The Diffiety Institute Preprint Series

Preprint DIPS–4/98 February 10, 1998

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Symmetries of equations containing a small parameter

by

Alexey V. Samokhin

SYMMETRIES OF EQUATIONS CONTAINING A SMALL PARAMETER

Alexey V. Samokhin^{1}

Moscow State University of Civil Aviation 20 Kronshtadtsky Blvd., Moscow 125838, Russia E-mail: asamokhin@glasnet.ru

Abstract For differential equations containing a small parameter ε symmetries are found in a form of series in powers of ε. The recurrent relation on summands of the series is obtained; the first summand is just a symmetry of the unperturbed equation (i.e., for an equation with ε = 0). Conservation laws of the unperturbed equation are not conserved in case ε ≠ 0 and the rate of their decay is explicitly determined. Illustrative examples include the Burgers equation and a system of magnetohydrodynamics equations.

1 Introduction

Most well known model equations of mathematical physics, such as Kortewegde Vries, Burgers, etc. are distinguished by their not-so-usual differential and algebraic properties. They have rich symmetry algebras and/or numerous conservation laws and so on. These properties are, however, regrettably unstable: few survive any perturbations of the equation. It is quite a disappointing situation from the point of view of physical applications: in physics, there is no such thing as an individual equation. Rather, the thing of prime interest is an underlying phenomenon, so there are ’clasters’ of equation describing it in varying assumptions concerning physically meaningful quantifiers; small perturbations of the ideal state equation is a universal way to take into account different changing characteristics.

This paper is an attempt to use symmetries and conservation laws of the ideal (non-perturbed) equation when the latter is perturbed. In the first section the recurrent relations for an expansion of a symmetry by a small parameter is obtained for an evolution equation with one spatial variable. Its truncated solution is used to construct approximate solutions of the perturbed equation starting with a solution of the unperturbed one. The Burgers equation is used for various examples.

In the second section we estimate the rate of ’decay’ of an ideal state conservation law in presence of a dissipative-like perturbations. Different rates of such a decay for different conservation laws produce quasistable solutions. The general argument is illustrated by a system of third order nonlinear magnetohydrodynamics equations with two dependent and three independent variables describing incompressible magnetofluids; this example may be found in [2] in more detail.

1The work was partially supported by RFBR grant 97-01-00462 and ISSEP grant d-87

2 Symmetries

Consider an equation

u_{t} = F (x, t, u, u_{x}, …) + εG(x, t, u, u_{x}, …),  (1)

where ε is a small parameter. Its symmetry may be chosen in the form

S = S(ε, x, t, u_{x}, u_{xx}, …).

Then it must satisfy the symmetry equation

∂_{t}S + {F, S} + ε{G, S},  (2)

where

{F, S} = (?_{F} − ℓ_{F})S = (ℓ_{S}−?_{S})F.

Here, by definition,

ℓ_{F} = ∑_{n=0}^{∞} \frac{∂F}{∂u_{n}} D^{n}

?_{F} = ∑_{n=0}^{∞} D^{n}(F)\frac{∂}{∂u_{n}}

and the total derivative D with respect to x is given by

D = ∑_{i=0}^{∞} u_{n+1} \frac{∂}{∂u_{n}};

u_{n} stands for u^{xxx...x}_{n} If we expand

S(ε, x, t, u, …) = ∑_{k=0}^{∞} S_{i}(x, t, u, …)ε^{i}

and substitute this expansion into (2), we obtain ε-less system of recurrent relations for S_{i}:

∂_{t}S_{i} + {F, S_{i}} + {G, S_{i−1}}.  (3)

Here S_{i} = 0 is assumed for i < 0. Thus S_{0} is a usual symmetry of the unperturbed equation u_{t} = F:

\frac{∂S_{0}}{∂t} + \{F, S_{0}\} = 0.

Even in case when the unperturbed equation possesses a rich nontrivial symmetry algebra, the perturbed equation (1) may has no symmetries. But if we calculate S_{i}, say, up to i = k, then we can obtain a certain k-approximation to a solution of the equation (1) in the following way.

Let u = f(x, t) be a solution of the unperturbed equation u_{t} = F. Consider a problem

u_{τ} = ∑_{i=0}^{k} S_{i}(x, t, u, u_{x} …)ε^{i},

u|_{τ=0} = f(x, t).  (4)

If v(τ, ε, x, t) is a solution of this problem, then

\frac{∂}{∂τ} |_{τ=0} (v_{t} − F − εG) =

\frac{∂}{∂t} ∑_{i=0}^{k} S_{i}(x, t, u, u_{x} …)ε^{i} + \{F, ∑_{i=0}^{k} S_{i}(x, t, u, u_{x} …)ε^{i}\} + \{G, ∑_{i=0}^{k} S_{i}(x, t, u, u_{x} …)ε^{i}\} =

ε^{k+1}{G, S_{k}}.  (5)

If, moreover, {G, S_{k}} is bounded, then (5) means that v is a solution of (1) up to O(ε^{k+1}).

Note that if S is a genuine symmetry of (1), then for any solution of the problem u_{τ} = S, u|_{τ=0} = f and for any τ_{0} the function u|_{τ=τ_{0}} is a solution of (1) as well as f. This procedure is known as a solution generating method. Hence, the solution of the problem (4) leads to a method of generating of approximate solutions.

Remark 1 If the addend G commutes with all symmetries from sym E, then the symmetry algebra of the perturbed equation includes the symmetry algebra of the unperturbed one:

{G, sym E} = 0 ⟹ sym E ⊂ sym E_{ε}

2.1 Examples

2.1.1 u_{t} symmetry

The chain of symmetry equations (3) has few chances to have a solution. Here is one trivial example when it do has a solution (and not as a series, but as a finite sum).

Suppose both F and G do not depend on t explicitly, ∂_{t}F = 0 and ∂_{t} = 0G. Then (3) comes to the system

{F, S_{0}} = 0, {F, S_{i}} = {S_{i−1}, G}, i > 0.

One can take S_{0} = F in the first equation of the system, S_{1} = G in the second one and S_{i} = 0 for i > 1. The resulting symmetry is S = F +εG, which coincides with an obvious symmetry u_{t} restricted to the equation u_{t} = F + εG.

2.1.2 Burgers equation.

Burgers equation has a form

u_{t} = u_{xx} + εuu_{x}.

In this case F = u_{xx}, G = uu_{x} and the symmetry equation is of the form

\frac{∂S_{i}}{∂t} + (∑_{j=0}^{∞} u_{j+2} \frac{∂}{∂u_{n}} − D^{2}) S_{i} = (uD + u_{1} − ∑_{j=0}^{∞} D^{j}(uu_{1})\frac{∂}{∂u_{j}} ) S_{i−1} (6)

The unperturbed equation in this case is simply the heat equation u_{t} = u_{xx}. It has a lot of symmetries. For instance, take S_{0} = u ≡ u_{0}. (This is a common symmetry for all linear equations.) Then we have the following equation on S_{1}:

\frac{∂S_{1}}{∂t} + (∑_{j=0}^{∞} u_{j+2} \frac{∂}{∂u_{j}} − D^{2}) S_{1} = (uD + u_{1} − ∑_{j=0}^{∞} D^{j}(uu_{1})\frac{∂}{∂u_{j}} ) u ≡ uu_{1} (7)

Suppose S_{1} = S_{1}(x, t, u_{0}). Then (7) comes to

\frac{∂S_{1}}{∂t} + u_{2} \frac{∂S_{1}}{∂u} − D^{2}S_{1} = uu_{1}

or

(S_{1})_{t} − (S_{1})_{xx} − 2u_{1}(S_{1})_{xu} − u^{2}(S_{1})_{uu} = uu_{1}.

This equation has no solutions, but the structure of the previous one prompts to take

S_{1} = S_{1}(x, t, u, w), where Dw = u or w = ∫ u dx.

It follows then that

w_{t} = \frac{∂}{∂t} ∫ u dx = ∫ u_{t} dx = ∫ (u_{2} + εuu_{1}) dx = u_{1} + \frac{1}{2}u^{2}

In other words we choose the covering over Burgers equation (see [1]), so that

D_{x} = D + u\frac{∂}{∂w}, D_{t} = ∑_{i=0}^{∞} D^{i}(u_{2} + εuu_{1}) + (u_{1} + \frac{ε}{2}u^{2})\frac{∂}{∂w}

Denote w = u_{−1} for convenience. The symmetry equation for such an S is of the form

\frac{∂S_{1}}{∂t} + ( ∑_{j=−1}^{∞} u_{j+2} \frac{∂}{∂u_{j}} − D^{2}) S_{1} = (uD + u_{1} − ∑_{j=−1}^{∞} D^{j}(uu_{1})\frac{∂}{∂u_{j}} ) S_{0}. (8)

For S_{0} = u it has a solution S_{1} = −\frac{1}{2}uw. Moreover, if S_{i} = Auw^{i}, then S_{i+1} may be chosen as −\frac{A}{2(i+1)}uw^{i+1}. Starting with S_{0} = u we obtain

S = u(1 − \frac{εw}{2} + \frac{1}{2} (\frac{εw}{2} )^{2} − \frac{1}{6} (\frac{εw}{2} )^{3} …) = ue^{− \frac{εw}{2}}

Of course, this result may be obtained in a more straightforward way, i.e., without expansion in series, see [1].

2.1.3 Solution generated symmetries

Here is one more example for Burgers equation. Since the heat equation is a linear one, any of its solutions u = f(x, t) is a symmetry. Take S_{0} = x which is an obvious solution. Assume S_{1} = S_{1}(x, t, u). Then

S_{1} = \frac{α}{4}(2t − x^{2})u + βxu + δu + g(x, t),

where α, β, δ are arbitrary constants and g(x, t) is an arbitrary solution of the heat equation. No S_{2} of the form S_{2}(x, t, u) exists here. But the solution generating process (4) with the help of

S = S_{0} + εS_{1} = x + ε(\frac{α}{4}(2t − x^{2})u + βxu + δu + g(x, t))

is a success up to O(ε^{2}).

3 Conservation laws

First we introduce notions and notations which are necessary to formulate results.

Let E = (E_{1}, …, E_{l}) = 0 be a N-th order system of l nonlinear differential equations on m-vector function (f^{1}, …, f^{m}) of n+1 independent variables (x_{0} = t, x_{1}, …, x_{n}).

We interpret the equation as a submanyfold in a jet space J^{N}(π), where π : IR^{m} × IR^{n+1} ⟶ IR^{n+1} is a trivial bundle. If x_{i} and u^{j} are base and fiber coordinates of π, then u^{j} = f^{j}(x_{0}, x_{1}, …, x_{n}), j = 1, …, m are sections of π denoted by j_{0}(f). The bundle π_{N} : J^{N}(π) ⟶ IR^{n+1} has x for base coordinates and u_{σ}^{j}, j = 1, …, m, σ = (i_{o}, …, i_{n}), |σ| ≤ N for its fiber coordinates. The sections j_{N}(f) of π_{N} are given by the formula

u_{σ}^{j} = \frac{∂^{|σ|}}{∂x^{σ}} f^{j}(x) = \frac{∂^{|σ|}}{∂x_{0}^{i0} ⋯ ∂x_{n}^{in}} f^{j}(x)

Now {E(x, u, …, u_{σ}) = 0} ⊂ J^{N}(π) defines a submanyfold in the jet space and we denote this submanyfold by E. Solutions of E = 0 are such f(x) that j_{N}(f)(IR^{n+1}) ⊂ E. Introduce the total differentiations D_{i} with respect to x_{i}:

D_{i} = \frac{∂}{∂x_{i}} + ∑_{j, σ} u_{σ+1_{i}}^{j} \frac{∂}{∂u_{σ}^{j}} where 1_{k} = (\underbrace{0, …, 0}_{k−1}, 1, 0, …, 0)

All differential prolongations of E i.e., the differential ideal J generated by E and the total differentiations D_{i}, define the submanyfold E^{∞} ⊂ J^{∞}(π).

Consider some l × m matrix ℓ_{E} (l is the codimension of E) in J^{N}(π)),

(ℓ_{E})_{rs} = ∑_{σ} \frac{∂E_{s}}{∂u_{σ}^{r}} D_{σ},

or

ℓ_{E} = ( \begin{matrix}∑_{σ} \frac{∂E_{1}}{∂u_{σ}^{1}} D_{σ} & … & ∑_{σ} \frac{∂E_{1}}{∂u_{σ}^{r}} D_{σ} & … & ∑_{σ} \frac{∂E_{1}}{∂u_{σ}^{m}} D_{σ} \\ ⋮ & & ⋮ & & ⋮ \\ ∑_{σ} \frac{∂E_{s}}{∂u_{σ}^{1}} D_{σ} & … & ∑_{σ} \frac{∂E_{s}}{∂u_{σ}^{r}} D_{σ} & … & ∑_{σ} \frac{∂E_{s}}{∂u_{σ}^{m}} D_{σ} \\ ⋮ & & ⋮ & & ⋮ \\ ∑_{σ} \frac{∂E_{l}}{∂u_{σ}^{1}} D_{σ} & … & ∑_{σ} \frac{∂E_{l}}{∂u_{σ}^{r}} D_{σ} & … & ∑_{σ} \frac{∂E_{l}}{∂u_{σ}^{m}} D_{σ} \end{matrix} )  (9)

A conservation law for the equation E is a differential n– form ω = ∑_{i=0}^{n} ω_{i}d̂x_{i} ,

such that dω = 0 on E^{∞} ; here d̂x_{i} = dx_{0} ∧ … ∧ dx_{i−1} ∧ dx_{i+1} ∧ … ∧ dx_{n} and ω_{i}’s are some functions on J^{∞}(π). ( In physics terminology ω_{0} is called a conserved density, while (−ω_{1}, ω_{2}, …, (−1)^{n}ω_{n}) is called a flux.)

The method for finding of conservation laws is as follows. Let ℓE^{∗} be a formal conjugate of ℓ_{E},

ℓE^{∗} = ( \begin{matrix}∑_{σ}(−1)^{|σ|}D_{σ} ∘ \frac{∂E_{1}}{∂u_{σ}^{1}} & … & ∑_{σ}(−1)^{|σ|}D_{σ} ∘ \frac{∂E_{r}}{∂u_{σ}^{1}} & … & ∑_{σ}(−1)^{|σ|}D_{σ} ∘ \frac{∂E_{l}}{∂u_{σ}^{1}} \\ ⋮ & & ⋮ & & ⋮ \\ ∑_{σ}(−1)^{|σ|}D_{σ} ∘ \frac{∂E_{1}}{∂u_{σ}^{s}} & … & ∑_{σ}(−1)^{|σ|}D_{σ} ∘ \frac{∂E_{r}}{∂u_{σ}^{s}} & … & ∑_{σ}(−1)^{|σ|}D_{σ} ∘ \frac{∂E_{l}}{∂u_{σ}^{s}} \\ ⋮ & & ⋮ & & ⋮ \\ ∑_{σ}(−1)^{|σ|}D_{σ} ∘ \frac{∂E_{1}}{∂u_{σ}^{m}} & … & ∑_{σ}(−1)^{|σ|}D_{σ} ∘ \frac{∂E_{r}}{∂u_{σ}^{m}} & … & ∑_{σ}(−1)^{|σ|}D_{σ} ∘ \frac{∂E_{l}}{∂u_{σ}^{m}} \end{matrix} )

(10)

Solutions ψ of the equation

ℓE^{∗}(ψ)|_{E^{∞}} = 0  (11)

are so called generating functions of conservation laws. They are connected to conservation laws themselves in a following way. By the definition of a conservation law, the equation dω|_{E^{∞}} = 0 holds, which is equivalent to dω = O(E)dx_{0} ∧ … ∧ dx_{n}, where O(E) ∈ J O(E) = ∑_{σ, r} O_{σ}D_{σ}(E_{r}). Now O^{∗}(1) is the generating function of this conservation law or a solution of (11) (^{∗} stand for formal conjugation). It remains however to find the conservation law itself and to check whether it is trivial (trivial by definition are conservation laws ω which are exact, i.e., ω = dw for some (n − 1)– differential form w).

The general procedure for finding the conservation law, starting with its generating function, is connected to C-spectral sequence of the equation; however, for a low order operator O (as in example discussed here) it is usually not hard to discover the conservation law corresponding to any given O^{∗}(1).

The main theorem is as follows (see [2])

THEOREM 1 Consider an equation E(η) depending on a small parameter η in such a way that E_{0} = E(0) is a non–dissipative system. Let ω = ∑_{0}^{n} ω_{i}d̂x_{i} be the

conservation law of E_{0}. Then the decay velocity of the conserved quantity ⟨ω_{0}⟩ in presence of dissipation is given by

\frac{d}{dt}⟨ω_{0}⟩|_{E(η)} = −η⟨O^{∗}(1) · \frac{∂E}{∂η}|_{η=0}⟩ up to O(η^{2})

3.1 Example: Conservation laws of MHD-equation

The 3-dimensional MHD–equation, describing incompressible magnetofluids in dimensionless variables may be taken in the following form, [2]:

\{ \begin{matrix}Δu_{t} + u_{x}Δu_{y} − u_{y}Δu_{x} + v_{y}Δv_{x} − v_{x}Δv_{y} = νΔ^{2}u \\ v_{t} + u_{x}v_{y} − u_{y}v_{x} = ηΔv \end{matrix}  (12)

Here subscripts mean partial differentiation: u_{x} = \frac{∂u}{∂x} and so on. The (12) equation will be denoted E(ν, η) onward. The ideal state is described by E(0, 0) and denoted by E_{0}.

To obtain conservation laws of the ideal state one starts by solving

ℓ_{E_{0}}^{∗}f|_{E_{0}^{∞}} = 0  (13)

Here f = ( \begin{matrix}S \\ T \end{matrix} ) is a possible generating function of a would be conservation

law; components S and T are some functions on J^{∞}(IR^{3}, IR^{2}). We remind that ^{∗} stands for formal conjugation. The universal linearization operator ℓ_{E_{0}} for (12) is given by the formula

ℓ_{E_{0}} = ( \begin{matrix}D_{t}Δ + u_{x}ΔD_{y} + Δu_{y} · D_{x}− & v_{y}ΔD_{x} − v_{x}ΔD_{y}+ \\ u_{y}ΔD_{x} − Δu_{x} · D_{y} & Δv_{x} · D_{y} − Δv_{y} · D_{x} \\ v_{y}D_{x} − v_{x}D_{y} & D_{t} + u_{x}D_{y} − u_{y}D_{x} \end{matrix} ) (14)

Now ℓ_{E_{0}}^{∗} is of the form

ℓ_{E_{0}}^{∗} = ( \begin{matrix}−D_{t}Δ − u_{x}D_{y}Δ + u_{y}D_{x}Δ+ & −v_{y}D_{x} + v_{x}D_{y} \\ 2(u_{yy} − u_{xx})D_{x}D_{y} + 2u_{xy}(D_{x}^{2} − D_{y}^{2}) \\ v_{x}D_{y}Δ − v_{y}D_{x}Δ− & −D_{t} − u_{x}D_{y} + u_{y}D_{x} \\ 2(v_{yy} − v_{xx})D_{x}D_{y} − 2v_{xy}(D_{x}^{2} − D_{y}^{2}) \end{matrix} ) (15)

We restrict ourselves to low order conservation laws, that is to such a f =

( \begin{matrix}S \\ T \end{matrix} ) in (13) that S and T are functions on J^{0}(IR^{3}, IR^{2}) and J^{2}(IR^{3}, IR^{2}) re-

spectively. This choice may be understood by considering the structure of ℓ_{E_{0}}^{∗} matrix: its second column is a first order operator while the first column is of third order. Solving the equation (13), which depends polinomially on higher derivatives u_{σ}, v_{σ}, is very tedious but straightforward job. We simply list the results.

The kernel of ℓ_{E_{0}}^{∗}|_{E_{0}^{∞}} is linearly generated by

( \begin{matrix}h(t) \\ 0 \end{matrix} ) , ( \begin{matrix}x^{2} + y^{2} \\ 0 \end{matrix} ) , ( \begin{matrix}p(t)x \\ 0 \end{matrix} ) , ( \begin{matrix}q(t)y \\ 0 \end{matrix} ) ,  (16)

( \begin{matrix}u \\ Δu \end{matrix} ) , ( \begin{matrix}f(v) \\ f′(v)Δv \end{matrix} ) , ( \begin{matrix}0 \\ Φ′(v) \end{matrix} )  (17)

where h, p, q, f and Φ are arbitrary functions. Most of them produce trivial conservation laws. There are three non–trivial conserved densities (two of them depending on arbitrary functions): the total energy E (magnetic plus kinetic energy), generalized ’cross helicity’ H_{c} and mean magnetic potential A,

\begin{aligned}E & = \frac{1}{2}⟨u_{x}^{2} + u_{y}^{2} + v_{x}^{2} + v_{y}^{2}⟩ \\ H_{c} & = ⟨f′(v) · (u_{x}v_{x} + u_{y}v_{y})⟩ \\ A & = ⟨Φ(v)⟩ \end{aligned}  (18)

Their generating functions are placed on the second line of (17) in respective order. Recall that f and Φ are arbitrary functions of v.

Let us calculate the decay rates of these conservation laws

Once dissipation coefficients ν or η are allowed to have small but finite values, quantities (17) are conserved no more. In accordance with general formulas of section 2 their decay rates are

\frac{dE}{dt} = −ν ∫_{S} uΔ^{2}u dxdy − η ∫_{S} (Δv)^{2} dxdy = − ∫_{S} [ν(Δu)^{2} + η(Δv)^{2}] dxdy ;
\frac{dH_{c}}{dt} = \frac{1}{2} ∫_{S} [νf(v)Δ^{2}u + ηf′(v)ΔuΔv] dxdy =
= −\frac{1}{2}(ν + η) ∫_{S} f′(v)ΔuΔv dxdy − \frac{1}{2}ν ∫_{S} f′′(v)Δu(v_{x}^{2} + v_{y}^{2}) dxdy ;
\frac{dA}{dt} = −η ∫_{S} Φ′(v)Δv dxdy = η ∫_{S} Φ′′(v)(v_{x}^{2} + v_{y}^{2}) dxdy

(19)

One can see that the decay of E is monotonic but those of H_{c} and A are not necessarily so. Such an inequality in decay rates leads to a distinct physical phenomenon of ’self–organization’ or quasi–stable states of plasma. Depending on initial conditions competing processes called ’selective decay’ or ’dynamic alignment’ occur: in selective decay energy decays relatively to mean potential, and in dynamic alignment energy decays relatively to cross helicity (velocity and magnetic field being aligned). There are also some more delicate possibilities of self–organization.

There exist a very simple procedure for finding solutions of the above described behavior. It was suggested in [3], and is known as ’Taylor trick’; it allows us to predict and calculate quasistable states. The procedure is as follows.

Taking into consideration their comparative decay rates let us minimize E with H_{c} and A as constrains. We put δ(E + λH_{c} + μA) = 0, A and H_{c} presumed constant, λ and μ being Lagrange multipliers. The Euler–Lagrange equations are

\{ \begin{matrix}Δ[u − F (v)] = 0 \\ Δv = f(v)Δu + g(v) , \end{matrix}  (20)

where F′ = f and g = ±Φ′.

The system (20) generally is not compatible with (12). But it is compatible if η = ν which is in particular true in the ideal case η = ν = 0. In this case, combining (12) and (20) we get

\{ \begin{matrix}Δ[u − F (v)] = 0 \\ Δv = \frac{ff′}{1−f^{2}} (v_{x}^{2} + v_{y}^{2}) + \frac{g}{1−f^{2}} \\ v_{t} = u_{y}v_{x} − u_{x}v_{y} \\ (u_{xy} − fv_{xy})(v_{x}^{2} − v_{y}^{2}) + [(u_{yy} − fv_{yy}) − (u_{xx} − fv_{xx})]v_{x}v_{y} = 0 \end{matrix}  (21)

Solutions of (21) describe the quasistationary states with remarkable accuracy as it was demonstrated numerically for special types of f and Φ in [4].

Remark 2 The first and the last equations of ((21) form the closed system

\{ \begin{matrix}Δw = 0 \\ z_{t} + w_{x}z_{y} − w_{y}z_{x} = 0 , \end{matrix}

where w = u − F (v) and z = v_{x}^{2} + v_{y}^{2}.

Remark 3 The second equation in ((21) may be written in a closed form ΔR = Ψ(R) where R = R(v), R′ = \sqrt{1 − f^{2}}

Remark 4 The case of u = F (v) in (21) is a generalization of dynamic alignment studied in [3] (aligned are gradients of u and v). It implies stationary solutions

\{ \begin{matrix}u = F (v) \\ v_{t} = 0 \\ ΔR = Ψ(R) , \end{matrix}

where R′(v) = \sqrt{1 − f^{2}(v)} as in previous remark.

References

[1] I.S. Krasi’shchik, A.M. Vinogradov Nonlocal symmetries and the theory of coverings, Acta Appl. Math., 2 (1989), p.79–86.

[2] A.V. Samokhin, Decay of conservation laws and their generating functions, Acta Appl. Math., 41 (1995), p.299–309.

[3] J.B. Taylor, Relaxation of toroidal plasma and generation of reverse magnetic field, Phys. Rev. Lett, 33 (1974), p. 1139–1141.

[4] A.C. Ting, M.H. Matthaeus, D. Montgomery, Turbulent relaxation processes in magnetohydrodynamics, Phys. Fluids 29(1986), p. 3261–3274.