The Diffiety Institute Preprint Series

Preprint DIPS–10/2001 June 21, 2001

Available via INTERNET: http://diffiety.ac.ru/

The Diffiety Institute

Polevaya 6-45, Pereslavl-Zalessky, 152140 Russia.

An example of (3 + 1)-dimensional integrable system

by

V. V. TRUSHKOV

An example of (3 + 1)-dimensional integrable system

V. V. TRUSHKOV

Abstract. An example of a (3+1)-dimensional integrable system is considered, infinite series of divergent forms are described. Classical symmetries for this system and self-similar exact solutions are found.

1. Introduction

A three-dimensional hydrodynamic model of the formation of sedimentary basins is considered. This model is described by quasi-steady equations of motion of incompressible fluid and by evolutionary equations of the density transfer:

∇p = μ△v⃗ + gρδ_{i3},

div v⃗ = 0, (1)

ρ_{t} + div(ρv⃗) = 0,

where ρ_{t} = ∂ρ/∂t.

In Section 2 of the paper theoretical background is introduced. In Section 3 symmetries of system (1) are described. In Section 5 conservation laws are described and integrability of system (1) is shown. In the fifth section some exact self-similar solutions are obtained.

2. Theoretical background

Consider the space J^{k}(n, m) with the coordinates

x_{1}, …, x_{n}, u^{1}, …, u^{m}, …, u_{σ}^{1}, …, u_{σ}^{m}, …,

where σ = (i_{1}, …, i_{n}) and |σ| ≤ k. This space is called the space of jets of order k. Then an equation E can be understood as a subset in J^{k}(n, m) defined by the relations

F^{1}(x, u, …, u_{σ}, …) = 0, …, F^{r}(x, u, …, u_{σ}, …) = 0. (2)

If all F^{i} are smooth functions and the system F^{1}, …, F^{r} is of maximal rank, then this subset is a smooth submanifold of codimension r. Thus, differential equations are smooth submanifolds in the jet spaces.

Key words and phrases. integrable equations, differential equation, classical symmetries, conservation laws.

This work was supported in part by the RFBR grants 99-01-01074 and 01-01-06055.

2000 Mathematics Subject Classification. 35Q30, 58J72, 76D07.

Received December 28, 2000; revised February 24, 2001.

Let f = (f^{1}(x), …, f^{m}(x)) be a smooth vector function. Then f determines an ndimensional submanifold in J^{k}(n, m) defined by

u_{σ}^{j} = \frac{∂^{|σ|}f^{j}}{∂x^{σ}} , j = 1, …, m, |σ| ≤ k,

(here and below we formally set u_{(0,...,0)} = u). This submanifold is called the k-jet of f and is denoted by j_{k}(f). It easily seen that f satisfies equation (2) if and only if j_{k}(f) lies in the corresponding submanifold in J^{k}(n, m). In other words, solutions of E are identified with jets lying in the equation manifold. Hence, the following definition is justified.

Definition 1. Let E ⊂ J^{k}(n, m) be a differential equation. A symmetry of E is a diffeomorphism φ: J^{k}(n, m) → J^{k}(n, m) such that

(i) φ(E) = E and (ii) φ(j_{k}(f)) is of the form j_{k}(f_{φ}) for every section f. A transformation is called a Lie transformation, if it satisfies condition (ii).

Definition 2. A vector field X on the manifold J^{k}(n, m) is called a Lie field, if the displacements along its trajectories are the Lie transformations.

Definition 3. A Lie field is called a classical infinitesimal symmetry of the equation E ⊂ J^{k}(n, m), if it is tangent to E.

We define the operators

D_{α}^{[k]} = \frac{∂}{∂x_{α}} + ∑_{j=1}^{m} ∑_{|σ|≤k} u_{σ+1_{α}}^{j} \frac{∂}{∂u_{σ}^{j}} . (3)

Then one can easily see that the system

F = 0, D_{1}^{[k]}(F) = 0, …, D_{n}^{[k]}(F) = 0 (4)

and initial equation (2) have the same solutions. If E ⊂ J^{k}(n, m) is the submanifold corresponding to (2), then the submanifold (probably, with singularities) in the jet space J^{k+1}(n, m) corresponding to (4) is denoted by E^{1} and called the first prolongation of E.

Now, we define by induction E^{l+1} = (E^{l})^{1} and see that the set E^{l} lies in J^{k+l}(n, m). We call E^{l} the l-th prolongation of E.

Let J^{∞}(n, m) be a space with coordinates

x_{1}, …, x_{n}, u^{1}, …, u^{m}, …, u_{σ}^{1}, …, u_{σ}^{m}, …,

where |σ| is unlimited. This space is called the space of infinite jets and the whole series of prolongations of the equation E determines a submanifold E^{∞} in J^{∞}(n, m) which is called the infinite prolongation of the equation E. The infinite prolongations are expressed by relations

D_{σ}(F) = 0, |σ| ≥ 0,

where D_{σ} = D_{1}^{i_{1}} ∘ ⋯ ∘ D_{n}^{i_{n}} and

D_{α} = \frac{∂}{∂x_{α}} + ∑_{j,σ} u_{σ+1_{α}}^{j} \frac{∂}{∂u_{σ}^{j}} (5)

are infinite counterparts of (3) and are called total derivatives.

Expressions of the form (5) can be considered as vector fields on J^{∞}(n, m). Thus, at any point θ ∈ J^{∞}(n, m) an n-dimensional plane C_{θ} arises spanned by vectors of these fields. It means that there exists a distribution C: θ ↦ C_{θ} on J^{∞}(n, m). We call it the Cartan distribution. Its importance to the theory of differential equations is explained by the following

Theorem 1 (see [1]). If E^{∞} ⊂ J^{∞}(n, m) is the infinite prolongation, then C_{θ} is tangent to E^{∞} at any point θ ∈ E^{∞}.

An n-dimensional submanifold in E^{∞} nondegenerately projecting to the space of independent variables is a solution of E if and only if it is a maximal integrable manifold of the Cartan distribution.

Hence, E^{∞} carries the structure which completely determines solutions of E and this structure is the Cartan distribution.

Consider the sets

CD(E^{∞}) = {X ∈ D(E^{∞}) | X(ω) = 0, ∀ω ∈ CΛ^{1}(E^{∞})},
D_{C}(E^{∞}) = {X ∈ D(E^{∞}) | [X, CD(E^{∞})] ⊂ CD(E^{∞})}.

Note that the set D_{C}(E^{∞}) is a subalgebra in the Lie algebra of vector fields on E^{∞} and CD(E^{∞}) is an ideal in D_{C}(E^{∞}).

The Lie ℝ-algebra

sym E = D_{C}(E^{∞})/CD(E^{∞})

is the Lie algebra of symmetries of the Cartan distribution on E^{∞}.

Definition 4. Elements of the Lie algebra sym E are called higher (infinitesimal) symmetries of the equation E.

Theorem 2 (see [1]). Let E ⊂ J^{k}(n, m) be an equation given by the system {F_{α} = 0} and the functions F_{α} be chosen in such a way that the graph of F = (…, F_{α}, …) intersects the space of independent variables transversally, then Lie algebra sym E is isomorphic to the Lie algebra of solutions of the equations

ℓ_{F_{α}}^{E}(φ) = 0,  α = 1, …, r,  φ = (φ^{1}, …, φ^{m}), (6)

where φ^{j} are smooth functions on E^{∞},

ℓ_{ψ} = ∑_{j,σ} \frac{∂ψ}{∂u_{σ}^{j}} D_{σ}^{(j)}

and ℓ_{ψ}^{E} is the restriction of ℓ_{ψ} to E^{∞}. This isomorphism is given by the correspondence

φ ↦ �_{φ} = ∑_{σ,j} D_{σ}(φ^{j})\frac{∂}{∂u_{σ}^{j}}

The function φ is callen the generating function corresponds to a symmetry of equation.

Generating functions of conservation laws correspond to solutions of the equation formally adjont to (6) (see [1]).

3. Symmetries

Consider a system of equations for the generating function of symmetries of equation (1)

D_{z}^{2}(φ_{u}) = −\frac{1}{μ}(D_{x}^{2}(φ_{u}) + D_{y}^{2}(φ_{u})) + D_{x}(φ_{p})),
D_{z}^{2}(φ_{v}) = −\frac{1}{μ}(D_{x}^{2}(φ_{v}) + D_{y}^{2}(φ_{v})) + D_{y}(φ_{p})),
D_{z}(φ_{p}) = μ(D_{x}^{2}(φ_{w}) + D_{y}^{2}(φ_{w}) − D_{x}D_{z}(φ_{u}) − D_{y}D_{z}(φ_{v})) + gφ_{ρ},
D_{z}(φ_{w}) = −D_{x}(φ_{x}) − D_{y}(φ_{y}),
D_{z}(φ_{ρ}) = −(\frac{ρ_{x}}{ρ_{z}} D_{x}(φ_{ρ}) + \frac{ρ_{y}}{ρ_{z}} D_{y}(φ_{ρ}) + \frac{ρ_{t}}{ρ_{z}} D_{t}(φ_{ρ}) + \frac{u}{ρ_{z}} φ_{u}

+ \frac{v}{ρ_{z}} φ_{v} + \frac{w}{ρ_{z}} φ_{w}).

Here D_{t}, D_{x}, D_{y}, D_{z} are the total derivatives restricted to the system.

We solve this system on the space of functions in the variables t, x, y, z, ρ , p, u, v, w, u_{t}, u_{x}, u_{y}, u_{z}, v_{t}, v_{x}, v_{y}, v_{z}, w_{t}, w_{x}, w_{y}, p_{t}, p_{x}, p_{y}, ρ_{t}, ρ_{y}, u_{xz}, v_{yz}.

Theorem 3 (Classical symmetries). The system

\frac{∂p}{∂x} = μ(\frac{∂^{2}u}{∂x^{2}} + \frac{∂^{2}u}{∂y^{2}} + \frac{∂^{2}u}{∂z^{2}}),

\frac{∂p}{∂y} = μ(\frac{∂^{2}v}{∂x^{2}} + \frac{∂^{2}v}{∂y^{2}} + \frac{∂^{2}v}{∂z^{2}} ),

\frac{∂p}{∂z} = μ(\frac{∂^{2}w}{∂x^{2}} + \frac{∂^{2}w}{∂y^{2}} + \frac{∂^{2}w}{∂z^{2}}) + gρ,

\frac{∂u}{∂x} + \frac{∂v}{∂y} + \frac{∂w}{∂z} = 0,

\frac{∂ρ}{∂t} + u\frac{∂ρ}{∂x} + v \frac{∂ρ}{∂y} + w\frac{∂ρ}{∂z} = 0

has the classical symmetries

(T)  ∂_{t},
(X_{α})  α(t)∂_{x} + α′(t)∂_{u},
(Y_{β})  β(t)∂_{y} + β′(t)∂_{v},
(Z_{γ})  γ(t)∂_{z} + γ′(t)∂_{w},
(P R)  ∂_{ρ} + gz∂_{p},
(P_{ε})  ε(t)∂_{p},
(M_{1})  t∂_{t} − ρ∂_{ρ} − p∂_{p} − u∂_{u} − v∂_{v} − w∂_{w},
(M_{2})  x∂_{x} + y∂_{y} + z∂_{z} − ρ∂_{ρ} + u∂_{u} + v∂_{v} + w∂_{w},
(Θ_{δ})  δ(t)(y∂_{x} − x∂_{y} + v∂_{u} − u∂_{v}) + δ′(t)(y∂_{u} − x∂_{v}).

T X_{α} Y_{β} Z_{γ} P_{ε} P R M_{1}  M_{2} Θ_{δ}
T 0 X_{α′} Y_{β′} Z_{γ′} P_{ε′} 0 T 0 Θ_{δ′}
X_{α}  0 0 0 0 0 −X_{tα′} X_{α} −Y_{αδ}
Y_{β}  0 0 0 0 −Y_{tβ′} Y_{β} X_{βδ}
Z_{γ}  0 0 P_{gγ} −Z_{tγ′} Z_{γ}  0
P_{ε}  0 0 −P_{ε+tε′} 0  0
P R  0 −P R −P R 0
M_{1}  0  0  Θ_{tδ′}
M_{2}  0  0
Θ_{δ}  0

Figure 1. Commutator relations between classical symmetries

The commutators of symmetries are presented on Fig. 1. In addition, one has

[P_{ε_{1}}, P_{ε_{2}}] = 0, [X_{α_{1}}, X_{α_{2}}] = 0, [Y_{β_{1}}, Y_{β_{2}}] = 0,
[Z_{γ_{1}}, Z_{γ_{2}}] = 0, [Θ_{δ_{1}}, Θ_{δ_{2}}] = 0.

The symmetry T corresponds to the translation in t. The symmetries X_{α}, Y_{β}, Z_{γ} correspond to generalized translations in x, y, z respectively. The symmetries P_{ε}, P R are gauges on trivial solution. The symmetries M_{1}, M_{2} are scale symmetries. The symmetry Θ_{δ} corresponds to generalized rotation.

Let X be an infinitesimal symmetry of the equation E ⊂ J^{k}(n, m), f_{0} be a solution of this equation, Γ_{f_{0}}^{k} = j_{k}(f_{0})(M) ⊂ E. Let {A_{t}} be the one-parameter group of the vector field X. Then the submanifold A_{t}(Γ_{f_{0}}^{k}) ⊂ J^{k}(n, m) is integrable for the Cartan distribution: if A_{t}(Γ_{f_{0}}^{k}) and Γ_{f_{0}}^{k} are a horizontal with respect to the projection on base, then A_{t}(Γ_{f_{0}}^{k}) = Γ_{f_{k}}^{k} for some section f_{t} and sufficiently small t. Moreover, we obtain from the definition of symmetries that A_{t}(Γ_{f_{0}}^{k}) ⊂ E. The translation of an initial solution f to a family of solutions f_{t} is called reproduction of solution the f under the action of X. If φ is the generating function of symmetry X = X_{φ} then f_{t} is a solution of the system of equations

\frac{∂f^{j}}{∂t} = φ^{j}|_{Γ_{f}^{k}}(x, t),  f^{j}(x, 0) = f_{0}^{j}(x),  j = 1, …, m.

If φ|_{Γ_{f}^{k}}(x, t) = 0 then f(x, t) = f(x, 0) for all t, i.e., f_{0} are stationary point for oneparameter group of transformations {A_{t}} under its action on section. In other words, the manifold Γ_{f_{0}}^{k} is invariant under the action of the vector field X_{φ}.

Definition 5. If Γ_{f_{0}}^{k} is invariant under the action of the vector field X_{φ}, then f_{0} is called a φ-invariant solution of E.

Applying this technique to (1), we obtain the following

6 V. V. TRUSHKOV

Theorem 4. Let ρ = ρ(t, x, y, z), p = p(t, x, y, z), u = u(t, x, y, z), v = v(t, x, y, z), w = w(t, x, y, z) be a solution of system (1). Then vector-functions

(T)  ρ = ρ(t + τ, x, y, z), p = p(t + τ, x, y, z)

u = u(t + τ, x, y, z), v = v(t + τ, x, y, z) w = w(t + τ, x, y, z)

(X_{α})  ρ = ρ(t, x + τα(t), y, z), p = p(t, x + τα(t), y, z)

u = u(t, x + τα(t), y, z) − τα′(t), v = v(t, x + τα(t), y, z) w = w(t, x + τα(t), y, z)

(Y_{β})  ρ = ρ(t, x, y + τβ(t), z), p = p(t, x, y + τβ(t), z)

u = u(t, x, y + τβ(t), z), v = v(t, x, y + τβ(t), z) − τβ′(t) w = w(t, x, y + τβ(t), z)

(Z_{γ})  ρ = ρ(t, x, y, z + τγ(t)), p = p(t, x, y, z + τγ(t))

u = u(t, x, y, z + τγ(t)), v = v(t, x, y, z + τγ(t)) w = w(t, x, y, z + τγ(t)) − τγ′(t)

(P_{ε})  ρ = ρ(t, x, y, z), p = p(t, x, y, z) + τε(t)

u = u(t, x, y, z), v = v(t, x, y, z) w = w(t, x, y, z)

(P R)  ρ = ρ(t, x, y, z) + τ, p = p(t, x, y, z) + gzτ

u = u(t, x, y, z), v = v(t, x, y, z) w = w(t, x, y, z)

(M_{1})  ρ = τρ(τt, x, y, z), p = τp(τt, x, y, z)

u = τu(τt, x, y, z), v = τv(τt, x, y, z) w = τw(τt, x, y, z)

(M_{2})  ρ = τρ(t, τx, τy, τz), p = p(t, τx, τy, τz)

u = u(t, τx, τy, τz)/τ, v = v(t, τx, τy, τz)/τ w = w(t, τx, τy, τz)/τ

(Θ_{δ})  ρ = ρ(t, r, θ − τδ, z), p = p(t, r, θ − τδ, z)

u = A cos(τ)u(t, r, θ − τδ, z) + B sin(τ)v(t, r, θ − τδ, z)

− \frac{δ′}{δ} rθ sin θ

v = −A sin(τ)u(t, r, θ − τδ, z) + B cos(τ)v(t, r, θ − τδ, z)

+ \frac{δ′}{δ} rθ cos θ

w = w(t, r, θ − τδ, z)

are solutions of the system too.

An example of (3 + 1)-dimensional integrable system 7

The following quotientequations correspond to the above considered symmetries:

(T)  \frac{∂p}{∂x} = μ(\frac{∂^{2}u}{∂x^{2}} + \frac{∂^{2}u}{∂y^{2}} + \frac{∂^{2}u}{∂z^{2}}),

\frac{∂p}{∂y} = μ(\frac{∂^{2}v}{∂x^{2}} + \frac{∂^{2}v}{∂y^{2}} + \frac{∂^{2}v}{∂z^{2}} ),

\frac{∂p}{∂z} = μ(\frac{∂^{2}w}{∂x^{2}} + \frac{∂^{2}w}{∂y^{2}} + \frac{∂^{2}w}{∂z^{2}}) + gρ,

\frac{∂u}{∂x} + \frac{∂v}{∂y} + \frac{∂w}{∂z} = 0,

u\frac{∂ρ}{∂x} + v \frac{∂ρ}{∂y} + w\frac{∂ρ}{∂z} = 0,

(X_{α})  μ(\frac{∂^{2}u}{∂y^{2}} + \frac{∂^{2}u}{∂z^{2}}) = 0,

\frac{∂p}{∂y} = μ(\frac{∂^{2}v}{∂y^{2}} + \frac{∂^{2}v}{∂z^{2}} ),

\frac{∂p}{∂z} = μ(\frac{∂^{2}w}{∂y^{2}} + \frac{∂^{2}w}{∂z^{2}}) + gρ,

\frac{α′(t)}{α(t)} + \frac{∂v}{∂y} + \frac{∂w}{∂z} = 0,

\frac{∂ρ}{∂t} + v \frac{∂ρ}{∂y} + w\frac{∂ρ}{∂z} = 0,

(Y_{β})  \frac{∂p}{∂x} = μ(\frac{∂^{2}u}{∂x^{2}} + \frac{∂^{2}u}{∂z^{2}}),

μ(\frac{∂^{2}v}{∂x^{2}} + \frac{∂^{2}v}{∂z^{2}} ) = 0,

\frac{∂p}{∂z} = μ(\frac{∂^{2}w}{∂x^{2}} + \frac{∂^{2}w}{∂z^{2}}) + gρ,

\frac{∂u}{∂x} + \frac{β′(t)}{β(t)} + \frac{∂w}{∂z} = 0,

\frac{∂ρ}{∂t} + u\frac{∂ρ}{∂x} + w\frac{∂ρ}{∂z} = 0,

(Z_{γ})  \frac{∂p}{∂x} = μ(\frac{∂^{2}u}{∂x^{2}} + \frac{∂^{2}u}{∂y^{2}}),

\frac{∂p}{∂y} = μ(\frac{∂^{2}v}{∂x^{2}} + \frac{∂^{2}v}{∂y^{2}} ),

μ(\frac{∂^{2}w}{∂x^{2}} + \frac{∂^{2}w}{∂y^{2}}) + gρ = 0,

\frac{∂u}{∂x} + \frac{∂v}{∂y} + \frac{γ′(t)}{γ(t)} = 0,

\frac{∂ρ}{∂t} + u\frac{∂ρ}{∂x} + v \frac{∂ρ}{∂y} = 0,

(M_{1})  \frac{∂p}{∂x} = μ(\frac{∂^{2}u}{∂x^{2}} + \frac{∂^{2}u}{∂y^{2}} + \frac{∂^{2}u}{∂z^{2}}),

\frac{∂p}{∂y} = μ(\frac{∂^{2}v}{∂x^{2}} + \frac{∂^{2}v}{∂y^{2}} + \frac{∂^{2}v}{∂z^{2}} ),

\frac{∂p}{∂z} = μ(\frac{∂^{2}w}{∂x^{2}} + \frac{∂^{2}w}{∂y^{2}} + \frac{∂^{2}w}{∂z^{2}}) + gρ,

\frac{∂u}{∂x} + \frac{∂v}{∂y} + \frac{∂w}{∂z} = 0,

u\frac{∂ρ}{∂x} + v \frac{∂ρ}{∂y} + w\frac{∂ρ}{∂z} = ρ,

(Θ_{δ})  v = A(t, r, z) cos(θ) + B(t, r, z) sin(θ) + \frac{δ′}{δ} rθ sin θ,

u = −A(t, r, z) sin(θ) + B(t, r, z) cos(θ) − \frac{δ′}{δ} rθ cos θ,

\frac{∂p}{∂r} = μ(\frac{∂^{2}B}{∂r^{2}} + \frac{1}{r} \frac{∂B}{∂r} − \frac{1}{r^{2}} B + \frac{∂^{2}B}{∂z^{2}} − \frac{δ′}{δ} \frac{2}{r}),

\frac{∂^{2}A}{∂r^{2}} + \frac{1}{r} \frac{∂A}{∂r} − \frac{1}{r^{2}} \frac{∂^{2}A}{∂z^{2}} = 0,

\frac{∂p}{∂z} = μ(\frac{∂^{2}w}{∂r^{2}} + \frac{1}{r} \frac{∂w}{∂r} + \frac{∂^{2}w}{∂z^{2}}) + gρ,

\frac{∂B}{∂r} + \frac{1}{r}B + 1 + \frac{∂w}{∂z} = 0,

\frac{∂ρ}{∂t} + B \frac{∂ρ}{∂r} + w\frac{∂ρ}{∂z} = 0.

4. Conservation laws

Theorem 5. System (1) has the following conservation laws with 0-order generating functions

ψ_{ρ} = (a′(ρ) + b(t))w,
ψ_{p} = −b′(t)/g,
ψ_{u} = c(t, x, y, z),
ψ_{v} = d(t, x, y, z),
ψ_{w} = a(ρ) + b(t)ρ + e(t, x, y),

where c, d, e are solutions of the system

\frac{∂^{2}c}{∂x^{2}} + \frac{∂^{2}c}{∂y^{2}} + \frac{∂^{2}c}{∂z^{2}} = \frac{∂e}{∂x},

\frac{∂^{2}d}{∂x^{2}} + \frac{∂^{2}d}{∂y^{2}} + \frac{∂^{2}d}{∂z^{2}} = \frac{∂e}{∂y},

Figure 2. The graph of the u function

\frac{∂c}{∂x} + \frac{∂d}{∂y} = 0.

Consider some divergent forms which corresponds to the generating functions in the last theorem:

(a(ρ))_{t} + (ua(ρ))_{x} + (va(ρ))_{y} + (wa(ρ))_{z} = 0,
(b(t)ρ)_{t} + (b(t)uρ)_{x} + (b(t)vρ)_{y} + (b(t)wρ)_{z} + \frac{μ}{g}△(b′(t)w)

− \frac{1}{g}(b′(t)p)_{z} = 0.

The main result of the paper is formulated in the following theorem.

Theorem 6. There exist the following higher-order generating functions of the conservation laws

ψ_{u} = (u_{y} − v_{x})_{σy}, ψ_{v} = −(u_{y} − v_{x})_{σx},

where σ = (σ_{x}, σ_{y}), σ_{x} + σ_{y} ≡ 1 mod 2.

This theorem shown that the system has divergent forms. Moreover, for every k there exists a divergent form, which contain derivatives of order higher than k.

5. Examples

Consider the invariant solution of symmetry Θ_{δ} and let δ = e^{t}, A = 0, B = 0, ρ = ρ(r), w = w(t, r, z), p = p(t, r). Then

u = −rθ sin θ,
v = rθ cos θ,
p = 2μln\frac{1}{r} + p_{0}(t),
ρ = 2ρ′_{0}(r^{2}),

(a)  (b)

Figure 3. The graphs of the v (a) and p (b) functions

w = −z − \frac{g}{μ} ∫_{0}^{r} \frac{ρ(r′^{2})}{r′}dr′ + w_{0}(t).

The graphs of functions u, v and p for p_{0} = 0, μ = 1/2 are at Fig. 2, and Fig. 3 a,b respectively.

This solution, being subjected to P R-deformation, transforms to:

u = −rθ sin θ = −y arctan \frac{y}{x},
v = rθ cos θ = x arctan \frac{y}{x},
p = 2μ ln \frac{1}{r} + p_{0}(t) + gzτ,
ρ = 2ρ′_{0}(r^{2}) + τ,
w = −z − \frac{g}{μ} ∫_{0}^{r} \frac{ρ(r′^{2})}{r′}dr′ + w_{0}(t).

After X_{−t}-deformation, we obtain solution with running singularity:

u = −y arctan \frac{y}{x − t} + 1,
v = (x − t) arctan \frac{y}{x − t},
p = μln\frac{1}{(x − t)^{2} + y^{2}} + p_{0}(t) + gzτ,
ρ = 2ρ′_{0}((x − t)^{2} + y^{2}) + τ,
w = −z − \frac{g}{μ} ∫_{0}^{(\sqrt{(x−t)^{2}+y^{2}})} \frac{ρ(r′^{2})}{r′}dr′ + w_{0}(t).

Acknowledgements. The author is grateful to Professor I.S. Krasil’shchik for his attention to this work.

References

[1] A.V. Bocharov, V.N. Chetverikov, S.V. Duzhin, N.G. Khor’kova, I.S. Krasil’shchik, A.V. Samokhin, Yu.N. Torkhov, A.M. Verbovetsky, and A.M. Vinogradov, Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. Amer. Math. Soc., Providence, RI, 1999. Edited and with a preface by Krasil’shchik and Vinogradov.

[2] B.M. Naimark, A.T. Ismail-Zadeh, A.I. Korortkii et al., Three-dimensional modelling of viscous flows in the uppermost mantle, Problems in Geodynamics and Seismology: Coll. sci. proc. Moscow, 1998. (Computational seismology. Iss. 30), 3–15.

Moscow State Institute Electronics and Mathematics, Independent University of Moscow and Independent University of Moscow, B. Vlasievskij per., 11, 121019 Moscow, Russia

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