#  Darboux integrability for diagonal systems of hydrodynamic type

Sergey I. Agafonov

Department of Mathematics,
São Paulo State University-UNESP,
São José do Rio Preto, Brazil
e-mail<EMAIL_ADDRESS>
###### Abstract

We prove that diagonal systems of hydrodynamic type are Darboux integrabe if
and only if the Laplace transformation sequences of the system for commuting
flows terminate, give geometric interpretation for Darboux integrability of
such systems in terms of congruences of lines and in terms of solution orbits
with respect to symmetry subalgebras, show that Darboux integrable systems are
necessarily semihamiltonian, and discuss known and new examples.

MSC: 35A30, 35L65, 53A25

Keywords: Darboux integrability, Laplace transformation, congruence of lines.

##  1 Introduction

The notion of integrability for partial differential equations evolved
dramatically over time: from ”explicit” solutions, like the d’Alambert formula
for one-dimensional wave equation, to highly sophisticated techniques such as
inverse scattaring transform for KdV and Schrödinger equations. One of the
classical approaches to integrability, worked out by Monge with ”intermediate
integrals” and further developed by Darboux, to deal with scalar hyperbolic
equations of 2nd order with two independent variables, is easily adaptable to
a larger class of systems. In this paper, we apply this approach to diagonal
systems of hydrodynamic type

$u^{i}_{t}=\lambda^{i}(u)u^{i}_{x},\ \ \ i=1,...,n,\ \ \ \mbox{(no
summation).}$ (1)

The Darboux method has its roots in the geometric theory of partial
differential equations [S-00, IL-03], whose foundations were laid by S.Lie,
E.Vessiot, and É. Cartan. It works for hyperbolic equations such that each its
Monge characteristic distribution has at least two independent Riemann
invariants.

System (1) is automatically hyperbolic and each its Monge distribution has at
least one Riemann invariant, namely $u^{i}$.

###### Definition 1

System (1) is Darboux integrable if each its Monge characteristic distribution
admits at least one extra Riemann invariant.

If the system is Darboux integrable then solving it reduces to integration of
a complete exterior differential system, i.e. to integration of ordinary
differential equations.

It turns out that the Darboux integrability criterion, obtained in the next
section, can be written in terms of the functions

$a_{ij}=\frac{\partial_{j}\lambda^{i}}{\lambda^{j}-\lambda^{i}},$ (2)

(where we use the notation $\partial_{i}=\partial_{u^{i}}$) and their
derivatives. In particular, Darboux integrable systems are necessary
semihamiltonian, i.e.

$\partial_{j}a_{ik}=\partial_{k}a_{ij},\ \ \ \ \forall\ i\neq k\neq j\neq i.$
(3)

Semihamiltonian diagonal systems (1) have very reach geometric and analytic
structure [T-00]. In particular, they have infinitely many conservation laws
and, the most remarkably, they can be linearized by the so-called generalized
hodograph method: any solution with $u^{i}_{x}\neq 0$ $\forall i$ has a local
representation in the following implicit form (see [T-90])

$\mu^{i}(u)=\lambda^{i}(u)t+x,\ \ \ \ i=1,...,n,$ (4)

where $\mu$ is a suitable solution of the linear system

$\partial_{k}\mu^{i}=a_{ik}(\mu^{k}-\mu^{i}),\ \ \ \ i\neq k,\ \ \ \mbox{(no
summation).}$ (5)

Conversely, a generic solution $\mu$ to (5) gives locally a solution to the
semihamiltonian system (1). Observe that $\lambda$-s also solve this system.
System (5) defines an infinite dimensional commutative algebra of flows,
commuting with (1) [T-90]: equations (1) and the following ones

$u^{i}_{\tau}=\mu^{i}(u)u^{i}_{x}$ (6)

are compatible, their common solution is a vector function $u(t,\tau,x)$.
Moreover, since the Darboux integrability is expressed in terms of $a_{ij}$,
if system (1) is Darboux integrable then any commuting system (6) is also
Darboux integrable.

If for some $i$ the coefficients $a_{ik}$ do not vanish, one can exclude all
$\mu^{k}$, $k\neq i$ from (5) and obtain a compatible linear system for
$\mu^{i}$. For such systems, Darboux generalized the Laplace method originally
developed for just one liner hyperbolic equation with two independent
variables (see [D-96] IV, p.274). We show that Laplace transformations for
this system are liftable to all $\mu$ (and therefore to all $\lambda$) and
that the $i$-th Monge distribution has an extra Riemann invariants if and only
if for the transformed system $\bar{a}_{im}=0,$ i.e. the $i$-th sequence of
Laplace transformations terminates after one step. This is completely
analogous to the results of [SZ-95, JA-97, AK-98, ZS-01]: a nonlinear 2nd
order hyperbolic equation with two independent variables is Darboux integrable
if and only if the Laplace transformation sequences of its linearizations
terminate. (For a linear equations, integrability for systems with such
terminating sequences was understood already by Laplace. Darboux (see [D-96]
II, p.23) describes this method though he did not mention any ”intermediate
integrals”!?) Remarkably, the Laplace transformation for semihamiltonian
systems (1) defined in this paper via linear system for commuting flows (5)
coincides with the one defined via the linear system for the densities of
conservation laws of (1) in [F-97] (see also [KT-97] and [AF-99]).

The set of Darboux integrable systems (1) is sufficiently large. In section 4,
we show that it can be parametrized by $n(n+1)$ functions of one variable and
discuss how integrate them by solving ODEs either via generalized hodograph
transform or directly constructing complete involutions for (1). For linearly
degenerate systems (i.e. such that $\lambda^{i}_{i}=0$), the formulas for such
involutions can be written explicitly and were found in [T-85, P-87]. In fact,
they are explicit because the second Riemann invariant can be computed
explicitly. We describe a pair of more general classes when this is the case.

Semihamiltonian systems can be defined as strictly hyperbolic systems of
conservation laws admitting $n$ Riemann invariants [S-94]. Choosing $n$
functionally independent densities $N^{i}(u)$, we rewrite (1) as a system of
conservation laws

$\frac{d}{dt}N^{i}(u)=\frac{d}{dx}M^{i}(u),~{}~{}~{}i=1,...,n.$ (7)

There is a natural projective geometry associated to a system of conservation
laws (7). Namely, a congruence of lines (see [AF-96]), i.e. an $n$-parameter
family of lines

$Y^{i}=N^{i}(u)\ Y^{0}+M^{i}(u)Y^{n+1},~{}~{}~{}i=1,...,n,$ (8)

in $(n+1)$-dimensional projective space $\mathbb{P}^{n+1}$ with homogeneous
coordinates $[Y^{0}:...:Y^{n+1}]$ carries all the information about the
system. The basic concepts of the theory of conservation law systems
(rarefaction curves, shock curves, reciprocal transformations, linear
degeneracy, etc.) translate in geometrical language [AF-96, AF-99]. In fact,
this language extends also to a geometric interpretation of solutions [A-20].

In section 5, we present geometric counterparts of Laplace transformations and
Darboux integrability. For better visualization, here in introduction we
exemplify them for $n=2$. Any line of the congruence (8) touches two so-called
focal surfaces $F_{1}$ and $F_{2}$ in $\mathbb{P}^{3}$. Coordinate lines
$u^{2}=const$ correspond to developable surface of the congruence with the
cuspidal edges on $F_{1}$. That means that the congruence lines are tangent to
$F_{1}$ along curves of some family $C_{1}$ as only the first coordinate
$u^{1}$ is varying. Similarly, the congruence lines are tangent to $F_{2}$
along curves of some family $C_{2}$ as only the second coordinate $u^{2}$
varies. If $u^{1}=const$ then the points, where the lines touches $F_{1}$,
trace curves of a family $K_{1}$, conjugate to ones in $C_{1}$. The family
$K_{2}$ on $F_{2}$ is defined similarly. Laplace transformations of the
congruence are defined as changing the role of the families $C_{i}$ and
$K_{i}$ on one of the focal surfaces: the lines of the transformed congruence
are tangent to the curves $K_{i}$ [F-97, AF-99]. For example, if $i=1$ then
one of the focal surfaces of the new congruence coincides with $F_{1}$ and the
second is formed by the cuspidal edges of the developable surfaces of the new
congruence, these surfaces being cut from the congruence by $u^{2}=const$.
This construction for line congruences in $\mathbb{P}^{3}$ was given by
Darboux (see [D-96], V2, book 4, cap. I) as a geometric interpretation for
Laplace transformation of the equation for the parametrization of the focal
surface by the conjugate coordinates $u^{1},u^{2}$. We show that, remarkably,
this construction provides also the geometric picture of Laplace
transformations for the system (5). Note that the equation for the
corresponding $\lambda$ is not the equation for the above parametrization of
the focal surface, therefore terminating of a sequence of Laplace
transformations for (5) is geometrically different from the one given by
Darboux: the curves, conjugated to the cuspidal edges on the new focal
surface, are cut by a pencil of planes, thus making impossible the next
Laplace transformation ”starting from” this new surface, while in the Darboux
picture the new focal surface of the transformed congruence degenerates into a
curve.

Another geometric aspect, discussed in this paper, is related to the
symmetries of (1). Darboux integrability, manifesting a kind of variable
separation (see [AFV-09]), is reflected also in the structure of the symmetry
algebra of commuting flows. Fixing a relation between two Riemann invariants
of the same Monge distribution, we get a differential relation compatible with
the system and depending on one arbitrary function of one variable. We show
that this relation fixes an orbit of a solution with respect to some
subalgebra of the symmetry algebra, defined by (5). Moreover, this subalgebra
is also cut by a differential relation, uniquely defined by the choice of
Monge distribution.

Even though Darboux method is potentially applicable to systems of equations
for vector-valued functions in any number of variables, most of the studies
were devoted to a single 2nd order equation for scalar funcion in two
variables (see [P-60, R-99] for generalization of the method to higher order
equations). This can be explained by the increasing complexity of calculations
needed to work out higher-dimensional cases. This complexity, probably,
explains also the scarcity of explicitly elaborated examples within prescribed
classes. We hope that this paper contributes to a better understanding of the
case of hydrodynamic systems.

A warning on terminology: systems (1) are ”hyperbolic” in the sense of the
theory developed in [BGH-95] only for $n=2$.

##  2 Monge characteristic distribution for diagonal systems

Let us rewrite equations (1) as an exterior differential system in
$\mathbb{R}^{2n+2}$ with coordinates $u^{i},u^{i}_{x},x,t$:

$\sigma_{i}:=u^{i}_{x}(dx+\lambda^{i}(u)dt)-du^{i}=0.$ (9)

Then solutions to (1) are integral manifolds of this system with $dx\wedge
dt\neq 0$. In the dual language, integration of (9) amounts to finding
2-dimensional integral manifolds tangent to the distribution $\cal{V}$ with
generators

$\mathcal{V}:=\left\langle\partial_{x}+\sum_{k}u^{k}_{x}\partial_{k},\
\partial_{t}+\sum_{k}\lambda^{k}u^{k}_{x}\partial_{k},\
\partial_{u^{i}_{x}}\right\rangle_{i=1,...,n}$ (10)

###### Definition 2

[S-00] A subdistribution $\mathcal{M}$ of $\mathcal{V}$ is Monge
characteristic distribution if

  1. 1.

$\mathcal{M}$ is singular: for any vector $m\in\mathcal{M}$ holds

$\dim\langle m\lrcorner d\sigma_{i},\
i=1,...,n\rangle|_{\mathcal{V}}<\max_{v\in\mathcal{V}}\dim\langle v\lrcorner
d\sigma_{i},\ i=1,...,n\rangle|_{\mathcal{V}},$

  2. 2.

$\mathcal{M}\cap\mathcal{I}\neq 0$ for any maximal involution $\mathcal{I}$ of
$\mathcal{V}$,

  3. 3.

$\mathcal{M}\cap\mathcal{W}$ is complete (i.e. integrable) for any maximal
complete subdistribution $\mathcal{W}$ of $\mathcal{M}$.

(Recall that involution is a subdistribution $\mathcal{I}$ of $\mathcal{V}$
such that $[\mathcal{I},\mathcal{I}]\in\mathcal{V}$.)

An easy exercise in linear algebra shows that (1) has $n$ Monge characteristic
distributions and the $i$-th one is cut by the following equations:

$\begin{array}[]{l}dx+\lambda^{i}dt=0,\\\ du^{i}=0,\\\
du^{j}+(\lambda^{i}-\lambda^{j})u^{j}_{x}dt=0,\ j\neq i\\\
du^{i}_{x}-D_{x}(\lambda^{i})u^{i}_{x}dt=0,\end{array}$ (11)

where $D_{x}=\partial_{x}+\sum_{k}u^{k}_{x}\partial_{k}+...$ is the total
derivative by $x$. It is immediate that this distribution has at least one
Riemann invariant (i.e. first integral of the Monge characteristic
distributions), namely $u^{i}$. The system is Darboux integrable if each Monge
distribution admits at least one extra Riemann invariant. Darboux inegrability
depends on the codimension of the exterior differential ideal generated by
(11).

One can consider $p$-th prolongations of system (1) and rewrite these
prolongations also as an exterior differential systems in
$\mathbb{R}^{(2+p)n+2}$ with coordinates $u^{i},u^{i}_{x},u^{i}_{xx},...,x,t$.
For these ”prolonged” exterior systems, Monge characteristic distributions are
defined in a similar way.

We say that a Riemann invariant (of a prolonged system) is of order 0 if it
does not depend on the derivatives of $u$ and of order $q$ if it depends on
derivatives of order $\leq q$ and is not of order $q-1$.

###### Lemma 3

Let the system (1) be strictly hyperbolic (i.e. $\lambda^{i}\neq\lambda^{j}$
for $i\neq j$) then

a) the $i$-th Monge distribution admits one extra Riemann invariant of order 0
if and only if

$\lambda^{i}_{k}=0\ \ \forall\ k\neq i,$

b) the $i$-th Monge distribution admits one Riemann invariant of order 1 and
only one Riemann invariant of order 0 if and only if the following conditions
hold true:

  * •

$\partial_{j}(a_{ik})=\partial_{k}(a_{ij})$ $\forall\ k\neq j\neq i\neq k,$

  * •

$\exists\ j\neq i:\ \lambda^{i}_{j}\neq 0,\ \
\partial_{j}b_{ij}+a_{ij}b_{ij}=0$, where
$b_{ij}=\frac{\lambda^{i}_{ij}}{\lambda^{i}_{j}}+\frac{\lambda^{i}_{i}}{\lambda^{j}-\lambda^{i}}=\frac{\partial_{i}a_{ij}}{a_{ij}}-a_{ji},$

  * •

if for $k\neq i$ holds true $\lambda^{i}_{k}\neq 0$ then $b_{ik}=b_{ij}$ and
$\partial_{k}b_{ik}+a_{ik}b_{ik}=0.$

Proof: In the dual language, the $i$-th Monge distribution is generated by the
vector fields
$\partial_{u^{k}_{x}}$ with $k\neq i$ and by

$\xi=\partial_{t}-\lambda^{i}\partial_{x}+\sum_{j\neq
i}(\lambda^{j}-\lambda^{i})u^{j}_{x}\partial_{j}+D_{x}(\lambda^{i})u^{i}_{x}\partial_{u^{i}_{x}}.$
(12)

Calculating commutators, we get $n-1$ vector fields of the distribution:

$\eta_{k}=\frac{1}{\lambda^{k}-\lambda^{i}}[\partial_{u^{k}_{x}},\xi]=\partial_{k}+a_{ik}u^{i}_{x}\partial_{u^{i}_{x}}.$

If there is an extra Riemann invariant of order 0 then, supplemented with
$\partial_{u^{i}_{x}}$, the distribution is involutive, $\eta_{k}$ reduces to
$\partial_{k}$ and we have
$[\partial_{k},\partial_{t}-\lambda^{i}\partial_{x}]=0$ by the Frobenius
Theorem. Hence a).

Suppose that $\lambda^{i}_{j}\neq 0$ for some $j\neq i$. Then there is only
one Riemann invariant of order 0. We use $\eta^{k}$ to kill coefficients of
$\partial_{j}$ and to reduce $\xi$ to

$\bar{\xi}=\partial_{t}-\lambda^{i}\partial_{x}+\lambda^{i}_{i}(u^{i}_{x})^{2}\partial_{u^{i}_{x}}.$

By the Frobenius theorem, there is a Riemann invariant of order 1 if and only
if the distribution
$\\{\bar{\xi},\partial_{u^{k}_{x}},\eta_{k},\zeta\\}_{k\neq i}$, where

$\zeta=\frac{1}{\lambda^{i}_{j}}[\eta_{j},\bar{\xi}]=-\partial_{x}+b_{ij}(u^{i}_{x})^{2}\partial_{u^{i}_{x}},$

is involutive. Computing commutators $[\eta_{j},\eta_{k}]$,
$[\eta_{j},\zeta]$, we get the first and the second relations of b), comparing
$\frac{1}{\lambda^{i}_{k}}[\eta_{k},\bar{\xi}]$ with $\zeta$ and computing
$[\eta_{k},\frac{1}{\lambda^{i}_{k}}[\eta_{k},\bar{\xi}]]$, we obtain the
third condition b). $\Box$

###### Theorem 4

If a strictly hyperbolic system (1) is Darboux integrable in order less then 3
then it is semihamiltonian, i.e. $\partial_{j}a_{ik}=\partial_{k}a_{ij}$
$\forall\ k\neq j\neq i.$

Proof: If the $i$-th Monge distribution possesses an extra Riemann invariant
of order less then 2 then $\partial_{j}a_{ik}=\partial_{k}a_{ij}$ by lemma 3.

In order 2, the $i$-th Monge distribution is given by forms (11) and the
following ones:

$\begin{array}[]{l}du^{j}_{x}+[(\lambda^{i}-\lambda^{j})u^{j}_{xx}-D_{x}(\lambda^{j})u^{j}_{x}]dt=0,\
\ \ j\neq i,\\\
du^{i}_{xx}-[2D_{x}(\lambda^{i})u^{i}_{xx}+D_{x}^{2}(\lambda^{i})u^{i}_{x}]dt=0.\end{array}$
(13)

Or, equivalently, it is generated by the vector fields $\partial_{u^{k}_{xx}}$
with $k\neq i$ and by

$\tau=\xi+\sum_{j\neq
i}[(\lambda^{j}-\lambda^{i})u^{j}_{xx}+D_{x}(\lambda^{j})u^{j}_{x}]\partial_{u^{j}_{x}}+[2D_{x}(\lambda^{i})u^{i}_{xx}+D_{x}^{2}(\lambda^{i})u^{i}_{x}]\partial_{u^{i}_{xx}},$

where $\xi$ is given by (12). We easily get $n-1$ more vector fields of the
distribution:

$\eta_{k}=\frac{1}{\lambda^{k}-\lambda^{i}}[\partial_{u^{k}_{xx}},\tau]=\partial_{u^{k}_{x}}+a_{ik}u^{i}_{x}\partial_{u^{i}_{xx}}.$

Replacing $\tau$ by $\bar{\tau}:=\tau-\sum_{j\neq
i}[(\lambda^{j}-\lambda^{i})u^{j}_{xx}+D_{x}(\lambda^{j})u^{j}_{x}]\eta_{j}$
we ”kill” the coefficients of $\partial_{u^{j}_{x}}$. With

$\zeta_{k}=\frac{1}{\lambda^{k}-\lambda^{i}}[\eta_{k},\bar{\tau}]=\partial_{k}+...,$

the commutators $[\eta_{j},\zeta_{k}]$ take the form $E\partial_{u^{i}_{xx}}$.
If the distribution does not have an extra Riemann invariant of order $1$ then
the expression $E$ must vanish. The equation $E=0$ is equivalent to
$\partial_{j}(a_{ik})=\partial_{k}(a_{ij})$ hence the system is
semihamiltonian. $\Box$

We believe that the claim of theorem 4 is true for any order.

Conjecture 1 Darboux integrable strictly hyperbolic system (1) is
semihamiltonian.

##  3 Laplace transformation for commuting flows and Darboux integrability

###### Theorem 5

If system (1) is Darboux integrable then any commuting system (6) is also
Darboux integrable.

Proof: The criterion given in lemma 3 is written in terms of $a_{ij}$, which
are the same for both systems. $\Box$

Let us consider system (5) on its own right, i.e. we do not suppose that
$a_{ij}$ are defined via $\lambda$-s by (2).

###### Lemma 6

[T-90] The system (5) is in involution if

$\partial_{j}a_{ki}=a_{ki}a_{ij}+a_{kj}a_{ji}-a_{ki}a_{kj},\ \ \ \ \forall\
j\neq k\neq i\neq j.$ (14)

Proof: Computing $\partial_{i}\partial_{j}\mu^{k}$ in two different ways and
equating the expressions obtained, we get a linear equation in
$\mu^{i},\mu^{j},\mu^{k}$. For the system in involution, the coefficients must
vanish. Vanishing of the coefficient of $\mu^{i}$ gives (14). The coefficient
of $\mu^{k}$ gives $\partial_{j}(a_{ik})=\partial_{k}(a_{ij})$, which follows
from relations (14). Finally, the coefficient ob $\mu^{j}$ gives again (14)
after permutation of indexes. $\Box$

Let us fix $i$ and suppose that $a_{ik}\neq 0$ for any $k\neq i$. Then we can
exclude all $\mu^{k}$ from system (5) and obtain a system of compatible PDEs
of second order for $\mu^{i}$ in the following way. Differentiating

$\mu^{k}=\frac{\partial_{k}\mu^{i}}{a_{ik}}+\mu^{i}$ (15)

by $u^{m}$ with $m\neq i$, $m\neq k$, taking into account (5), and using
subindexes for partial derivatives of $\mu$, we obtain

$\mu^{i}_{km}+\left(a_{km}-\frac{\partial_{m}a_{ik}}{a_{ik}}\right)\mu^{i}_{k}+a_{ik}\left(1-\frac{a_{km}}{a_{im}}\right)\mu^{i}_{m}=0.$
(16)

Similarly, differentiating (15) by $u^{i}$ we get

$\mu^{i}_{ki}+\left(a_{ki}-\frac{\partial_{i}a_{ik}}{a_{ik}}\right)\mu^{i}_{k}+a_{ik}\mu^{i}_{i}=0.$
(17)

Thus for $U=\mu^{i}$ we have a system of linear PDEs of the form

$U_{ij}+A_{ij}U_{i}+A_{ji}U_{j}=0,\ \ \ \ \ i\neq j,$ (18)

where again we use the notation $U_{i}=\partial_{i}U$,
$U_{ij}=\partial_{i}\partial_{j}U$.

Conversely, given $i$, we recover system (5) from (18) for

$\mu^{i}=U,\ \ \
\mu^{k}=\frac{\partial_{k}U}{A_{ki}+\frac{\partial_{i}A_{ik}}{A_{ik}}}+U,$

with

$a_{km}=A_{km}+\frac{\partial_{m}A_{ik}}{A_{ik}}=A_{im}\left(1-\frac{A_{mk}}{A_{ik}}\right),\
\ \mbox{if}\ \ \ i\neq m\neq k\neq i$ (19)

and

$a_{ik}=A_{ik},\ \ \ \ a_{ki}=A_{ki}+\frac{\partial_{i}A_{ik}}{A_{ik}},\ \
\mbox{if}\ \ i\neq k.$ (20)

The two expressions for $a_{km}$ manifest the compatibility conditions for
(18).

A crucial observation is that if we pass from (18) to (5) via (20,19) for one
$i$ and then return from (5) to another system of type (18) via (15) fixing
another index $l\neq i$, we get a Laplace transformation of (18) for the
chosen pair $i,l$. Thus, the system for commuting flows (5) can be viewed as
the set of possible Laplace transformations. But we can as well start with the
system (5).

###### Definition 7

Laplace transformation of system (5) for a pair of indexes $i,j$ is obtained
by passing to (18) for $U=\mu^{i}$ and returning to (5) via (20,19) with
$i=j$:

$\bar{\mu}^{j}=\mu^{i},\ \ \
\bar{\mu}^{i}=\frac{\mu^{i}_{i}}{a_{ji}-\frac{\partial_{i}a_{ij}}{a_{ij}}}+\mu^{i},\
\ \
\bar{\mu}^{k}=\frac{\mu^{i}_{k}}{a_{jk}-\frac{\partial_{k}a_{ij}}{a_{ij}}}+\mu^{i}=\frac{a_{ij}\mu^{k}-a_{kj}\mu^{i}}{a_{ij}-a_{kj}},\
\ \ k\neq i\neq j\neq k.$ (21)

Applying formulas (16,17,19,20), one computes all the coefficients
$\bar{a}_{km}$ of the transformed system.

Since $\lambda$ is a solution to system (5) the Laplace transformation acts
also on solutions to system (1): one applies formulas (21) to $\lambda$-s and
invokes the implicit form (4).

###### Theorem 8

Suppose that, for a fixed pair of indexes $i,j$, a semihamiltonian system (1)
has non-vanishing coefficients $a_{im}$ and pairwise distinct coefficients
$a_{mj}$. Then its $i$-th Monge distribution has an extra Riemann invariant of
order 1 if and only if all the coefficients $\bar{a}_{im}$ of the Laplace
transformation with indexes $i,j$ of the correpondent system (5) vanish:
$\bar{a}_{im}=0$.

Proof: Two systems (5) are related by the $ij$\- Laplace transformation if
equations (18) for $U=\mu^{i}$ and $U=\bar{\mu}^{j}$ are the same. Thus, by
(19,20) we have for $m\neq j$:

$\bar{a}_{im}=A_{im}+\frac{\partial_{m}A_{ji}}{A_{ji}}=a_{im}+\frac{\partial_{m}(a_{ji}-\frac{\partial_{i}a_{ij}}{a_{ij}})}{a_{ji}-\frac{\partial_{i}a_{ij}}{a_{ij}}}.$

On the other hand, using another expression for $\bar{a}_{im}$, we also have

$\bar{a}_{im}=A_{jm}\left(1-\frac{A_{mi}}{A_{ji}}\right)=a_{im}\left(1-\frac{a_{mj}}{a_{ij}}\right)\left(1-\frac{a_{mi}-\frac{\partial_{i}a_{im}}{a_{im}}}{a_{ji}-\frac{\partial_{i}a_{ij}}{a_{ij}}}\right).$

Similarly,

$\bar{a}_{ij}=A_{ij}+\frac{\partial_{j}A_{ji}}{A_{ji}}=a_{ij}+\frac{\partial_{j}(a_{ji}-\frac{\partial_{i}a_{ij}}{a_{ij}})}{a_{ji}-\frac{\partial_{i}a_{ij}}{a_{ij}}}.$

Thus, vanishing of $\bar{a}_{im}$ for $m\neq j$ gives

$a_{im}+\frac{\partial_{m}b_{ij}}{b_{ij}}=0,\ \ \ \ b_{im}=b_{ij},$

vanishing of $\bar{a}_{ij}$ yields

$a_{ij}+\frac{\partial_{j}b_{ij}}{b_{ij}}=0,$

and the theorem follows from lemma 3. $\Box$

The above theorem motivates the following definition.

###### Definition 9

We say that $i$-th sequence of Laplace transformation for a semihamiltonian
system $S_{0}$ terminates after $p$ steps if there is a sequence
$j_{1},j_{2},...,j_{p}\in\mathbb{N}$ of indexes and a sequence
$S_{1},S_{2},...,S_{p}$ of semihamiltonian systems such that $S_{q}$ is a
$ij_{q}$-Laplace transformation of $S_{q-1}$ and for the system $S_{p}$ holds
true $a_{im}=0,\ \forall m\neq i$.

We believe that the following generalization of theorem 8 is true.

Conjecture 2 The $i$-th Monge distribution of a semihamiltonian system (1) has
an extra Riemann invariant of order $p$ if and only if its $i$-th sequence of
Laplace transformation terminates after $p$ steps.

This conjecture is supported by the case $n=2$: the hodograph transform
reduces a two-component system (1) to a single equation of form (18), for
which the result is well-known.

##  4 Darboux integrable diagonal systems of hydrodynamic type

System (1) is Darboux integrable in order 1 if and only if its characteristic
speeds $\lambda^{i}$ satisfy the conditions b) of lemma (3) for all $i$, these
conditions forming a system of differential equations. It turns out that the
space of solutions to this system is infinite dimensional. Lets us consider a
generic solution and suppose that $\lambda^{i}_{j}\neq 0,$ whenever $j\neq i$.

###### Theorem 10

Let $\Lambda^{i}_{j}$, $\beta_{i}$, $0\leq i,j\leq n$ be germs of real
analytic functions of one variables at $u_{0}^{i}$. Then there is (a germ of)
system (1), Darboux integrable in order 1, such that

$\lambda^{i}_{j}|_{u^{k}=u^{k}_{0},\ k\neq i}=\Lambda^{i}_{j},\ \ \ \
b^{i}_{j}|_{u^{k}=u^{k}_{0},\ k\neq i}=\beta_{i}.$

Proof: From semihamiltonian property we have $\forall\ k\neq j\neq i\neq k,$

$\lambda^{i}_{jk}=\frac{(2\lambda^{i}-\lambda^{j}-\lambda^{k})\lambda^{i}_{j}\lambda^{i}_{k}}{(\lambda^{i}-\lambda^{k})}+\frac{(\lambda^{k}-\lambda^{i})\lambda^{i}_{j}\lambda^{j}_{k}}{(\lambda^{k}-\lambda^{j})(\lambda^{j}-\lambda^{i})}+\frac{(\lambda^{j}-\lambda^{i})\lambda^{i}_{k}\lambda^{k}_{j}}{(\lambda^{j}-\lambda^{k})(\lambda^{k}-\lambda^{i})}.$

For any $i$, the last conditions b) of lemma (3) allow to introduce functions
$B_{i}=b_{ik}$, satisfying

$\partial_{k}B_{i}=-a_{ik}B_{i}$

for $k\neq i$. Definition of $b_{ij}$ gives

$\lambda^{i}_{ik}=\lambda^{i}_{k}B_{i}-\lambda^{i}_{i}a_{ik}$

for $k\neq i$. Invoking the definitions of $a_{ij}$, we see that we have a
system of partial differential equations for functions $\lambda^{i},\
\lambda^{i}_{j}$, and $B_{i}$. By direct calculation one shows that this
exterior system is in involution and, by Riquier’s theory [R-10], the initial
conditions specified by the theorem fix a local analytic solution. $\Box$

Solution of a Darboux integrable semihamiltonian system reduces to solving a
system of ordinary differential equations, like for any other Darboux
integrable equation or system.

Consider first a generic case with $b_{ij}\neq 0$. Then by theorem 8 and
formulas (21) we have

$d\mu^{i}=b_{ij_{i}}(\mu^{i}-\varphi_{i}(u^{i}))du^{i}+\sum_{k\neq
i}a_{ik}(\mu^{k}-\mu^{i})du^{k},$ (22)

on solutions to (5), where $i=1,...,n,$ $\varphi_{i}$ is a suitable function
of one variable, and $j_{i}$ is an index such that $ij_{i}$-Lablace
transformation yields $\bar{a}_{im}=0$.

###### Proposition 11

If all Monge distributions of system (1) have an extra Riemann invariant then
the exterior system (22), where $i=1,...,n,$ is completely integrable (in the
sense of Frobenius) for any choice of smooth functions $\varphi_{i}$.

Proof: One checks by direct calculation that the right hand sides of (22) are
closed modulo (22) due to lemma (3). $\Box$

Having a solution $\mu$ to (22) with some choice of $\varphi$-s, one writes
solution to (1) in the implicit form (4). This approach directly generalizes
to semihamiltonian systems such that, for each $i$, its $i$-th sequence of
Laplace transformation terminates after $p_{i}$ steps. In fact, applying
formulas (21), we get

$\sum_{k=0}^{p_{i}}\alpha^{i}_{k}(u)\partial^{k}_{i}(\mu^{i})=\varphi_{i}(u_{i})$
(23)

for solutions of (5), where the form of the coefficients $\alpha^{i}_{k}$ is
not now important and $\alpha^{i}_{p_{i}}\neq 0$. Thus, with $\varphi_{i}$
fixed, the system (5,23) determine a completely integrable system.

Note that we can reduce integration of a semihamiltonian system (1) to
ordinary differential equations also in the case when all but one Monge
distributions have an extra Riemann invariant. Let us consider the case of
order 1.

###### Proposition 12

If all but $n$-th Monge distributions of real analytic system (1) have an
extra Riemann invariant then the exterior system (22), where $i=1,...,n-1$,
supplemented by the following equation

$d\mu^{n}=\sum_{k=1}^{n-1}a_{nk}(\mu^{k}-\mu^{n})du^{k}+u^{n}_{x}dz^{n}$ (24)

admits a Cauchy characteristic for any choice of smooth functions
$\varphi_{i}$.

Proof: By a little of linear algebra one shows that any integral element of
our system satisfying the following 2 equations

$du^{n}=0,\ \ \
du^{n}_{x}=\sum_{m=1}^{n-1}[\partial_{n}(a_{nm})(\mu^{m}-\mu^{n})+a_{nm}(a_{mn}(\mu^{n}-\mu^{m})-u^{n}_{x})]du^{m}$

is a Cauchy characteristic. $\Box$
Thus we can integrate the system (22, i=1,…n-1) by the method of
characteristics.

If $b_{im}=0$ for some fixed $i$ and all $m\neq i$, the system (22) is not
closed (i.e. at least $\partial_{i}\mu^{i}$ is not defined). In this case one
always can apply a general procedure: find a second Riemann invariant
$I^{i}(x,t,u,u^{i}_{x})$ for each Monge distribution, set

$I^{i}(x,t,u,u^{i}_{x})=\varphi_{i}(u^{i}),$ (25)

where $\varphi_{i}$ are arbitrary, resolve these equations for $u^{i}_{x}$:
$u^{i}_{x}=P^{i}(x,t,u,\varphi_{i}(u^{i}))$ and integrate the exterior system

$du^{i}=P^{i}(x,t,u,\varphi_{i}(u^{i}))(dx+\lambda^{i}(u)dx).$ (26)

###### Proposition 13

If all Monge distributions of system (1) have an extra Riemann invariant then
the exterior system (26), where $i=1,...,n,$ is completely integrable (in the
sense of Frobenius) for any choice of smooth functions $\varphi_{i}$.

Moreover, if all but $n$-th Monge distributions have an extra Riemann
invariant then the exterior system (22), where $i=1,...,n-1,$ supplemented by
one equation

$du^{n}=u^{n}_{x}(dx+\lambda^{n}(u)dt),$

admits a Cauchy characteristic for any choice of real analytic functions
$\varphi_{i}$.

Proof: By differentiation of the identity $I^{i}(x,t,u,P^{i}(x,t,u,p))=p$ with
respect to $x$, $t$, $p$, and all $u^{k}$, one finds partial derivatives of
$P^{i}$. The condition that $I^{i}$ is invariant along the $i$-th Monge
distribution implies that the system (26) with $i=1,...,n$ is in involution.

One proves the second claim by checking that the equations

$dx+\lambda^{n}dt=0,\ \ \
du^{n}_{x}=u^{n}_{x}\left(\lambda^{n}_{n}u^{n}_{x}+\sum_{k=1}^{n-1}\lambda^{n}_{k}P^{k}\right)dt$

define a Cauchy characteristic. $\Box$

Remark 1. The second claim of proposition 13 follows from the general theory
(see [V-28, S-00]): if $\dim\mathcal{V}$ is odd and
$\dim[\mathcal{V},\mathcal{V}]=\dim\mathcal{V}+1$ there is at least one Cauchy
characteristic.

Example 1: $\bf b_{ik}=0,\ \forall\ k\neq i$. In this case one can find the
second Riemann invariant by quadratures. The $i$-th Monge distribution is
spanned by the following set of vector fields

$\\{\partial_{x},\
\partial_{t}+(u^{i}_{x})^{2}\lambda^{i}_{i}\partial_{u^{i}_{x}},\
\partial_{k}+u^{i}_{x}a_{ik}\partial_{u^{i}_{x}}\\}_{k\neq i}.$

Then the $i$-th distribution has the invariant
$\left(\frac{1}{u^{i}_{x}}+\lambda^{i}_{i}(u)t\right)H_{i}(u)$, where the
functions $H_{i}$, called Lame coefficients, are solutions to the following
system of PDEs

$\partial_{k}\ln(H_{i})=a_{ik},\ k\neq i,$

compatible due to (14). This system is integrable by quadratures. In fact, let
$i=1$ and $u_{0}=(u^{1}_{0},...,u^{n}_{0})$. Then $\ln(H_{1})=\int
a_{1n}du^{n}+\ln(\tilde{H}_{1}(u^{1},u^{2},...,u^{n-1}))$ for some
$\tilde{H}_{1}$. For $k\neq 1,n$ we have
$\partial_{k}\ln(H_{1})=a_{1k}=\int\partial_{k}(a_{1n})du^{n}+\partial_{k}\ln(\tilde{H}_{1})=\int\partial_{n}(a_{1k})du^{n}+\partial_{k}\ln(\tilde{H}_{1})=a_{1k}-a_{1k}|_{u^{n}=u^{n}_{0}}+\partial_{k}\ln(\tilde{H}_{1})$.
With $\tilde{a}_{1k}(u^{1},u^{2},...,u^{n-1})=a_{1k}|_{u^{n}=u^{n}_{0}}$ one
has $\partial_{k}\ln(\tilde{H_{1}})=\tilde{a}_{1k}$ and the claim follows by
induction. General solution $H_{i}$ is defined up to multiplication by an
arbitrary function of $u^{i}$.

Thus for the system (26), we find

$P^{i}=\frac{H_{i}(u)}{\varphi_{i}(u^{i})-t\lambda^{i}_{i}(u)H_{i}(u)}.$

Example 2: $\bf n=2,$ $\bf b_{12}=b_{21}=0$. Resolving (26) with respect to
$dt$ and $dx$, we get:

$\displaystyle
dt=\frac{\varphi_{1}du^{1}}{H_{1}(\lambda^{1}-\lambda^{2})}+\frac{\varphi_{2}du^{2}}{H_{2}(\lambda^{2}-\lambda^{1})}+\left(\frac{\lambda^{1}_{1}du^{1}}{\lambda^{2}-\lambda^{1}}+\frac{\lambda^{2}_{2}du^{2}}{\lambda^{1}-\lambda^{2}}\right)t,$
(27) $\displaystyle
dx=\frac{\lambda^{2}\varphi_{1}du^{1}}{H_{1}(\lambda^{2}-\lambda^{1})}+\frac{\lambda^{1}\varphi_{2}du^{2}}{H_{2}(\lambda^{1}-\lambda^{2})}+\left(\frac{\lambda^{2}\lambda^{1}_{1}du^{1}}{\lambda^{1}-\lambda^{2}}+\frac{\lambda^{1}\lambda^{2}_{2}du^{2}}{\lambda^{2}-\lambda^{1}}\right)t.$
(28)

One checks that the form-valued coefficient
$\omega=\frac{\lambda^{1}_{1}du^{1}}{\lambda^{2}-\lambda^{1}}+\frac{\lambda^{2}_{2}du^{2}}{\lambda^{1}-\lambda^{2}}$
by $t$ in the first equation is closed due to $b_{12}=b_{21}=0$. Applying a
textbook technique of integration of linear ODEs, we get

$t=W(u)\int\frac{1}{W}\left(\frac{\varphi_{1}du^{1}}{H_{1}(\lambda^{1}-\lambda^{2})}+\frac{\varphi_{2}du^{2}}{H_{2}(\lambda^{2}-\lambda^{1})}\right),$

where $W=\exp(\int\omega)$. With this $t$, the form for $dx$ does not depend
on $x$, therefore the general solution is obtained by quadratures.

Remark 2. Linearly degenerate systems, defined as satisfying
$\lambda^{i}_{i}=0\ \forall\ i$, are Darboux integrable, provided that they
are semihamiltonian, since for such systems $b_{ik}=0$. Moreover, the
expressions for $P^{i}$ can be rewritten as
$P^{i}=\tilde{\varphi}_{i}(u^{i})H_{i}(u)$. These formulas appeared in [T-85],
where such system were integrated via (26). A different approach, restricted
to hamiltonian systems (see [DN-83] for definitions), was given in [P-87]: it
applies (4) and takes advantage of particular properties of conservation laws
for linearly degenerate systems. Independent method of integration was
described in [F-90]. It has a definite geometric flavor: integration is based
on the fact that the $(n+2)$-web, formed by characteristics and by coordinate
lines of independent variables $x,t$, has rank $n$ on each solution.

##  5 Line congruence geometry of semihamiltonian systems with terminating
sequences of Laplace transformations

Semihamiltonian systems (1) admit infinitely many conservation laws [T-90],
i.e. differential one-forms $N(u)dx+M(u)dt$, closed on any solution. One can
choose $n$ functionally independent densities $N^{i}(u)$ and rewrite (1) as a
system of conservation laws (7).

A conservation low $N(u)dx+M(u)dt$ of (1) satisfies the system

$\partial_{i}M=\lambda^{i}\partial_{i}N,~{}~{}~{}i=1,...,n,\ \ \ \mbox{(no
summation)}.$

Excluding the flow $M$, we have a consistent system of linear PDE for the
density $N$:

$N_{ij}=a_{ij}N_{i}+a_{ji}N_{j}.$ (29)

Thus we can apply Laplace transformations to this system. These
transformations are liftable to the characteristic speeds $\lambda^{s}$
[AF-99] and give formulas (21) that we obtained earlier starting from (5).

Laplace transformations for (1) can be defined geometrically via the
corresponding line congruence [AF-99]. For any choice of $n$ independent
conservation laws, a line of the congruence (8), corresponding to a
semihamiltonian system (1), is tangent to $n$ focal submanifolds [AF-96],
labeled by $i=1,...,n$:

$\begin{array}[]{l}Y^{0}=-\lambda^{i}Y^{n+1}\\\
Y^{m}=(M^{k}(u)-\lambda^{i}N^{k}(u))Y^{n+1},\ \ \ \ k=1,...,n.\end{array}$

Passing to the affine chart $y^{k}=-\frac{Y^{k}}{Y^{n+1}}$, we get a
parametrization $r^{i}(u)$ of the $i$-th focal submanifold:

$\begin{array}[]{l}y^{0}=\lambda^{i},\\\ y^{k}=\lambda^{i}N^{k}(u)-M^{k}(u),\
\ \ \ k=1,...,n.\end{array}$ (30)

Now fix $i,j$ and consider $n$-parameter family of lines passing trough
$r^{i}(u)$ in the direction of $\partial_{j}r^{i}(u)$:

$\begin{array}[]{l}y^{0}=\lambda^{i}+s\lambda^{i}_{j},\\\
y^{k}=\lambda^{i}N^{k}-M^{k}+s[\lambda^{i}_{j}N^{k}+(\lambda^{i}-\lambda^{j})\partial_{j}N^{k}],\
\ \ k=1,...,n.\end{array}$ (31)

Excluding the parameter $s$, we get a line congruence corresponding to the
system of conservation laws $\bar{N}^{k}(u)dx+\bar{M}(u)^{k}dt$ with

$\bar{N}^{k}(u)=N^{k}-\frac{\partial_{j}N^{k}}{a_{ij}},\ \ \ \
\bar{M}^{k}=M^{k}-\frac{\lambda^{i}\partial_{j}N^{k}}{a_{ij}}.$ (32)

It is quite natural to call (32) Laplace transformation of the congruence (8).

For any $i\neq k\neq j\neq i$ we check by direct computation, using (29) and
(14), that

$\partial_{k}\bar{M}=\frac{a_{ij}\lambda^{k}-a_{ki}\lambda^{i}}{a_{ij}-a_{ki}}\partial_{k}\bar{N}.$

Similarly, (29) implies

$\partial_{i}\bar{M}=\left(\lambda^{i}-\frac{\lambda^{i}_{i}}{\frac{\partial_{i}a_{ij}}{a_{ij}}-a_{ji}}\right)\partial_{i}\bar{N}$

and

$\partial_{j}\bar{M}=\lambda^{i}\partial_{j}\bar{N}.$

Thus the characteristic speeds of the transformed system of conservation laws
do not depend on the choice of $N^{i}$ and coincide with ones of (1) after
applying the Laplace transformation (21) to $\lambda$-s.

Remark 1. In fact, it is not difficult to see that semihamiltonian systems can
be defined as strictly hyperbolic systems of conservation laws admitting $n$
Riemann invariants [S-94]: the compatibility conditions (14) for the commuting
flows are also the compatibility conditions for the system (29) on the
densities of conservation laws for the diagonal system (1).

Let us translate Darboux integrability into the geometric language. To this
end we recall the notion of reciprocal transformation for the system (1):
given 2 conservation laws $Bdx+Adt$, $Ndx+Mdt$, we introduce new independent
variables $X,T$ by

$\begin{array}[]{c}dX=B(u)dx+A(u)dt,\\\ dT=N(u)dx+M(u)dt\end{array}$ (33)

and rewrite (1) as

$u^{i}_{T}=\left(\frac{B\lambda^{i}-A}{M-N\lambda^{i}}\right)u^{i}_{X}.$ (34)

Reciprocal transformations respect the property of being semihamiltonian, but
not the property of being Darboux integrable. One easily sees this already for
extra Riemann invariants of order 0: the condition $\lambda^{i}_{k}=0,\
\forall k\neq i$ does not imply
$\partial_{k}\left(\frac{B\lambda^{i}-A}{M-N\lambda^{i}}\right)=0,\ \forall
k\neq i$. The reason is that an extra Riemann invariant depends on $x,t$,
which became nonlocal variables (i.e. recoverable via integration of forms
closed on solutions) after applying a nontrivial reciprocal transformation. If
one uses only linear combinations of $dx,dt$ and of the ”coordinate”
conservation laws $N^{i}(u)dx+M^{i}(u)dt$ then the reciprocal transformations
form the projective group $PGL_{n+2}$ acting in the projective space of the
congruence (8) [AF-96].

To understand the geometry of Darboux integrability it is enough to interpret,
for a fixed $i$, the condition $a_{im}=0,$ $\forall\ m\neq i$ since it is the
form of the system after a (sequence of) Laplace transformations of the
congruence.

A line of the congruence (8) is tangent to the $i$-th focal submanifold
$F_{i}$, parametrized by (30), at some point
$r^{i}(u)=(\lambda^{i},\lambda^{1}N^{1}-M^{1},...,\lambda^{1}N^{n}(u)-M^{n})$.
If $u$ moves along the $k$-th coordinate line, given by $u^{l}=const,\ \forall
l\neq k$, the point $r^{i}(u)$ traces a curve on the focal submanifold.
Repeating these construction for all $r^{i}(u)$ and all $k$, we obtain the so-
called focal net. Note that the congruence lines are tangent to the images of
the $i$-th coordinate lines on the $i$-th focal submanifold $F_{i}$. The
$ij$-Laplace transformation replaces the congruence lines by the lines tangent
to the images of the $j$-th coordinate lines. The crucial fact about
diagonalizable systems of conservation laws is that the curves of focal net
are pairwise conjugate [AF-99], which translates analytically as

$\partial_{k}\partial_{l}r^{i}(u)\in T_{r^{i}}F_{i}.$

Thus the families of tangent planes to $F_{i}$ along the images of $i$-th
coordinate lines envelope a ruled hypersurfaces, whose $(n-1)$-dimensional
generators are spanned by the vectors $\partial_{k}r^{i}(u),\ k\neq i$. These
generators determine an integrable $(n-1)$-dimensional distribution on
$F_{i}$. Let as call the integral submanifolds of this distribution conjugate
to the congruence with respect to the $i$-th focal submanifold.

###### Theorem 14

Suppose that, for a semihamiltonian system (1), the $i$-th focal submanifold
of the corresponding congruence is $n$-dimensional. If the $i$-th Monge
characteristic distribution has an extra Riemann invariant of order 0 then the
submaifolds, conjugate to the congruence with respect to the $i$-th focal
submanifold $F_{i}$, are cut on $F_{i}$ by a pencil of hyperplanes.

Conversely, if for some conservative form (7) of (1) the submaifolds,
conjugate to the congruence with respect to the $i$-th focal submanifold
$F_{i}$, are cut on $F_{i}$ by a pencil of hyperplanes then there is a
reciprocal transformation of (1), whose $i$-th Monge characteristic
distribution has an extra Riemann invariant of order 0.

Proof: If there is a Riemann invariant of order 0, then
$\partial_{k}\lambda^{i}=0,\forall k\neq i$ and the conjugate submanifolds are
cut on $F_{i}$, parametrized as in (30), by the pencil of hyperplanes
$y^{0}=\lambda^{i}(u).$

To prove the converse claim, we choose homogeneous coordinates so that the
pencil of hyperplanes are spanned by $Y^{0}=0$ and $Y^{n+1}=0.$ Changing the
homogeneous coordinates corresponds to applying a reciprocal transformation to
(1). $\Box$

Remark 2. The above theorem gives a geometric criterion for a semihamiltonian
systems (1) to be Darboux integrable up to a reciprocal transformation.

Remark 3. The conjugacy of the directions along $k$-th and $l$-th coordinate
lines can be written as

$\partial_{k}\partial_{l}r^{i}(u)=\sum_{m=1}^{n}C_{im}(u)\partial_{m}r^{i}(u).$

For $n=2$ the congruence sits in $\mathbb{P}^{3}$ and all components $y(u)$ of
the parametrization (30) of the focal surface $F_{i}$ satisfy a scalar
equation of the form (18). There is a nice geometric interpretation of
terminating in one step of Laplace transformation of the corresponding system
(18): the focal surfaces of the congruence, obtained by Laplace
transformation, degenerate to curves (see [D-96], V2, book 4, cap.I). This
interpretation is not valid for the Laplace transformation of (1). The reason
is that the equation for $\lambda$ is different from the one for all the
coordinates of the corresponding focal surface, therefore the Laplace
transformation, given geometrically by the same construction, is applied to
different systems!

Vanishing of $b_{ij}\ \forall j\neq i$ for some fixed $i$ implies existence of
Riemann invariant of order 1 (see lemma 3) for the $i$-th Monge distribution.
This ”degenerate” case has the following geometric interpretation.

###### Theorem 15

The envelopes of tangent planes to the $i$-th focal hypersurface along $i$-th
coordinate lines $u^{k}=const\ \forall k\neq i$ are cones over the images of
these lines on the focal hypersurface with a $(n-3)$-dimensional vertex if and
only if $b_{ij}=0\ \forall j\neq i$.

Proof: The envelopes are swept by $(n-1)$-dimensional affine planes, generated
by vectors $\vec{v}_{j}-a_{ij}\vec{v}_{0},\ {k\neq i}$ at
$r^{i}(u)=(\lambda^{i},\lambda^{1}N^{1}-M^{1},...,\lambda^{1}N^{n}(u)-M^{n})$,
where $\vec{v}_{0}=(1,N^{1},...,N^{n})$,
$\vec{v}_{j}=\partial_{j}\vec{v}_{0}$. Using (29), one checks that
$\partial_{i}(\vec{v}_{j}-a_{ij}\vec{v}_{0})\in\langle\vec{v}_{k}-a_{ik}\vec{v}_{0}\rangle_{k\neq
i}$ if and only if $\partial_{i}a_{ij}-a_{ji}a_{ij}=a_{ij}b_{ij}=0$. This
means that the vector space
$\langle\vec{v}_{k}-a_{ik}\vec{v}_{0}\rangle_{k\neq i}$ is stable along the
$i$-th coordinate line $u^{k}=const,\ k\neq i$ and all the affine generators
intersect at some $(n-3)$-dimensional space at the infinite hyperplane. $\Box$

##  6 Subalgebras of commuting flows and Darboux integrability

Let as fix $i$ and look for subalgebras of the commuting flow algebra (5) cut
by a linear differential relation of the form

$\mu^{i}_{i}=\sum_{k\neq i}K_{k}(u)\mu^{k}_{k}+\sum_{j}L_{j}(u)\mu^{j},$ (35)

compatible with (5). Note that up to index permutation this is a generic form
of one linear (homogeneous) differential relation of first order.

###### Theorem 16

Equations (5) for commuting flows of a semihamiltonian system (1) are
compatible with the differential relation (35) if and only if $K_{k}=0\
\forall k$, $L_{j}=0,\ \forall j\neq i$, $L_{i}=b_{ij}\ \forall j\neq i$ and
the $i$-th Monge characteristic distribution admits an extra Riemann invariant
of order 1, i.e. $\partial_{j}b_{ij}+a_{ij}b_{ij}=0\ \ \forall\ j\neq i$.

Proof: For any $j\neq i$ we can compute the second derivatives $\mu^{i}_{ij}$
in two different ways: either differentiating (35) by $u^{j}$ or
differentiating $a_{ij}(\mu^{j}-\mu^{i})$ by $u^{i}$. Simplifying modulo
(5,35), we see that $\mu^{i}_{ii}$ appears only once with the coefficient
$K_{j}$. Therefore $K_{j}=0$, since we are not allowed to get any further
relations for (35) compatible with (5). Similarly, the coefficient of
$\mu^{j}_{j}$ yields $L_{j}=0$, vanishing of the coefficient of $\mu^{j}$
implies $L_{i}=b_{ij}$ , and, finally, on computing the coefficient of
$\mu^{i}$ we conclude $\partial_{j}L_{i}=-a_{ij}+a_{ij}a_{ji}$, hence
$\partial_{j}b_{ij}+a_{ij}b_{ij}=0$. $\Box$

Even though the infinitesimal action of the symmetry algebra of commuting
flows is defined only on solutions of (1) via (6) (i.e. solutions to (5)
provide not classical but generalized symmetry operators [O-86]) we still can
talk about the orbits of a solution.

###### Theorem 17

Suppose that the $i$-th Monge characteristic distribution of a semihamiltonian
system (1) admits an extra Riemann invariant of order 1. Then the orbit of the
subalgebra of (5) defined by the differential relation

$\mu^{i}_{i}=b_{ij}\mu^{i},$ (36)

is given by the differential relation (25) and is fixed by the choice of
$\varphi_{i}$.

Proof: Following the general approach (see for instance [O-82]), we compute a
differential equation that defines the orbit of a generic solution. Let $\mu$
be a solution to (5). Then the operator

$\sum_{j}\mu^{j}u^{j}_{x}\partial_{j}+\sum_{j}D_{x}(\mu^{j}u^{j}_{x})\partial_{u^{j}_{x}}+\sum_{j}D_{t}(\mu^{j}u^{j}_{x})\partial_{u^{j}_{t}}-\mu^{i}D_{x}$
(37)

gives a generalized symmetry of (1). (We used the fact that the total
differentiations $D_{x},D_{t}$ generate the ideal of generalized symmetries of
any system [O-86].) Let us consider the action of (37) on differential
functions of $x,t,u,u_{x}$. Substituting $\mu^{j}_{k}$ for $k\neq j$ from (5)
we get

$-\mu^{i}(\partial_{x}+u^{i}_{x}\partial_{i}+u^{i}_{xx}\partial_{u^{i}_{x}})+\sum_{j}\mu^{j}(u^{j}_{x}\partial_{j}+u^{j}_{xx}\partial_{u^{j}_{x}})+\sum_{j}u^{j}_{x}\left(\mu^{j}_{j}u^{j}_{x}+\sum_{k\neq
j}a_{jk}(\mu^{k}-\mu^{j})u^{k}_{x}\right)\partial_{u^{j}_{x}}.$

Collecting the (operator-valued) coefficients of $\mu^{j}$, $\mu^{j}_{j}$ and
substituting $\mu^{i}_{i}$ yields:

$\displaystyle\mu^{i}\left(-\partial_{x}+b_{ij}(u^{i}_{x})^{2}\partial_{u^{i}_{x}}-\sum_{j\neq
i}u^{j}_{x}\left(\partial_{j}+a_{ij}u^{i}_{x}\partial_{u^{i}_{x}}\right)\right)+$
(38) $\displaystyle\sum_{j\neq
i}\mu^{j}_{j}(u^{j}_{x})^{2}\partial_{u^{j}_{x}}+\sum_{j\neq
i}\mu^{j}\left(u^{j}_{x}\left(\partial_{j}+a_{ij}u^{i}_{x}\partial_{u^{i}_{x}}\right)+u^{j}_{xx}\partial_{u^{j}_{x}}\right).$
(39)

To write down an equation for the orbit, we need invariants of our subalgebra,
depending on $x,t,u,u_{x}$. On the ”tangent space” of the orbit at a ”point”
(which is a solution to (1)), the initial values of $\mu^{k}$, $k=1,...,n$ and
$\mu^{k}_{k}$, $k\neq i$ are arbitrary. Thus, splitting with respect to these
variables, we see that the invariants are the first integrals of the
distribution generated by the following set of vector fields

$\\{\partial_{u^{j}_{x}},\ \partial_{j}+a_{ij}u^{i}_{x}\partial_{u^{i}_{x}},\
\ -\partial_{x}+b_{ik}(u^{i}_{x})^{2}\partial_{u^{i}_{x}}\\}_{j\neq i}.$ (40)

Note that $b_{ik}=b_{ij}$ for any $j\neq i\neq k\neq j$, this distribution is
involutive by lemma 3, and its integral manifolds are of codimension 3. A
solution to (1) gives a 2-dimensional surface in $(2n+2)$-dimensional space
with the coordinates $x,t,u,u_{x}$. If this solution is generic then its
orbit, being also the orbit of the Lie pseudogroup with the
$(2n-1)$-dimensional algebra (40), is $(2n+1)$ dimensional. Being invariant of
codimension 1 in $(2n+2)$-dimensional space, the orbit is given by a single
relation between invariants. Thus, one can define the orbit by a single
equation

$F(x,t,u,u^{i}_{x})=0,$ (41)

where $F$ is an invariant of (40) and this equation is compatible with (1).
(For more detail on the construction of orbits, i.e. automorphic PDEs, see
[O-82]).

Now we use the compatibility. To this end we substitute the differentials

$du^{i}_{x}=-\frac{1}{\partial_{u^{i}_{x}}F}\left(\partial_{t}(F)dt+\partial_{x}(F)dx+\sum_{k}\partial_{i}(F)du^{k}\right),\
\ \ \ du^{k}=u^{k}_{x}(dx+\lambda^{k}dt).\\\ $

into

$d(du^{i})=du^{i}_{x}\wedge(dx+\lambda^{i}dt)+u^{i}_{x}\sum_{k}\lambda^{i}_{k}u^{k}_{x}dx\wedge
dt=0.$

The resulting equation

$\\{-\partial_{t}+\lambda^{i}\partial_{x}-\lambda^{i}_{i}(u^{i}_{x})^{2}\partial_{u^{i}_{x}}\\}F+\sum_{k\neq
i}u^{k}_{x}(\lambda^{i}-\lambda^{k})\\{\partial_{k}+a_{ik}u^{i}_{x}\partial_{u^{i}_{x}}\\}F=0$

cannot impose any additional relation on the orbit. Since the operators
$\partial_{k}+a_{ik}u^{i}_{x}\partial_{u^{i}_{x}}$ kill $F$ for any $k\neq i$
the function $F$ is also invariant along the vector field
$-\partial_{t}+\lambda^{i}\partial_{x}-\lambda^{i}_{i}(u^{i}_{x})^{2}\partial_{u^{i}_{x}}$.
Invoking lemma 3 we conclude that $F$ is a Riemann invariant of the $i$-th
Monge distribution. Since the integral manifolds of this distribution are of
codimension 2, any single relation (41) may be rewritten as (25). $\Box$

##  7 Concluding remarks

###  7.1 Classification

Vessiot [V-39] showed that Goursat’s classification [G-98] of Darboux
integrable scalar equations of second order with two independent variables is
governed by Lie groups (see also [S-00] for modern exposition and [V-01] for
generalizations). Is it possible to find a similar classification for Darboux
integrable systems (1)?

###  7.2 Algebraic structure

Laplace transformations were extended for systems of linear hyperbolic PDEs
with several dependent and two independent variables via generalized
factorization in [T-05] (see also [R-99]). Is an analogous algebraic approach
relevant to the linear system of commuting flows (5)?

###  7.3 Web rank

Darboux integrabe semihamiltonian linearly degenerate systems (1) have a nice
geometric property, put in the language of the web theory [BB-38] as follows:
on any solution, the Monge characteristic distributions and the coordinate
lines $x=const$, $t=const$ cut an $(n+2)$-web of rank $n$ [F-90]. For the
general case of Darboux integrable systems (1), no web-geometric
interpretation is known.

###  7.4 Nondiagonalizable systems

The well-known hamiltonian system

$\frac{\partial}{\partial t}\left(\begin{array}[]{l}u_{1}\\\ u_{2}\\\
u_{3}\end{array}\right)=\left(\begin{array}[]{ccc}-1&1&1\\\ 1&-1&1\\\
1&1&-1\end{array}\right)\frac{d}{dx}\left(\begin{array}[]{l}\partial_{u_{1}}H(u)\\\
\partial_{u_{2}}H(u)\\\ \partial_{u_{3}}H(u)\end{array}\right)$

with the Hamiltonian density $H(u)=\frac{1}{2}u_{1}u_{2}u_{3}$ is
nondiagonalizable, linearly degenerate and ”integrable” in the sense of having
infinitely many higher conservation laws and admitting a Lax pair. It does not
possesses Riemann invariants of order 0 or 1. Its characteristic speeds can be
made constant by a reciprocal transformation [F-93], thus giving Riemann
invariants of order 0 for the transformed system. The author does not know if
this system has Riemann invariants of higher order. It seems that no example
of Darboux integrable non-diagonalizable system of hydrodynamic type ever
appeared in literature.

## Acknowledgements

This research was supported by FAPESP grant #2018/20009-6.

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