#  Contact germs and partial differential equations

O.V. Kaptsov
Institute of Computaional Modelling SD RAS,
Krasnoyarsk, Russia
E-mail<EMAIL_ADDRESS>
The article introduces contact germs that transform solutions of some partial
differential equations into solutions of other equations. Parametric
symmetries of differential equations generalizing point and contact symmetries
are defined. New transformations and symmetries may depend on derivatives of
arbitrary but finite order. The stationary Schrödinger equations, acoustics
and gas dynamics equations are considered as examples.

Key words: germs, symmetry, differential equations.

##  1 Introduction

The theory of point symmetries is now widely used to find solutions to
differential equations. There are many monographs describing the theory and
its applications in detail [1, 2, 3, 4]. In the second half of the last
century, generalizations of point symmetries began to appear. Among them we
mention, first of all, higher, conditional and asymptotic symmetries [5, 6, 7,
8]. These symmetries were found for the KdV, sine-Gordon, KP and others
equations. Higher symmetries make it possible to identify equations that have
a large number of solutions expressed by means of elementary and special
functions. However, it is rarely possible to directly apply higher symmetries
to construct solutions of nonlinear equations. On the other hand, operator
symmetries of the first and second orders are actively used to find solutions
of linear equations [9, 10]. In particular, symmetries allow us to find
coordinate systems in which the equations admit the separation of variables.

Most problems in point symmetry theory are well understood, but there are open
questions. These include, for example, the problem of constructively
describing non-isomorphic subalgebras of a Lie algebra admitted by
differential equations. Note that the transformation groups, in the general
case, are local. Obviously, there is no need to distinguish between the
neighborhoods of the point in which these groups are defined. Therefore, it
makes sense to immediately consider the germs of transformations, which is
what we do in this article.

The article has the following structure. In the second section, basic concepts
are introduced such as prolonged spaces, contact modules of differential
1-forms, differential families of germs, the differential ring of infinite
germs and ideals generated by differential families. Differential equations
are defined using differential families of germs. In the next section, we
introduce contact germs and prove the lifting lemma, which allows one to
construct prolongations of germs to contact ones. The found prolongation
formula generalizes the classical one for point mappings. The fourth section
gives examples of contact germs that transform solutions of some differential
equations into solutions of other equations. As examples, we consider the one-
dimensional acoustic equation and the three-dimensional Schrödinger equation.
In the next section, parametric germs and symmetries are introduced. The new
symmetries generalize classical point symmetries. Transformations defining
symmetries can depend on derivatives above the first order and, as a rule, are
irreversible. This allows us to avoid the restrictions of Bäcklund’s theorem
[11]. Examples of such symmetries for one parabolic equation and for a system
of gas dynamics equations are given.

##  2 Basic concepts

Let $\mathbb{N}$ be the set of non-negative integers, $\mathbb{R}$ be the
field of real numbers, and $P_{i,n}$ is the space of homogeneous polynomials
of degree $i$ in $n$ variables with coefficients in $\mathbb{R}$. We call the
set $J^{k}=\mathbb{R}^{m}\times\prod_{i=1}^{k}P_{i,n}$ the space of $k$-jets,
and $P^{k}=\mathbb{R}^{n}\times J^{k}$ the $k$ times prolonged space
$P^{0}=\mathbb{R}^{n+m}$ or the $k$-th prolongation of $P^{0}$. Let
$P^{\infty}$ be the space $P^{0}\times\prod_{i=1}^{\infty}P_{i,n},$
$\pi^{j}_{i}$ the projection of $P^{j}$ onto $P^{i}$, and $\pi_{i}$ the
projection of $P^{\infty}$ onto $P^{i}$. The coordinate functions on
$P^{\infty}$ are denoted by $x_{k},u^{l}_{\alpha}$, where $k=1,\dots,n$,
$l=1,\dots,m$, $\alpha\in\mathbb{N}^{n}$.

Let $a_{i}=\pi_{i}(a)$ be the projection of a point $a\in P^{\infty}$,
$\mathcal{F}_{a_{i}}$ the ring of germs of smooth functions at point $a_{i}$
for every $i\in\mathbb{N}$, and
$\nu^{j}_{i}:\mathcal{F}_{a_{i}}\longrightarrow\mathcal{F}_{a_{j}}$ the the
canonical embedding ($i\leq j$). The family of rings
$(\mathcal{F}_{a_{i}})_{i\in\mathbb{N}}$ together with the embeddings
$\nu^{j}_{i}$ forms a directed system [12]. The direct limit of
$\mathcal{F}_{a}^{\infty}=\underrightarrow{\lim}\mathcal{F}_{a_{i}}$ is called
the ring of $\infty$-germs at point $a$, with embeddings
$\nu_{i}:\mathcal{F}_{a_{i}}\rightarrow\mathcal{F}_{a}^{\infty}$. We introduce
the total derivatives (derivation operators)

$D_{x_{k}}(f)=\frac{\partial f}{\partial x_{k}}+\sum_{\begin{subarray}{c}1\leq
j\leq m\\\ \ \alpha\in\mathbb{N}^{n}\end{subarray}}\frac{\partial f}{\partial
u^{j}_{\alpha}}u^{j}_{\alpha+1_{k}}\quad\forall f\in\mathcal{F}_{a},\quad
k=1,\dots n,$

where $1_{k}$ is an $n$-dimensional vector with all coordinates zero except
the $k$-th coordinate, which is equal to $1$. Then $\mathcal{F}_{a}$ becomes a
differential ring.

We denote by $\Omega_{a_{i}}$ the module over the ring $\mathcal{F}_{a_{i+1}}$
generated by the differentials $dx_{i},du^{j}_{\alpha}$, where $i=1,\dots,n$,
$j=1,\dots,m$, $\alpha\in\mathbb{N}^{n}$. The elements of this module are
called germs of differential 1-forms. For any $r,s\in\mathbb{N}$ such that
$r\leq s$, there are linear maps
$\eta^{s}_{r}:\Omega_{a_{r}}\longrightarrow\Omega_{a_{s}}$, given by the
formulas $\eta^{s}_{r}(fdg)=\nu^{s}_{r}(f)dg$. This is how a directed system
$\\{\Omega_{a_{r}},\nu^{s}_{r}\\}$ arises, which has a direct limit
$\Omega_{a}^{\infty}=\underrightarrow{\lim}\Omega_{a_{r}}$.

Definition. A submodule $C_{a_{k}}$ of the module $\Omega_{a_{k}}$ generated
by differential forms of the form

$\omega^{j}_{\alpha}=du^{j}_{\alpha}-\sum_{1\leq i\leq
n}u^{j}_{\alpha+1_{i}}dx_{i},\qquad|\alpha|\leq k$

is called contact.

Definition. The finite set
$E=\\{f_{1},\dots,f_{l}\\}\subset\mathcal{F}_{a_{k}}$ is called a differential
family, and the expression

$f_{1}=\cdots=f_{l}=0$ (1)

by a system of differential equations.

Any differential family $E$ generates a differential ideal $<E>$ of the ring
$\mathcal{F}_{a}^{\infty}$. The elements of an ideal are represented as a sum
of germs of the form $g_{\alpha}D^{\alpha}f_{i}$, where
$\alpha=(\alpha_{1},\dots,\alpha_{n})$,
$D^{\alpha}=D_{x_{1}}^{\alpha_{1}}\cdots D_{x_{n}}^{\alpha_{n}}$ , $f_{i}\in
E$, $g_{\alpha}\in\mathcal{F}_{a}^{\infty}$. The germ of a smooth mapping
$f:\mathbb{R}^{k_{1}}\longrightarrow\mathbb{R}^{k_{2}}$ at a point
$p\in\mathbb{R}^{k_{1}}$ is denoted by
$f:(\mathbb{R}^{k_{1}},p)\longrightarrow\mathbb{R}^{k_{2}}$.

Definition. If $s_{0}:(\mathbb{R}^{n},c)\longrightarrow\mathbb{R}^{m}$ is a
germ, then its $k$-th prolongation is the germ
$s_{k}:(\mathbb{R}^{n},c)\longrightarrow P^{k}$ given by
$s_{k}=(q_{0},q_{1},...,q_{k})$, where $q_{0}=s_{0}$,
$q_{i}=\frac{\partial^{\alpha}(s_{0})}{\partial
x_{1}^{\alpha_{1}}\dots\partial x_{n}^{\alpha_{n}}}$, $|\alpha|=i>0$. An
infinite prolongation of the germ $s_{0}$ is the sequence
$s_{\infty}=(q_{i})^{\infty}_{i=0}$

Definition. Let $E=(f_{i})_{i=1}^{l}\subset\mathcal{F}_{a_{k}}$ be a
differential family. A germ
$s_{0}:(\mathbb{R}^{n},c)\longrightarrow\mathbb{R}^{m}$ is called a solution
to the system (1), if its $k$-th prolongation annihilates the family, i.e.
$f_{i}\circ s_{k}=0$ for $i=1,\dots,n$. We will say that $s_{\infty}$
annihilates the ideal $<E>$ if $f\circ s_{\infty}=0$ $\forall f\in<E>$.

Remark. The germ $s_{k}$ annihilates the family $E$ if and only if
$s_{\infty}$ annihilates the ideal $<E>$. This follows from the well-known
property of solutions of the differential equations [1]: if $s_{k}$
annihilates $f$, then $s_{k+1}$ annihilates $D_{x_{i}}(f)$ for any
$i=1,...,n$.

##  3 Contact germs

In this section we introduce contact germs of smooth maps of prolonged spaces
acting on solutions of differential equations.

Let $\phi:(P^{j},a_{j})\longrightarrow P^{i}$ be a germ, where $j\geq i$ and
$\phi(a_{j})=b_{i}$. It induces a ring homomorphism
$\phi^{\star}:\mathcal{F}_{b_{i}}\longrightarrow\mathcal{F}_{a_{j}}$ with

$\phi^{\star}(f)=f(\phi)\quad\forall f\in\mathcal{F}_{b_{i}}$

and a linear map of the modules
$\psi^{\star}:\Omega_{b_{i}}\longrightarrow\Omega_{a_{j}}$, given by

$\psi^{\star}(fdg)=\phi^{\star}(f)d\phi^{\star}(g)\quad\forall
f,g\in\mathcal{F}_{b_{i}}$

Matrix $\mathfrak{D}\phi$ with elements $D_{x_{i}}(\phi_{k})$, where
$\phi_{k}$ is the $k$-th component of the germ and $i=1,\dots n$, is called
the total Jacobian matrix of the germ $\phi$.

Definition. A germ $\phi:(P^{j},a_{j})\longrightarrow(P^{i},b_{i})$ is called
contact of order $i$ if the mapping $\psi^{\star}$ preserves contact modules,
i.e. $\psi^{\star}(C_{b_{i}})\subset C_{a_{j}}$.

The following lifting lemma describes a way of constructing contact germs. The
classical statement can be found in [1, 13].

Lemma 1. Let $\Phi_{0}:(P^{k},a_{k})\longrightarrow P^{0}$ be a germ of the
form

$y_{i}=f_{i}(x_{1},\dots,x_{n},u^{1},\dots,u^{m}_{\alpha}),\qquad 1\leq i\leq
n,\ |\alpha|\leq k,$
$\tilde{u}_{j}=g_{j}(x_{1},\dots,x_{n},u^{1},\dots,u^{m}_{\alpha}),\qquad
1\leq j\leq m,$

and the total Jacobian matrix $\mathfrak{D}f$ is invertible. Then for any
$s\geq 1$ there is a contact germ $\Phi_{s}:(P^{k+s},a_{k+s})\longrightarrow
P^{s}$ of the form $(v_{0},\dots,v_{s})$, where $v_{0}=\Phi_{0}$,
$v_{1}=(\mathfrak{D}g)(\mathfrak{D}f)^{-1}$, and the remaining components are
given by the formula

$v_{i+1}=(\mathfrak{D}v_{i})(\mathfrak{D}f)^{-1}.$ (2)

Proof. Let us introduce the notations $f=(f_{1},\dots,f_{n})$,
$g=(g_{1},\dots,g_{m})$, $dx=(dx_{1},\dots,dx_{n})$,
$du_{\alpha}=(du^{1}_{\alpha},\dots,du^{m}_{\alpha})$,
$u_{\alpha+1}=\mathfrak{D}u_{\alpha}$. To construct a first-order contact
germ, we need to find matrices $v_{1},B_{0},\dots,B_{k}$ that satisfy the
condition

$dg-v_{1}df=B_{0}(du-
u_{1}dx)+\cdots+B_{k}(du_{\alpha}-u_{\alpha+1}dx),\qquad|\alpha|=k.$ (3)

The left side of the last formula is written as

$g_{x}dx+g_{u}du+\dots+g_{\alpha}du_{\alpha}-v_{1}(f_{x}dx+f_{u}du+\dots+f_{\alpha}du_{\alpha}),$

where $g_{x}=(\frac{\partial g_{j}}{\partial x_{i}}),\ (\frac{\partial
g_{j}}{\partial u^{i}}),\dots,f_{u_{\alpha}}=(\frac{\partial f_{j}}{\partial
u^{i}_{\alpha}})$ are standard Jacobian matrices.

From the coefficients of $dx,du,...,du_{\alpha}$ in (3) we get

$g_{x}-v_{1}f_{x}+B_{0}u_{1}+\dots+B_{k}u_{\alpha+1}=0,\qquad|\alpha|=k$
$g_{u}-v_{1}f_{u}=B_{0},\quad\cdots,\quad
g_{u_{\alpha}}-v_{1}f_{u_{\alpha}}=B_{k}.$

Substituting $B_{0},...,B_{k}$ into the first equation of this system, we have

$\mathfrak{D}g=v_{1}\mathfrak{D}f.$

By condition, the matrix $\mathfrak{D}f$ is invertible. This means that the
prolongation formula has the form

$v_{1}=\mathfrak{D}g(\mathfrak{D}f)^{-1}.$ (4)

Thus, a contact germ $\Phi_{1}=(v_{0},v_{1})$ of the first order is obtained.
The formula (2) is proved in a similar way.

The germ $\Phi_{k}$ constructed in the previous lemma will be called the
$k$-th prolongation of the germ $\Phi_{0}$. We will call an infinite sequence
$\Phi_{\infty}=\\{v_{i}\\}_{i\geq 0}$ a $\infty$-contact germ.

Let us define the homomorphism
$\Phi_{\infty}^{\star}:\mathcal{F}_{b}^{\infty}\longrightarrow\mathcal{F}_{a}^{\infty}$
by the formula $\Phi_{\infty}^{\star}(f)=f(\Phi_{\infty})$ for any
$f\in\mathcal{F}_{b}^{\infty}$.

Proposition. Let $E_{1}\subset\mathcal{F}_{a_{k}}$ and
$E_{2}\subset\mathcal{F}_{b_{k}}$ be differential families,
$\Phi_{\infty}:P_{a}^{\infty}\longrightarrow P^{\infty}_{b}$ be an
$\infty$-contact germ, and $s_{\infty}$ annihilates the ideal $<E_{1}>$. If
$\Phi_{\infty}^{\star}(<E_{2}>)\subset<E_{1}>$, then $\Phi_{\infty}\circ
s_{\infty}$ annihilates the ideal $<E_{2}>$.

Indeed, according to the conditions, $g\circ\Phi_{\infty}\in<E_{1}>$ for any
$g\in<E_{2}>$ and $f\circ s_{\infty}=0$ $\forall f\in<E_{2}>$ . Then the
following is true

$g\circ\Phi_{\infty}\circ s_{\infty}=0.$

Thus, contact germs act on solutions of differential systems.

Definition. Differential families $E_{1}\subset\mathcal{F}_{a_{k}}$ and
$E_{2}\subset\mathcal{F}_{b_{k}}$ are called equivalent if there exist
$\infty$-contact germs $\Phi_{\infty}:P_{a}^{\infty}\longrightarrow
P^{\infty}_{b}$ , $\Psi_{\infty}:P_{b}^{\infty}\longrightarrow P^{\infty}_{a}$
such that $\Phi_{\infty}^{\star}(<E_{2}>)\subset<E_{1}>$ and
$\Psi_{\infty}^{\star}(<E_{1}>)\subset<E_{2}>$.

Definition. Let $E=(f_{i})_{i=1}^{l}\subset\mathcal{F}_{a_{k}}$ be a
differential family. If there are $\infty$-contact germs
$\Phi_{\infty}:P_{a}^{\infty}\longrightarrow P^{\infty}_{a}$ such that
$\Phi_{\infty}^{\star}(<E>)\subset<E>$, then $\Phi_{\infty}$ is called the
symmetry of the system

$f_{1}=0,\cdots,f_{l}=0.$

##  4 Actions of germs on solutions

This section provides examples of contact germs acting on solutions of
differential equations. For convenience, classical notations are used. So
instead of $u^{j}_{\alpha}$ we will use the notation
$\frac{\partial^{\alpha}u^{j}}{\partial x_{1}^{\alpha_{1}}\cdots\partial
x_{n}^{\alpha_{n}}}$ or even shorter ones like $u_{tt},u_{xx}$ .

As a first example, consider the equation

$u_{tt}=(x^{2}u_{x})_{x},$ (5)

such equations were studied in [14] in connection with applications to
acoustics.

We will seek germs of the form

$\tau=t,\quad y=h(x),\quad v=f(x)u_{x}+g(x)u$ (6)

whose prolongations are symmetries of the equation (5). Here $f,g,h$ are germs
depending on $x$. According to Lemma 1, the first and second prolongations of
the germ (6) are given by the formulas

$v_{t}=fu_{tx}+gu_{t},\qquad v_{y}=\frac{D_{x}(fu_{x}+gu)}{h^{\prime}},$
$v_{tt}=fu_{ttx}+gu_{tt}\ ,\qquad v_{yy}=\frac{D_{x}(v_{y})}{h^{\prime}}\ .$

Substituting these formulas into the left side of the equation

$v_{tt}-(y^{2}v_{y})_{y}=0,$

we get an expression containing $u_{tt},u_{ttx}$. We calculate these
derivatives using the equation (5). As a result, we have a new expression,
which is a polynomial of the first degree in $u_{xxx},u_{xx},u_{x},u$.
Equating the coefficients of the polynomial to zero, we obtain a system of
ordinary differential equations for the functions $f,g,h$. Solving the
obtained system, we find the transformation of the form

$\tau=t,\quad y=\frac{c_{0}}{x},\quad v=c_{1}x^{2}u_{x}+c_{2}xu,$

where $c_{0},c_{1},c_{2}$ are arbitrary constants.

Remark. Equation (5) by substitution

$u=\frac{1}{\sqrt{x}}w(t,\ln x)$

is reduced to the Klein-Gordon equation

$w_{tt}=w_{zz}-\frac{1}{4}w.$

Let us now proceed to construct the transformations connecting the three-
dimensional stationary Schrödinger equation

$\Delta\psi+U(x_{1},x_{2},x_{3})\psi=0,$ (7)

with Laplace’s equation.

We will need the following statement [13]. Given a linear partial differential
equation

$u_{xx}+H(x)u+Lu=0,$ (8)

where $L$ is an operator of the form

$L=\sum_{|\alpha|\geq 0}^{k}a_{\alpha}(y)\frac{\partial^{\alpha}}{\partial
y_{1}^{\alpha_{1}}\cdots\partial
y_{n}^{\alpha_{n}}},\qquad\alpha=(\alpha_{1},\dots,\alpha_{n}),$

then the Euler-Darboux transformation

$v=u_{x}-\frac{h^{\prime}(x)}{h(x)}u$ (9)

maps solutions of the equation (8) into solutions of the equation

$v_{xx}+H_{1}(x)v+Lv=0,$

if $h$ satisfies the ordinary differential equation

$h^{\prime\prime}+(H(x)+c)h=0,\qquad c\in\mathbb{R},$ (10)

and $H_{1}$ has the form

$H_{1}=H+2(\ln h)^{\prime\prime}.$

Consider Laplace’s equation

$u_{xx}+u_{yy}+u_{zz}=0$

and apply the Euler-Darboux transformation (9) to it. Then the function
$v(x,y,z)$ satisfies the Schrödinger equation of the form

$v_{xx}+v_{yy}+v_{zz}+2(\ln h)^{\prime\prime}v=0,$ (11)

where $h$ is a solution of the equation

$h^{\prime\prime}+ch=0,\quad c\in\mathbb{R}.$

The solutions of the last equation have the form

$h=c_{1}x+c_{2},\quad(c=0),$
$h=c_{1}\exp(kx)+c_{2}\exp(-kx),\quad(c=-k^{2}<0),$
$h=c_{1}\sin(kx)+c_{2}\cos(-kx),\quad(c=k^{2}>0).$

As a result, the function $H_{1}$ has one of three forms

$\frac{-2c_{1}^{2}}{(c_{1}x+c_{2})^{2}},\qquad\frac{8c_{1}c_{2}k^{2}}{(c_{1}\exp(kx)+c_{2}\exp(-kx))^{2}},\qquad\frac{-8c_{1}c_{2}k^{2}}{(c_{1}\sin(kx)+c_{2}\cos(-kx))^{2}}.$

Next, we apply the Euler-Darboux transformation to the equation (11)

$v^{1}=v_{y}-v\frac{h^{\prime}(y)}{h(y)}$

and we arrive at the Schrödinger equation

$\Delta v^{1}+(H_{1}(x)+H_{2}(y))v^{1}=0,$

where $H_{2}(y)$ has one of the three forms indicated above, with $x$ replaced
by $y$. If we now perform the Euler-Darboux transformation on $z$, we obtain
the equation

$\Delta v^{2}+(H_{1}(x)+H_{2}(y)+H_{3}(z))v^{2}=0.$

To the last equation we can again apply the Euler-Darboux transformations in
$x$, $y$, $z$ and obtain new equations with new potentials. In this case, it
is necessary to find solutions to the corresponding equations of the form
(10).

Another way to transform the Schrödinger equation (7) is to apply a group of
three-dimensional conformal transformations to it, leaving the Laplace
equation invariant. For example, the inversion transformation

$x_{1}^{\prime}=\frac{x_{1}}{r},\qquad x_{2}^{\prime}=\frac{x_{2}}{r},\qquad
x_{3}^{\prime}=\frac{x_{3}}{r},\qquad\psi_{1}^{\prime}=\frac{\psi}{r},$

where $r^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$, maps solutions of the equation
(7) into solutions of the equation

$\Delta\psi^{\prime}+\frac{1}{(r\prime)^{4}}U(x_{1}^{\prime},x_{2}^{\prime},x_{3}^{\prime})\psi^{\prime}=0,$

where
$(r\prime)^{2}=(x_{1}^{\prime})^{2}+(x_{2}^{\prime})^{2}+(x_{3}^{\prime})^{2}$.
Similar arguments can easily be transferred to the $n$-dimensional Schrödinger
equation.

##  5 Parametric symmetries

The search for contact germs preserving the differential ideals of equations
is in general a difficult problem. As is well known, Sophus Lie proposed the
use of transformations acting on solutions of differential equations and
forming local groups [1, 3]. In the paper [15] it was proposed to abandon some
of the limitations of Lie theory. It was assumed that transformations are of
the form

$\tilde{x}_{i}=x_{i}+\sum_{k=1}^{\infty}a^{k}f_{ik}\qquad\tilde{u}^{j}=u^{j}+\sum_{k=1}^{\infty}a^{k}g_{jk},$

where $f_{ik},g_{jk}$ are analytic functions of $x_{l},u^{s}_{\alpha}$ (
$1\leq l\leq n$, $1\leq s\leq m$, $0\leq|\alpha|\leq p$). Such transformations
do not necessarily form a local group since they may be irreversible. In
particular, it was shown in [15] that Burgers’ equation

$u_{t}=u_{xx}+uu_{x}.$

is invariant under the transformation

$t^{\prime}=t,\qquad x^{\prime}=x,\qquad u^{\prime}=u+2D_{x}(\ln(1+au)),\qquad
a\in\mathbb{R}$

and an infinite number of other transformations depending on higher
derivatives

We now recall the construction from [15] and assume for simplicity that $m=1$,
$n=2$. Let us consider a formal parametric mapping defined by power series in
the parameter $a$

$\tilde{x}=x+\sum_{k=1}^{\infty}a^{k}x_{k},\qquad\tilde{y}=y+\sum_{k=1}^{\infty}a^{k}y_{k},\qquad\tilde{u}=u+\sum_{k=1}^{\infty}a^{k}u_{k},$
(12)

where $x_{k},y_{k},u_{k}$ are germs depending on $x,y,u,u_{10},u_{01}$. We
further use the classical notation $p=u_{x},\ q=u_{y},\ r=u_{xx},\ s=u_{xy},\
t=u_{yy}$.

Suppose that the first prolongation of the formal mapping (12) has the form

$\tilde{p}=p+\sum_{k=1}^{\infty}a^{k}p_{k},\qquad\tilde{q}=q+\sum_{k=1}^{\infty}a^{k}q_{k},$
(13)

where $p_{k},q_{k}$ are germs depending on $x,y,u,p,q$. We require that the
Pfaffian equation

$d\tilde{u}-\tilde{p}d\tilde{x}-\tilde{q}d\tilde{y}=0$ (14)

was a consequence of the system

$du=pdx+qdy,\quad dp=rdx+sdy,\quad dq=sdx+tdy.$ (15)

Substituting the series (12), (13) into the equation (14) and using the system
(15), we obtain the Pfaffian equation of the form

$Adx+Bdy=0,$

where $A,B$ are power series in $a$, all coefficients of which must be equal
to zero. Thus, collecting the terms containing $a^{1}$ into $A$ and $B$, we
obtain

$p_{1}=D_{x}(u_{1})-pD_{x}(x_{1})-qD_{x}(y_{1}),\quad
q_{1}=D_{y}(u_{1})-pD_{y}(x_{1})-qD_{y}(y_{1}).$ (16)

These are the classical formulas for coefficients of the prolonged
infinitesimal operator in Lie theory [1, 2].

Equating coefficients of $a^{2}$ into $A$ and $B$ to zero yields

$\displaystyle
p_{2}=D_{x}(u_{2})-pD_{x}(x_{2})-p_{1}D_{x}(x_{1})-qD_{x}(y_{2})-q_{1}D_{x}(y_{1}),$
$\displaystyle
q_{2}=D_{y}(u_{2})-pD_{y}(x_{2})-p_{1}D_{y}(x_{1})-qD_{y}(y_{2})-q_{1}D_{y}(y_{1}).$

Note that, in contrast to the group analysis of differential equations, the
germs $x_{2},y_{2},u_{2}$ are not expressed in terms of $x_{1},y_{1},u_{1}$.

The identical vanishing of coefficients of $a^{n}$ implies

$\displaystyle
p_{n}=D_{x}(u_{n})-\sum_{i=0}^{n-1}(p_{i}D_{x}(x_{n-i})+q_{i}D_{x}(y_{n-i})),$
$\displaystyle
q_{n}=D_{y}(u_{n})-\sum_{i=0}^{n-1}(p_{i}D_{y}(x_{n-i})+q_{i}D_{y}(y_{n-i})),$

with $p_{0}=p,\ q_{0}=q$.

Consider now the Pfaffian equations

$d\tilde{p}=\tilde{r}d\tilde{x}+\tilde{s}d\tilde{y},\qquad
d\tilde{q}=\tilde{s}d\tilde{x}+\tilde{t}d\tilde{y},$ (17)

where $\tilde{r},\tilde{s},\tilde{t}$ are represented by

$\tilde{r}=r+\sum_{k=1}^{\infty}a^{k}r_{k},\qquad\tilde{s}=s+\sum_{k=1}^{\infty}a^{k}s_{k},\qquad\tilde{t}=t+\sum_{k=1}^{\infty}a^{k}t_{k}.$
(18)

The equations (17) must be consequences of the system

$\displaystyle dp=rdx+sdy,\qquad dq=sdx+tdy,$ $\displaystyle
dr=u_{30}dx+u_{21}dy,\quad ds=u_{21}dx+u_{12}dy,\quad dt=u_{12}dx+u_{03}dy.$

Substituting (13), (18) into (17) and using the latter system, we get two
expressions of the form

$A_{1}dx+B_{1}dy=0,\qquad A_{2}dx+B_{2}dy=0$

where $A_{1},B_{1},A_{2},B_{2}$ are power series in $a$, all coefficients of
which must vanish identically. Calculating the coefficients of these series in
$a,a^{2}$, we have

$\displaystyle r_{1}=D_{x}(p_{1})-rD_{x}(x_{1})-sD_{x}(y_{1}),$ $\displaystyle
s_{1}=D_{y}(p_{1})-rD_{y}(x_{1})-sD_{y}(y_{1}),$ $\displaystyle
t_{1}=D_{y}(q_{1})-sD_{y}(x_{1})-tD_{y}(y_{1}),$ $\displaystyle
r_{2}=D_{x}(p_{2})-rD_{x}(x_{2})-r_{1}D_{x}(x_{1})-sD_{x}(y_{2})-s_{1}D_{x}(y_{1}),$
$\displaystyle
s_{2}=D_{y}(p_{2})-rD_{y}(x_{2})-r_{1}D_{y}(x_{1})-sD_{y}(y_{2})-s_{1}D_{y}(y_{1}),$
$\displaystyle
t_{2}=D_{y}(q_{2})-sD_{y}(x_{2})-s_{1}D_{y}(x_{1})-tD_{y}(y_{2})-t_{1}D_{y}(y_{1}).$

If necessary, the calculations are easy to continue.

As a first example, consider the equation

$u_{y}-u^{2}u_{xx}=0.$ (19)

Let’s find new symmetries of this equation. For this purpose one requires that
the expression

$\tilde{u}_{\tilde{y}}-\tilde{u}^{2}\tilde{u}_{\tilde{x}\tilde{x}}=\tilde{q}-\tilde{u}^{2}\tilde{r}$
(20)

lies in the differential ideal $<u_{y}-u^{2}u_{xx}>$. We first substitute
expressions for $\tilde{q},\ \tilde{u},\ \tilde{r}$ into (20) and obtain a
power series in $a$. Equating coefficient of $a$ to zero leads to equation

$q_{1}-u^{2}r_{1}-2uru_{1}=0.$ (21)

The left side of (21) must lie in the differential ideal
$<u_{y}-u^{2}u_{xx}>$. Note that (21) is the defining equation for the
infinitesimal symmetries of the equation (19). The equation (21) reduces to a
linear overdetermined partial differential system for the functions
$x_{1},y_{1},u_{1}$. If we assume that these functions depend on
$x,y,u,u_{x}$, then the solution of this system has the form

$x_{1}=f,\quad y_{1}=c_{1}y+c_{2},\quad
u_{1}=u_{x}f+(c_{3}x+c_{4})u_{x}-u(c_{3}+c_{1}/2),$ (22)

where $f$ is an arbitrary smooth function of $x,y,u,u_{x}$, and
$c_{1},\dots,c_{4}$ are arbitrary constants.

When we find the coefficient of $a^{2}$ and equate it to zero, we obtain the
second defining equation for the functions $x_{2},y_{2},u_{2}$

$q_{2}-u^{2}r_{2}-2ur_{1}u_{1}-ru_{1}^{2}-2ruu_{2}=0,$ (23)

which should also be a differential consequence of the equation (19). If the
function $f$ in (22) vanishes identically, then it is not possible to find
non-trivial solutions to the second defining equation (23).

Suppose that $f\neq 0$ and $c_{1}=\cdots c_{4}=0$, then a nontrivial solution
of the equation (23) can exist if

$f=u\sqrt{b_{1}y+b_{2}-2y_{2}},\qquad
u_{2}=(x_{2}+b_{3}x+b_{4})u_{x}-(b_{3}+b_{4}/4)u,$

where $x_{2},y_{2}$ are arbitrary functions of $x,y,u,u_{x}$ and
$b_{1},b_{2},b_{3},b_{4}$ are arbitrary constants. Assuming
$x_{2}=y_{2}=b_{1}=b_{3}=b_{4}=0$, and $b_{2}=1$, we obtain the mapping

$\tilde{x}=x+au,\qquad\tilde{y}=y,\qquad\tilde{u}=u+auu_{x},$ (24)

generating the non Lie symmetry of the equation (19). Repeated application of
the map (24) leads to a symmetry depending on $u_{xx}$. This process can be
repeated any number of times and new symmetries can be obtained.

Let us now consider a system of linear partial differential equations

$L(x,\partial)u=0,\qquad L(x,\partial)=\sum_{|\alpha|\geq
0}^{k}A_{\alpha}\partial^{\alpha},$ (25)

where $u=(u^{1},\dots,u^{m})$,
$\partial^{\alpha}=\frac{\partial^{\alpha}}{\partial
x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{n}}}$,
$\alpha=(\alpha_{1},\dots,\alpha_{n})$, and $A_{\alpha}$ are $m\times m$ are
matrices depending on $x=(x_{1},\dots,x_{n})$, . Suppose this system admits a
symmetry operator written in the canonical form [2, 3]

$X=\sum_{i=1}^{m}\eta_{i}\frac{\partial}{\partial
u^{i}}+\sum_{\begin{subarray}{c}1\leq j\leq m\\\
\alpha\in\mathbb{N}^{n}\end{subarray}}D^{\alpha}\eta_{i}\frac{\partial}{\partial
u^{i}_{\alpha}}$

Then the collection $\eta=(\eta_{1},\dots,\eta_{m})$ must satisfy the
condition

$L(x,D)\eta\in<L(x,\partial)u>.$

Consequently, if $u$ is a solution to the system (25), then $\eta$ is also a
solution to this system.

Consider now the one-dimensional gas dynamics equations

$\displaystyle
u_{t}+uu_{x}+\rho^{n}\rho_{x}=0,\qquad\rho_{t}+u\rho_{x}+u_{x}\rho=0,$

where $u$ is the gas velocity, and $\rho$ is the density. We assume that
entropy remains constant and $n\in\mathbb{R}$. We use the hodograph
transformation [16], and interchange the roles of dependent and independent
variables. This leads to a system

$x_{\rho}-ut_{\rho}+\rho^{n}t_{u}=0,\qquad x_{u}-ut_{u}+\rho t_{\rho}=0.$ (26)

One of the admited infinitesimal operators of the last system in canonical
form is

$(bt+at_{u})\frac{\partial}{\partial
t}+(bx+a(x_{u}-t))\frac{\partial}{\partial x},\qquad\forall a,b\in\mathbb{R}.$

So the transformation

$\tilde{u}=u,\quad\tilde{\rho}=\rho,\quad\tilde{t}=bt+at_{u},\quad\tilde{x}=bx+a(x_{u}-t)$

maps solutions of a system (26) into solutions of the same system, i.e. this
transformation is a symmetry. This map in the original variables has the form

$\tilde{u}=u,\quad\tilde{\rho}=\rho,\quad\tilde{t}=bt+\frac{a\rho_{x}}{\Delta}\
,\quad\tilde{x}=bx-a\frac{\rho_{t}+t}{\Delta}\ ,$

with $\Delta=u_{t}\rho_{x}-u_{x}\rho_{t}$. It is a symmetry of the gas
dynamics equations. In this way we can construct a series of symmetries.

##  6 Conclusion

The paper proposes a new approach to the study of symmetries of differential
equations. To present the classical theory and its generalizations, it turns
out to be convenient to use germs of functions and differential forms.
Parametric symmetries are of great interest for further research, since this
makes it possible to use parameter expansion and sequentially solve the
defining equations. Parametric families of germs are, in general, not
invertible and can depend on derivatives of finite order. To show the
effectiveness of this approach it is necessary to find new examples of
differential equations admitting parametric symmetries.

This work is supported by the Krasnoyarsk Mathematical Center and financed by
the Ministry of Science and Higher Education of the Russian Federation in the
framework of the establishment and development of regional Centers for
Mathematics Research and Education (Agreement No. 075-02-2024-1378).

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