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EDN: DCHWJH УДК 517.9+514.7
Contact Mappings of Jet Spaces
Oleg V. Kaptsov*
Institute of Computational Modelling SB RAS Krasnoyarsk, Russian Federation
Received 10.03.2023, received in revised form 15.05.2023, accepted 04.07.2023
Abstract. In this paper we consider mappings of jet spaces that preserve the module of canonical Pfaffian forms, but are not generally invertible. These mappings are called contact. A lemma on the prolongation of contact mappings is proved. Conditions are found under which these mappings transform solutions of some partial differential equations into solutions of other equations. Examples of contact mappings of differential equations are given. We consider contact mappings depending on a parameter and give example of differential equation invariant under the maps.
Keywords: jets, canonical differential forms, invariant solutions.
Citation: O.V. Kaptsov, Contact Mappings of Jet Spaces, J. Sib. Fed. Univ. Math. Phys., 2023, 16(5), 583-590. EDN: DCHWJH.
Introduction
As is well known, contact transformations are used to solve problems of classical mechanics and equations of mathematical physics [1-3]. The most known examples of such transformations are the Legendre and Ampere transformations. The theory of contact transformations was developed by S.Lie. At present, there are numerous sources devoted to these issues [4-7]. The contact transformations are diffeomorphisms of the jet space that preserve the contact structure. To integrate differential equations, it is useful to find contact transformations that leave these equations invariant.
However, not only contact transformations are applied to integrate differential equations. Leonhard Euler started using differential substitutions, which are not diffeomorphisms, to integrate linear partial differential equations [8]. Now these substitutions are called the Euler-Darboux transformation [9] or simply the Darboux transformation [10].
In this paper, we consider analytic mappings of jet spaces that preserve the modulus of canonical differential forms and call these mappings contact. We prove a lifting lemma that shows how to construct a contact mapping. For applications to differential equations, the mappings are required to transform solutions of the equations into solutions of other equations or act on solutions of given equations. Examples of second-order partial differential equations connected by contact mappings are given.
We also study contact mappings depending on a parameter. It is easier to look for such mappings in the form of series in powers of the parameter. As an example, we consider the Burgers equation. Parametric contact mappings are found that act on solutions of this equation. These mappings have no inversional maps.
* kaptsov@icm.krasn.ru https://orcid.org/0000-0002-9562-9092 © Siberian Federal University. All rights reserved
1. Contact mappings of the jet space
We begin with notation and definitions. Denote by Z^0 the non-negative integer numbers and by Nn the set of natural numbers 1,... ,n. The p-th-order jet space [5] with coordinates {xi,uja : i e Nn,j e Nm,a e Z"0} is denoted by Jp(Rn, Rm) or simply Jp. We suppose that J0 = Rn(x) x Rm(u).
Denote by Jw the space of infinite jets and by np the projection np : Jw —> Jp. Consider a point a e Jw and a ring Ap of convergent power series centered at the point ap = np(a). We write
Aa = U Apa.
p=0
Recall that an operator D on a ring is called a derivation operator if it satisfies the conditions:
D(a + b) = Da + Db, D(ab) = D(a)b + aD(b)
for all elements a, b of the ring. We say that a derivation operator Dk (k e Nn) on the ring Aa is the total derivative when the following conditions is satisfied
Dk (xi) = Sik , Dk (uja) = K+ik,
where Sik is the Kronecker delta and 1k = (Sik,..., Snk) e Z"0. So Aa is a differential ring with set A of derivation operators Di,..., Dn; its elements are called differential power series. Any ideal of Aa stable under A is called a differential ideal of Aa. The differential ideal of the ring Aa generated by the set E C Aa is denoted by < E >. The set of differentials
{dxi,duja : i e Nn,j e Nm; a e Z^, M < p}
generates a left module Qap of differential 1-forms over the ring Aap. As usual, we say that differential forms
n
j = duja Di(w>a)dxi, j e Nm, a e Z^ (1)
i=i
are canonical.
Definition. A submodule of the left module Qap generated by canonical forms (where |a| < p) is denoted by Cp and is called the contact submodule.
We describe below a dual transformation of forms [11]. Let A(W) be the ring of analytic functions on an open set W C Rk and let Q(W) be the left module of differential 1-forms on W. Suppose Wi C Rkl, W2 C Rk2 are open sets. Then any analytic mapping 4 : Wi —> W2 induces a homomorphism
4* : A(W2) ^A(Wi), 4*(f )= f (4),
and a linear map 4* : Q(W2) ^ il(Wi), given by
( Jk2. \ k2 <H E fi(y)d'Vi) =J2 4*(fi)d4i,
^i=i ' i=i
where 4i is component of 4. It is convenient to think of 4* as a module homomorphism over different rings connected by the homomorphism of rings. The maps 4* and 4* are usually not distinguished.
We now generalize the classical contact transformations [3].
Definition. Let U be a neighborhood of a point a € Jp and let V be a neighborhood of a point b € Jq (q < p). An analytic mapping $ : U ^ V is called a contact mapping if the module homomorphism <f)k maps the contact submodule Cq into the contact submodule Cp.
The following lifting lemma shows how to construct a contact mapping.
Lemma. Let U be a neighborhood of the point a € Jp and let $ : U ^ J° be an analytic mapping of the form
y = f (x,...,ua), u = g(x,...,ua), \a\ < p
such that the matrix Df = (Difjis invertible at the point a. Then there exists an open set U1 C Jp+1 and a unique contact mapping 4*1 : U1 ^ J1 coinciding with $ on U.
Proof. In what follows, we use the following notation
dx = (dxi,... ,dxn), dua = (du^, ■ ■ ■, du™), Ua+1 = (Di(uja))i^i^n, l^j^m-
According to the definition of a contact mapping, the differential form dv — v1dx must be represented as
dg — V1df = Bo(du — u1dx) + • • • + Bp(dua — ua+1dx), \a\ = p, (2)
where B0,... ,Bp are m x m matrices.
The left-hand side of equation (2) is written as
gxdx + gudu +-----+ gua dua — V1(fx dx + fudu +-----+ fua dua),
where gx = (^j), (iu),..., fUa = (f) are the corresponding Jacobian matrices. We collect together the coefficients of similar differential terms in (2) and set all of them equal to zero. The result is a system of matrix equations
gx — vfx + B0u1 +-----+ Bpua+1 = 0,
gu — V1fu = B0, ••• , gua — v 1 fua = Bp. Substituting B0,... ,Bp into the first equation of this system, we have
Dg = vDf,
with matrices Dg = (Digj), Df = (Difk) where j € Nm and i,k € Nn. By the hypotheses of our lemma, the matrix Df is invertible, so the lifting formula (first prolongation) is
v1 = (Dg) ◦ (Df)-1. (3)
The lifting to J2,..., Jk+1 is carried out in a similar way. The recurrent formula has the form
vk+1 = (Dvk) ◦ (Df )-1. (4)
These formulas are generalizations of the well-known formulas for the lifting (prolongation) of point transformations [1,9].
Definition. If E = {fi}ieNk is a family of differential series of the ring Apa, then the expression
fi =0, 1 < i < k is called a system of differential equations and denoted by sys[E].
Definition. Let E be a family of differential series in the ring Apa and let V be an open set in Rn. We say that a smooth mapping s : V ^ Jp annihilates the family E if
s*(E) = 0, S*(Cp) = 0 (5)
and the rank of s is n at every point of V. If n0 is the projection of Jp onto J0, then the composition n o s is called a solution of sys[E].
The mapping s is lifted so that it annihilates the canonical forms (1) as described above. The lifted map is denoted by s.
Proposition. Let Ei, E2 be two families of differential series of rings Ap and Ap respectively. Let U be a neighborhood of the point ap £ Jp and let V be an open set in 1". Assume that a mapping s : V ^ U annihilates Ei and $ : U ^ Jq (q ^ p) is a contact mapping such that $*(E2) C< Ei >, then $ o s annihilates E2.
Proof. Since
$*(E2) c< Ei >, (6)
it is clear that s*($*(E2)) = 0. It follows that
s*(E2($)) = E2($ o s) = ($ o s)*(E2)=0.
The equality ($ o s)*(Cp) = 0 follows in the same way.
Remarks. To put it simply, the contact mapping $ maps solutions of sys[E\] to solutions of sys[E2] if $*(E2) c< E1 >. If we extend the homomorphism $* to the ideal < E2 >, then the condition (6) can be written more invariantly
$*(< E2 >) C< Ei > .
Definition. Let E be a family of differential series of the ring Ap and let U be a neighborhood of the point ap £ Jp. A contact mapping $ : U ^ Jq (q ^ p) such that $*(E) C< E > is called a symmetry of sys[E] .
Let us give examples of contact mappings connecting partial differential equations. We now use the classical notation. Consider two equations
utt = x"uxx, n £ N, (7)
m
vtt = vyy + yVy, m £ 1. (8)
We want to find a contact mapping that transforms solutions of the equation (7) to solutions of the equation (8). Consider a mapping $ : J1 ^ J0 of the form
t' = t, y = h(x), v = f (x)ux + g(x)u, (9)
where h, f, g are some smooth functions. We will lift this mapping according to the formulas
(3), (4)
Vt = Dtv = futx + gut, Vtt = futtx + gutt, _Dx(v)_ Dx(fux + gu) _ Dx(vy)
y Dxh h' yy h'
Substituting the found expression for vtt into (8), we have
m
futtx + gutt = vyy + yvy.
We can express utt ,uttx by using (10) and obtain a new equation
1 n (Dx(fux + gu)\ mDx(fux + gu)
(10)
Xuxx)xf + xnuxxg + hDX^XKJ + ' ^ = (11)
The left-hand side of this equation is a polynomial 1ÏÏ. 'U'xxx, u*xx, U*x, U* *
Collecting the coefficients
of similar terms in the polynomial and setting all of them equal to zero, we obtain four equations for the functions f, h, g. The two shortest equations are
xn(h')2 = 1, m(h')2g' + hh'g'' — hh''g' = 0.
Integrating these equations for n = 2, we find
2 2-n -I
h = ±--x 2 + co, g = c- + c2h- m,
2-n
where c0,ci, c2 are arbitrary constants. The remaining two equations for the function f are easy to integrate. The following two cases arise: ci = 0, c2 =0 and ci = 0, c2 = 0. In the first case, the function f is equal to ax (a G R). Then the transformation
y = ±-x 2 , v = a(xux + (n — 1)u), a G R
2-n
maps solutions of the equation (7) into ones of equation
3n — 4
Vtt = Vyy +
2—n
In the second case, the transformation
2 2
y = ±--x 2 , v = ax2n-3(xux + (n — 1)u)
2—n
maps solutions of the equation (7) into ones of equation
5n — 8
Vtt = Vyy +
n2
2. Parametric contact mappings
It is well known that finding symmetries of differential equations can be simplified if we restrict ourselves to the search for one-parameter groups of transformations that leave the equations invariant. In this section, it is assumed that contact mappings depend on the parameter a. More precisely, we seek an expansion of the mappings in powers of a.
Next we restrict ourselves to to the case n = 2, m =1 and use the classical notation for coordinates in the jet spaces J0(x,y,u), J 1(x,y,u,p,q), J2(x,y,u,p, q,r,s,t).
Consider a mapping of the form
— 2 3
x = x + axi + a X2 + a X3 + ... ,
y = y + ayi + a2y2 + a3y3 + .. . , (12)
— 2 3
u = u + aui + a U2 + a U3 + ... ,
where x1,x2,x3,y1,... ,u3 are functions of x,y,... ,ua. To find the first prolongation of the mapping (12)
23
p = p + api + a p2 + a p3 + ... , 23
q = q + aqi + a q2 + a q3 + ... , it is necessary that the differential form
cdo = du — pdx — qdy (13)
y
vanishes when the Pfaff equation
= du — pdx — qdy = 0 (14)
is satisfied.
Substituting the expressions (12) into the form os0, by using the equality (14), and collecting together all terms that contain a, we obtain the well-known the first prolongation formulas [1]
Pi = Dx(ui) — pDx(xi) — qDx(yi), qi = Dy (ui) — pDy (xi) — qDy (yi ).
Collecting together all terms that contain a2, we find that
P2 = Dx(u2) — pDx(x2) — PiDx(xi) — qDx(y2) — qiDx (yi), q2 = Dy (u2 ) — pDy (x2 ) — piDy (xi) — qDy (y2 ) — qiDy (yi).
It is important to remark that x2, y2,u2 are an arbitrary functions. When we collect together all terms that contain a3 this leads to
p3 = Dx(u3) — pDx(x3) — piDx(x2) — p2Dx(xi) — qDx(y3) — q1Dx(y2) — q2Dx(yi), q3 = Dy (u3) — pDy (x3) — piDy (x2 ) — p2Dy (xi) — qDy (y3) — qi Dy y) — q2Dy (yi).
Similar formulas are valid for pn, qn (n > 3).
It is easy to find formulas for the second prolongation
2 3 2 3
r = r + ari + a r'2 + a r3 + ... , s = s + asi + a S2 + a S3 + ... , t = t + ati + a2t2 + a3t3 + ....
For this to be accomplished, it is necessary that the differential forms
tSjio = dps — rdx — Sdy tSjoi = dq — Sdx — tdy
vanish if
= dp — rdx — .sdy = 0, = dq — sdx — tdy = 0.
Using arguments similar to those given above, it is easy to obtain the following formulas
ri = Dx(pi) — rDx(xi) — sDx (yi), si = Dy (pi) — rDy (xi) — sDy (yi), ti = Dy (qi) — sDy (xi) — tDy (yi), r2 = Dxp) — rDx(x2) — riDx(xi) — sDx(y2) — siDx(yi), s2 = Dy (p2 ) — rDy (x2 ) — riDy (xi ) — sDy y) — siDy (yi), t2 = Dy (q2) — sDy (x2 ) — siDy (xi) — tDy (y2 ) — ti Dy (yi).
As example, consider the Burgers equation
uy — uxx — uux = 0. (15)
We look for contact mappings such that (15) is invariant under the ones. The symmetry condition implies that the expression
us y uSxx usus x
lies in the ideal < ut — uxx — uux > .
The simplest of these mappings has the form
2aux
x = x, y = y, u = u +--, a G R.
au + 1
This mapping satisfies the second-order differential equation
_ = 2 ua(aua — u)
uaa 2 _ ,
au +1
with initial conditions: u(0) = u, ua(0) = ux. Recall that in Lie theory, symmetry transformations satisfy first order ordinary differential equations [1].
A more general symmetry mapping is given by the formulas
x = x, y = y, u = u + 2Dx(log h),
where the function h satisfies the condition
Dyh — DXh — uDxh £< uy — uxx — uux > . (16)
More precisely, the following statement is true.
Proposition. Let u be a solution to the equation (15), and let the differential series h satisfy the condition (16). Then the function
v = u + 2Dx(log h) (17)
is also a solution to the Burgers equation
vt — vxx — vvx = 0.
Indeed, substituting the function v given by (17) into the left-hand side of the last equation, we obtain an expression that can be represented as
_ _ + 2n ( Dy h — Dxxh — uDxh uy uxx uux + 2Dx 1 h
Thus the Proposition follows from (16).
It is important to note that if h satisfies the condition (16), then n = Dxh is a solution of the determining equations for the symmetry generator. Therefore, knowing the symmetries of the equation it is easy to find h.
In particular, the condition (16) is satisfied by h of the form
h = so + a[s-(2ux + u2) + s2u + s3(yu + x) + s^(2yux + yu2 + xu) + + S5(y2(4ux + 2u2) + 2xyu + x2 + 2y)],
where a,s0,..., s5 are arbitrary constants. If s0 = 0, then the function u is represented by power series in a. The condition (16) is equivalent to a new determining equation
Dy h — Dx h — uDxh = 0.
In this case, the last equation should follow from Eq. (15).
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. 075-02-2023-912).
References
[1] L.V.Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982.
[2] V.I.Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1978.
[3] N.H.Ibragimov, Transformation Groups Applied to Mathematical Physics, Reidel, Boston, 1985.
[4] G.Bluman, S.Kumei, Symmetries and Differential Equations, NY, Springer, 1989.
[5] P.Olver, Applications of Lie Groups to Differential Equations, Springer, NY, 2000.
[6] N.H.Ibragimov (Editor), CRC Handbook of Lie Group Analysis of Differential Equations. Vol. I-III, CRC Press, Boca Raton, 1995.
[7] I.S.Krasilshchik, A.M.Vinogradov (eds.), Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, AMS, 1999.
[8] L.Euler, Foundations of Integral Calculus, Vol. 3, GIFML, Moscow, 1958 (in Russian).
[9] O.V.Kaptsov, Methods for integrating partial differential equations, Moscow, Science, 2009 (in Russian).
[10] V.Matveev, M.Salle, Darboux Transformations and Solitons, Berlin, Springer-Verlag, 1991.
[11] J.R.Munkres, Analysis on Manifolds, CRC Press, Boca Raton, 1991.
[12] O.Stormark, Lie's structural approach to PDE systems, Cambridge University Press, 2000.
Контактные отображения пространства джетов
Олег В. Капцов
Институт вычислительного моделирования СО РАН Красноярск, Российская Федерация
Аннотация. В работе рассматриваются отображения пространств джетов, сохраняющие контактную структуру — канонические дифференциальные формы Пфаффа. В общем случае они не являются обратимыми, и мы называем их контактными отображениями. Доказывается лемма о поднятии контактных отображений. Найдены условия, гарантирующие, что контактные отображения переводят решения одних уравнений с частными производными в решения других уравнений. Рассматриваются контактные отображения, зависящие от параметра. Приводятся примеры контактных отображений, связывающих решения дифференциальных уравнений, и примеры новых симметрий уравнений.
Ключевые слова: пространства джетов, канонические дифференциальные формы, инвариантные решения.
