On Beltrami fields with nonconstant proportionality factor on the plane
On Beltrami fields with nonconstant
proportionality factor on the plane
Vladislav V. Kravchenko1, Héctor Oviedo2
Department of Mathematics, CINVESTAV del IPN, Unidad Queretaro,
Libramiento Norponiente No. 2000, Fracc. Real de Juriquilla, Queretaro,
Qro. C.P. 76230 MEXICO e-mail: vkravchenko@qro.cinvestav.mx∗
SEPI, ESIME Zacatenco, Instituto Politécnico Nacional, Av. IPN S/N,
C.P. 07738, D.F. MEXICO†
November 2, 2018
Abstract
We consider the equation
B + α
B = 0 (1)
on the plane with α being a real-valued function and show that it can
be reduced to a Vekua equation of a special form. In the case when α
depends on one Cartesian variable a complete system of exact solutions
of the Vekua equation and hence of equation (1) is constructed based
on L. Bers’ theory of pseudoanalytic formal powers.
1 Introduction
Solutions of the equation
B + α
B = 0 (2)
∗Research was supported by CONACYT, Mexico via the research project 50424.
†During the preparation of this work the second-named author was supported by
CONACYT on a postdoctoral stay at the Department of Mathematics, CINVESTAV
del IPN, Unidad Querétaro
http://arxiv.org/abs/0704.1488v2
where α is a scalar function of space coordinates are known as Beltrami
fields and are of fundamental importance in different branches of modern
physics (see, e.g., [22], [18], [7], [21], [1], [9], [8], [11]). For simplicity, in this
work we consider the real-valued proportionality factor α and real-valued
solutions of (2), though the presented approach is applicable in a complex-
valued situation as well with a considerable complication of mathematical
techniques involved (instead of complex Vekua equations their bicomplex
generalizations should be considered [5], [14]). We consider equation (2) on a
plane of the variables x and y, that is α and
B are functions of two Cartesian
variables only. In this case as we show in section 3 equation (2) reduces to
the equation
+ αu = 0. (3)
This second-order equation can be reduced (see [14]) to a corresponding
Vekua equation (describing generalized analytic functions) of a special form.
This reduction under quite general conditions allows us to construct a com-
plete system of exact solutions of (3) explicitly (see [13] and [15]). For the
reduction of (3) to a Vekua equation it is sufficient to find a particular solu-
tion of (3). In the present work (section 4) we show that in a very important
for applications case of α being a function of one Cartesian variable a particu-
lar solution of (3) is always available in a simple explicit form. This situation
corresponds to models describing waves propagating in stratified media (see,
e.g., [16]). As a result in this case we are able to construct a complete system
of solutions explicitly which for many purposes means a general solution. We
give an example of such construction.
We show in this work that when α = α(y) (of course in a similar way
the case α = α(x) can be considered) equation (3) and hence equation (2)
reduce to the Vekua equation of the following form
∂zW (x, y) =
if ′(y)
2f(y)
W (x, y) (4)
where
sinA+ c2√
cosA;
A is an antiderivative of α with respect to y, c1 and c2 are arbitrary real con-
stants, z = x + iy and ∂z =
(∂x + i∂y). A complete (in a compact uniform
convergence topology) system of exact solutions to (4) can be constructed
explicitly. The system represents a set of formal powers [3], [6] which gen-
eralize the usual analytic complex powers (z − z0)n, n = 0, 1, 2, . . . and in
a sense give us a general solution of (4). Thus, in the case when α is a
function of one Cartesian variable the Vekua equation equivalent to (2) in a
two-dimensional situation can be solved and a complete system of solutions
of (2) is obtained.
2 Preliminaries
We need the following definition. Consider the equation
∂zϕ = Φ (5)
on a whole complex plane or on a convex domain, where Φ = Φ1 + iΦ2 is a
given complex valued function such that its real part Φ1 and imaginary part
Φ2 satisfy the equation
∂yΦ1 − ∂xΦ2 = 0 (6)
then there exist real valued solutions of (5) which can be easily constructed
in the following way
ϕ(x, y) = 2
Φ1(η, y)dη +
Φ2(x0, ξ)dξ
+ c (7)
where (x0, y0) is an arbitrary fixed point in the domain of interest and c is
an arbitrary real constant.
By A we denote the integral operator in (7):
A[Φ](x, y) = 2
Φ1(η, y)dη +
Φ2(x0, ξ)dξ
Note that formula (7) can be extended to any simply connected domain by
considering the integral along an arbitrary rectifiable curve Γ leading from
(x0, y0) to (x, y)
ϕ(x, y) = 2
Φ1dx+ Φ2dy
Thus if Φ satisfies (6), there exists a family of real valued functions ϕ such
that ∂zϕ = Φ, given by the formula ϕ = A[Φ].
Let f denote a given positive twice continuously differentiable function
defined on a domain Ω ⊂ C. Consider the following Vekua equation
W in Ω (8)
where the subindex z means the application of the operator ∂z , W is a
complex-valued function and W is its complex conjugate function. As was
shown in [12], [13], [14], [15], [17] equation (8) is closely related to the second-
order equation of the form
(div p grad+q)u = 0 in Ω (9)
where p and q are real-valued functions. In particular the following state-
ments are valid.
Theorem 1 [14] Let u0 be a positive solution of (9). Assume that f =
and W is any solution of (8). Then u = 1√
ReW is a solution of (9) and
p ImW is a solution of
grad+q1)v = 0 in Ω (10)
where
q1 = −
. (11)
Theorem 2 [14] Let Ω be a simply connected domain, u0 be a positive so-
lution of (9) and f = p1/2u0. Assume that u is a solution of (9). Then a
solution v of (10) with q1 defined by (11) such that W = p
1/2u+ ip−1/2v is a
solution of (8) is constructed according to the formula
v = u−10 A(ipu
0∂z(u
0 u)) (12)
and vice versa, let v be a solution of (10), then the corresponding solution
u of (9) such that W = p1/2u + ip−1/2v is a solution of (8), is constructed
according to the formula
u = −u0A(ip−1u−20 ∂z(u0v)). (13)
Thus the relation between (8) and (9) is very similar to that between
the Cauchy-Riemann system and the Laplace equation. Moreover, choosing
p ≡ 1, q ≡ 0 and u0 ≡ 1 we obtain that (12) and (13) become the well known
formulas from the classical complex analysis for constructing conjugate har-
monic functions.
For a Vekua equation of the form
Wz = aW + bW
where a and b are arbitrary complex-valued functions from an appropriate
function space [20] a well developed theory of Taylor and Laurent series in
formal powers was created (see [3], [4]) containing among others the expan-
sion and the Runge theorems as well as more precise convergence results
(see, e.g., [19]) and a general simple algorithm [15] for explicit construction
of formal powers for the Vekua equation of the form (8).
3 Reduction of (2) to a Vekua equation
We consider equation (2) where both α and
B are supposed to be dependent
on two Cartesian variables x and y. Then equation (2) can be written as the
following system
∂yB3 + αB1 = 0 (14)
− ∂xB3 + αB2 = 0 (15)
∂xB2 − ∂yB1 + αB3 = 0.
Solving this system for B3 leads to the equation
∆B3 −
+ α2B3 = 0 (16)
where 〈·, ·〉 denotes the usual scalar product of two vectors.
Note that
α div
= ∆B3 −
and hence (16) can be rewritten as follows
+ αB3 = 0. (17)
Thus equation (2) reduces to an equation of the form (9) with p = 1/α and
q = α.
Let us notice that (see, e.g., [14])
∇+ α = 1√
(∆− r) 1√
where
r = −1
− α2. (18)
That is B3 is a solution of (17) iff the function f = B3/
α is a solution of
the stationary Schrödinger equation
(−∆+ r) f = 0 (19)
with r defined by (18). As was explained in section 2, given its particular
solution this equation reduces to the Vekua equation (8). Unfortunately, in
general we are not able to propose a particular solution of (17). Nevertheless
in an important special case when α depends on one Cartesian variable, a
particular solution of (17) is always available in explicit form. We give this
result in the next section.
4 Solution in the case when α is a function of
one Cartesian variable
Let us consider equation (19) where α = α(y). We assume that α is a
nonvanishing function and look for a solution of the corresponding ordinary
differential equation
f0 = 0.
Its general solution is known (see [10, 2.162 (14)]) and is given by the ex-
pression
f0(y) =
sinA(y) +
cosA(y) (20)
where A is an antiderivative of α and c1, c2 are arbitrary real constants.
Choosing, e.g., c1 = 1, c2 = 0 and calculating the coefficient (∂zf0) /f0
we arrive at the following Vekua equation which is equivalent to (2) in the
case under consideration (and which is considered in any simply connected
domain where sinA(y) does not vanish),
∂zW (x, y) =
α(y) cotA(y)− α
2α(y)
W (x, y).
Note that F = f0 =
sinA(y)√
and G = i
sinA(y) represent a generating
pair for this Vekua equation (see [13], [15]) and hence if W is its solution,
the corresponding pseudoanalytic function of the second kind ω = 1
ReW +
if0 ImW satisfies the equation
1− f 20
1 + f 20
ωz (21)
which can be written in the form of the following system
ψy, φy = −
where φ = Reω and ψ = Imω.
For f0 being representable in a separable form f0(x, y) = X(x)Y (y) the
formulas for constructing corresponding formal powers explicitly were pre-
sented already by L. Bers and A. Gelbart (see [3] and [6]). Using them we
obtain the following representation for the formal powers corresponding to
(n)(a, z0; z) = a1
(x− x0)(n−k)ikY k
+ ia2
(x− x0)(n−k)ikỸ k
(we preserve the notations from [3]) where z0 = x0+ iy0 is an arbitrary point
of the domain of interest, a is an arbitrary complex number: a = a1 + ia2,
Y k and Ỹ k are constructed as follows
Y (0)(y0, y) = Ỹ
(0)(y0, y) = 1
and for n = 1, 2, . . .
Y (n)(y0, y) = n
Y (n−1)(y0, η)f
0 (η)dη n odd
Y (n)(y0, y) = n
Y (n−1)(y0, η)
f 20 (η)
n even
Ỹ (n)(x0, x) = n
Ỹ (n−1)(y0, η)
f 20 (η)
n odd
Ỹ (n)(x0, x) = n
Ỹ (n−1)(y0, η)f
0 (η)dη n even.
The system
(n)(1, z0; z), ∗Z
(n)(i, z0; z)
represents a complete (in a
compact uniform convergence topology [2]) system of solutions of (21) that
means that any solution ω of (21) in a simply connected domain Ω can be
represented as a series
ω(z) =
(n)(an, z0; z) =
a′n ∗Z
(n)(1, z0; z) + a
(n)(i, z0; z)
where a′n = Re an, a
n = Im an and the series converges normally (uniformly
on any compact subset of Ω). Consequently the system of functions
f0(y) Re(∗Z
(n)(1, z0; z)), f0(y) Re(∗Z
(n)(i, z0; z))
represents in the same sense a complete system of solutions of (19) with r
defined by (18), and
α(y)f0(y) Re(∗Z
(n)(1, z0; z)),
α(y)f0(y) Re(∗Z
(n)(i, z0; z))
is a complete system of solutions of (17). Thus in the case under consideration
any solution B3 of (17) can be represented in the form
B3(x, y) =
(an sinA(y) Re(∗Z(n)(1, z0; z)) + bn sinA(y) Re(∗Z(n)(i, z0; z)))
where an and bn are real constants.
The other two components of the vector
B are obtained from (14) and
(15):
B1 = −
∂yB3 and B2 =
∂xB3 (23)
that gives us a complete system of solutions of (2) in the case under consid-
eration. On the following example we explain how this procedure works.
Example 3 Let us consider the following relatively simple situation in which
the corresponding integrals are not difficult to evaluate. Let
α(y) =
1− y2
and Ω be an open unitary disk with a center in the origin. We take in (20)
c1 = 0 and c2 = 1. Then it is easy to verify that
f0(y) = (1− y2)
The first three formal powers with a centre in the origin can be calculated as
follows
(1)(1, 0; z) = x+ i[
y(1− y2) 32
3y(1− y2) 12
arcsin y],
(1)(i, 0; z) = − y
(1 − y2) 12
+ ix,
(2)(1, 0; z) = x2 − 1
y2 − 3
y arcsin y
(1− y2) 12
+ 2ix
y(1− y2) 32
3y(1− y2) 12
arcsin y
(2)(i, 0; z) = − 2xy
(1− y2) 12
x2 − y2 − 1
(3)(1, 0; z) = x3 − 3x
y arcsin y
(1− y2) 12
+ 3ix2
y(1− y2) 32
3y(1− y2) 12
arcsin y
− i(− 3
y(1− y2)
y(1− y2)
2 + y(1− y2)
(1− y2)2 arcsin y + 33
arcsin y),
(3)(i, 0; z) = −
(1− y2) 12
y(1 + y2)
(1− y2) 12
arcsin y+ix
x2 − 3
Now taking the real parts of these formal powers and multiplying them by the
factor
αf0 (see (22)) we obtain the first elements of the complete system
of solutions of (17), that is any solution B3 of (17) in a simply connected
domain can be represented as an infinite linear combination of the functions
{(1− y2)
2 , x(1 − y2)
2 , −y, (1− y2)
x2 − 1
y2 − 3
y arcsin y
(1− y2) 12
− 2xy, (1− y2)
x3 − 3x
y arcsin y
(1− y2) 12
− 3x2y +
y(1 + y2)−
(1− y2)
2 arcsin y, ...}
and the corresponding series converges normally.
From (23) it is easy to calculate the corresponding components B1 and B2
respectively,
{y, xy, (1− y2)
(1− y2)
2 arcsin y + y(x2 − 3
2x(1− y2)
x(1 − y2)
2 arcsin y + y
x3 − 9
xy2 +
y2 − 3x2)(1− y2)
2 − 3
y arcsin y, ...}
{0, (1− y2), 0, 2x(1− y2), −2y(1− y2)
(3x2 −
y2)(1− y2)−
y(1− y2)
2 arcsin y, −6xy(1− y2)
2 , ...}.
Thus, we obtain the following complete system of solutions of (2) with the
proportionality factor α defined by (24),
B 0 =
(1− y2) 12
B 1 =
(1− y2)
x(1 − y2) 12
B 2 =
(1− y2) 12
B 3 =
(1− y2) 12 arcsin y + y(x2 − 3
y2 + 5
2x(1− y2)
(1− y2) 12
x2 − 1
y2 − 3
y arcsin y
(1−y2)
 ,
B 4 =
2x(1− y2) 12
−2y(1− y2) 12
B 5 =
x(1 − y2) 12 arcsin y + y
x3 − 9
xy2 + 15
(3x2 − 3
y2)(1− y2)− 9
y(1− y2) 12 arcsin y
(1− y2) 12
x3 − 3x
y2 + 3
y arcsin y
(1−y2)
 ,
B 6 =
y2 − 3x2)(1− y2) 12 − 3
y arcsin y
−6xy(1− y2) 12
−3x2y + 3
y(1 + y2)− 3
(1− y2) 12 arcsin y
. . . .
5 Concluding remarks
We presented two new results.
1. We showed that the system of equations describing Beltrami fields on
the plane can be reduced to a Vekua equation of the form (8) whenever
any particular solution of the corresponding second-order equation of
the form (17) or (19) is known.
2. In the case when the proportionality factor α depends on one Carte-
sian variable we obtain a particular solution explicitly, construct the
corresponding Vekua equation (section 4), and solve it in the sense that
a complete system of solutions is obtained which gives us a complete
system of solutions of the original vector equation describing Beltrami
fields.
Of course not always the integrals involved in the construction of the
complete system of solutions are sufficiently easy to evaluate explicitly as
in the example 3. Nevertheless our numerical experiments confirm that in
general the formal powers and hence the solutions of (2) can be calculated
with a remarkable accuracy. For example, the vector
B 40 (see notations in
the example 3) in the Matlab 7 package on a usual PC can be calculated with
a precision of the order 10−4. Thus, the use of formal powers for numerical
solution of boundary value problems corresponding to (2) and more generally
to equations of the form (9) is really promising. The work in this direction
will be reported elsewhere.
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	Introduction
	Preliminaries
	Reduction of (??) to a Vekua equation
	Solution in the case when  is a function of one Cartesian variable
	Concluding remarks
