + 0 structures.

^{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.

## II. NON-HOMOGENEOUS HYDRODYNAMIC OPERATORS

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.

*ω*

^{ij}depend on field variables

*u*. Then, the underlying non-homogeneous local Poisson structure of hydrodynamic type is defined as

*ω*

^{ij}of order 0. Conditions for

*C*

^{ij}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.

*ω*

^{ij}(

*u*) is Hamiltonian if and only if it forms a finite-dimensional Poisson structure, i.e., it satisfies the following conditions:

We remark that in the non-degenerate case, i.e., det *ω*^{ij} ≠ 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.

*A*

^{ij}is Hamiltonian if and only if

*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.

*ω*

^{ij}is Hamiltonian, and the compatibility conditions are satisfied,

^{ijk}is the (3, 0)-tensor,

## III. CLASSIFICATION FOR SYSTEMS IN TWO AND THREE COMPONENTS

Savoldi^{17} 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 *ω*^{ij} by means of Theorem II.3, it is sufficient to solve conditions (26) and (27) with fixed tensors *g*^{ij} and $bkij$, giving rise to an overdetermined system of PDEs. In addition, we require the ultralocal operator *ω*^{ij} 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}

### A. Systems in *n* = 2 components

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 *g*^{ij} can be classified by its rank, with $rankgij=m<n$. 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.

*n*= 2, rank(

*g*

^{ij}) is in {0, 1}. The only solution for the case rank(

*g*

^{ij}) = 0 is the trivial one; then, the operator reduces to a symplectic form. In the case rank(

*g*

^{ij}) = 1, we can construct two different operators,

*f*(

*v*) is an arbitrary function depending only on the variable

*v*.

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 $C2,2ij$.

### B. Systems in *n* = 3 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.

- $rankgij=0$:(31)$C3,1ij=0wx0\u2212wx00000+0f(u,v,w)0\u2212f(u,v,w)00000.$
- $rankgij=1$:Here, the function(32)$C3,2ij=\u2202x00000000+0f(v,w)g(v,w)\u2212f(v,w)0h(v,w)\u2212g(v,w)\u2212h(v,w)0.$
*f*(*v*,*w*) is expressed in terms of the functions*h*(*v*,*w*) and*g*(*v*,*w*) as(33)$f(v,w)=h(v,w)l(w)+\u222b1vg(s,w)\u2202wh(s,w)\u2212h(s,w)\u2202wg(s,w)h(s,w)2ds,$(34)$C3,3ij=\u2202x00000000+0wx0\u2212wx00000+0f(v,w)0\u2212f(v,w)00000,$(35)$C3,4ij=\u2202x00000000+00\u2212wxu000wxu00+00f(v,w)u000\u2212f(v,w)u00,$with(36)$C3,5ij=\u2202x00000000+0\u2212vxu\u2212wxuvxu00wxu00+0f(v,w)ug(v,w)u\u2212f(v,w)u0h(v,w)u\u2212g(v,w)u\u2212h(v,w)u0,$*f*(*v*,*w*) given in (33). - $rankgij=2$:(37)$C3,6ij=\u2202x000\u2202x0000+0f(w)g(w)\u2212f(w)0cg(w)\u2212g(w)\u2212cg(w)0,$(38)$C3,7ij=\u2202x000\u2202x0000+00000\u2212wxv0wxv0+00cf(w)00(1\u2212cu)f(w)v\u2212cf(w)\u2212(1\u2212cu)f(w)v0,$(39)$C3,8ij=\u2202x000\u2202x0000+00\u2212wwxuw\u2212v00wxuw\u2212vwwxuw\u2212v\u2212wxuw\u2212v0+(1+w2)f(w)01(1+w2)w\u2212cv1+w2uw\u2212v\u22121(1+w2)0\u22121\u2212cu1+w2uw\u2212v\u2212w\u2212cv1+w2uw\u2212v1\u2212cu1+w2uw\u2212v0,$(40)$C3,9ij=0\u2202x0\u2202x00000+0f(w)cg(w)\u2212f(w)0g(w)\u2212cg(w)\u2212g(w)0,$with the additional condition(41)$C3,10ij=0\u2202x0\u2202x00000+00\u2212wxv000wxv00+0f(w)h(w)\u2212ug(w)v\u2212f(w)0g(w)\u2212h(w)\u2212ug(w)v\u2212g(w)0,$(42)$h(w)g\u2032(w)\u2212g(w)f(w)+h\u2032(w)=0,$(43)$C3,11ij=0\u2202x0\u2202x00000+00wxuw\u2212v00\u2212wwxuw\u2212v\u2212wxuw\u2212vwwxuw\u2212v0+f(w)0cwuw\u22122cwuw\u2212v\u2212cw0\u2212wv\u22122cwuw\u2212v\u2212uw\u22122cwuw\u2212vwwv\u22122cwuw\u2212v0.$

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

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.

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.

## IV. APPLICATIONS

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.

*(two-wave interaction system).*Mokhov

^{5}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*),

*a*being a constant. The system admits a Hamiltonian formulation, with the operator

*u*↔

*v*, it is evident that the operator found by Mokhov is of type $C2,1ij$ in (29).

*(Sinh–Gordon equation)*. Let us consider the Sinh–Gordon equation,

*φ*= 2 log

*u*, we have

*v*= 2

*u*

_{τ}/

*u*and considering the light-cone coordinates

*τ*=

*t*,

*ξ*=

*t*−

*x*,

*u*↔

*v*and the function

*f*(

*u*) =

*u*/2,

### A. Inverted Hamiltonian systems

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

*u*

_{t}=

*A*

^{ij}

*δH*/

*δu*

^{j}is a functional defining a

*x*-translation,

*i*= 1, …,

*n*. Tsarev

^{14}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*,

*u*

_{σ}) such that

*p*

_{t}=

*q*

_{x}. Then, one can choose

*H*′ =

*∫*

*q*(

*u*,

*u*

_{σ})

*dt*as the Hamiltonian functional of the inverted system.

*local*Hamiltonian structure. Indeed, by introducing the new set of variables,

*x*, obtaining the

*inverted system*.

*k*has the form

*F*

_{1},

*F*

_{2}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

*u*

_{kx}in order to conserve linearity in

*u*

_{t}once inverted.

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

Let us consider the evolutionary equation *u*_{t} = *F*(*u*, *u*_{σ}) 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.

*p*

_{t}is of order $\u2264k$, at most equal to the order of the equation, and so is

*q*

_{x}. Hence,

*q*(

*u*,

*u*

_{σ}) is of order at most

*k*− 1. This implies that the Hamiltonian

*H*′ =

*∫q*(

*u*

^{1}, …,

*u*

^{k−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

*B*

^{ij}in

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.

*(KdV equation-I)*. Let us consider the KdV equation,

*x*in three components

*u*

^{1}(

*x*,

*t*),

*u*

^{2}(

*x*,

*t*), and

*u*

^{3}(

*x*,

*t*) defined as

*u*=

*u*

^{1},

*u*

_{x}=

*u*

^{2},

*u*

_{xx}=

*u*

^{3}, yielding the following non-homogeneous system of hydrodynamic type:

^{14}

*g*

^{ij}being degenerate. It is easy to see that applying the change of variables

*u*

^{1}=

*w*, we again obtain operator (32), where

*(KdV equation-II)*. Mokhov

^{20}found a transformation of variables (also known as

*local quadratic unimodular change*),

*g*

^{ij}) = 1. The Hamiltonian given in terms of the new variables is

*(generalized KdV equation)*. Let us consider the generalized KdV equation,

*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 $\u2202x3+6\u2202xu\u2202x\u22121u\u2202x$. The Hamiltonians associated with mKdV are

*u*

^{1}=

*u*,

*u*

^{2}=

*u*

_{x},

*u*

^{3}=

*u*

_{xx}so that the equation reads as a quasilinear system of first order PDEs,

*u*

^{1}↔

*u*

^{3}and

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.

^{14}

^{16,21}

*u*

_{x}=

*u*

^{2},

*u*

_{xx}=

*u*

^{3}, the inverted system is

*P*associated with the operator

*A*

_{2}is

*H*′ as a functional in the new variables is

## V. CONCLUSIONS

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 geometry^{27–30} 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.

## ACKNOWLEDGMENTS

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.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**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 AVAILABILITY

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

### APPENDIX: DERIVATION OF THE OPERATORS IN THEOREM III.1 AND THEOREM III.2

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.

*C*

_{3,2}. We start by considering the degenerate operator of order 1 for a system in three components in Savoldi’s classification,

^{17}

*ω*an ultralocal tail, its elements

*ω*

^{ij}are functions at most depending on the three variables of the system

*u*,

*v*,

*w*.

*ω*is Hamiltonian if it fulfills Theorem II.1. We reduce the number of free functions in (A2) by using the skew-symmetry property,

^{ijk}for case (A1),

*s*is intended via repeated indices. For three components,

*s*= 1, 2, 3 and we use the notation for the variables

*u*

^{1}=

*u*,

*u*

^{2}=

*v*, and

*u*

^{3}=

*w*. The only nonzero element in

*g*is

*g*

^{11}= 1; hence, constraint (26) on the non-zero elements of the tensor Φ

^{ijk}takes the form Φ

^{1jk}= Φ

^{k1j}, with

*u*. We introduce the notation

*f*(

*v*,

*w*). Observing that

*f*(

*v*,

*w*) given in (33). The operator is then

*g*

^{ij}, the resulting conditions strongly depend on the structure of $bkij$. For instance, for the operator $C3,3ij$, we have the same operator

*g*

^{ij}as in (A1), but different $bkij$,

*ω*after considering the skew-symmetry property; hence, we have three field variables

*ω*

^{12},

*ω*

^{13},

*ω*

^{23}. The first conditions are imposed by comparing the tensors

*ω*

^{23}= 0. Solving the remaining equations, we find the dependence of the fields

*ω*

^{12},

*ω*

^{13}on the variables

*u*,

*v*,

*w*,

*g*(

*v*,

*w*) = 0. The corresponding operator is then

*g*

^{ij}and $bkij$, obtaining the above-mentioned classification.

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