PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 77(91) (2005), 29–51

SYMMETRIES OF CONSERVATION LAWS

Sanja Konjik

Communicated by Stevan Pilipović

Abstract. We apply techniques of symmetry group analysis in solving two systems of conservation laws: a model of two strictly hyperbolic conservation laws and a zero pressure gas dynamics model, which both have no global solution, but whose solution consists of singular shock waves. We show that these shock waves are solutions in the sense of 1-strong association. Also, we compute all projectable symmetry groups and show that they are 1-strongly associated, hence transform existing solutions in the sense of 1-strong association into other solutions.

The concept of classical symmetry groups offers a large number of possibilities in studying differential equations, in particular in constructing explicit solutions to linear and nonlinear differential equations or determining and classifying invariance properties [16, 17]. In various problems of mathematical physics the classical theory turns out to be insufficient, due to singular objects (like distributions or discontinuous nonlinearities) which can occur in the equation or equations with solutions in a weak sense, i.e., weak solutions (distributional, generalized or in the sense of association). Therefore, the methods of classical symmetry group analysis of differential equations have been extended to linear equations in the class of distributions [1, 2], as well as to equations involving generalized functions [5, 6, 10, 11, 9].

The aim of this paper is to apply techniques of symmetry group analysis in solving two systems of differential equations given in the form of conservation laws. The paper is divided into two parts. Section 1 provides a brief overview of the basic definitions and theorems which are going to be used for studying conservation laws. We start by recalling some facts on symmetry group analysis, which are in detail carried out in [16] (see also [17]). Then we turn to symmetries in the generalized setting, precisely to associated ones. As we will see later, the reason for this lies in the fact that the solutions of the conservation laws we consider are, under certain assumptions, shock waves. Lack of space prevents us from also giving a short introduction to generalized functions. Therefore, for the notations and properties of the Colombeau algebra of generalized functions we recommend [6, 9] or [15]; in particular, definitions of generalized step-functions, splitted delta functions and m− and m′−singular delta functions are provided in [13]. We close the introductory part by a short overview of conservation laws. Based on [3, 4, 7, 12, 18] we fix notations and present the general solution of the Riemann problem. Motivated by [8] and [13] we proceed in section 2 by investigating two systems of conservation laws: a model of two strictly hyperbolic conservation laws which is genuinely nonlinear but for which the Riemann problem has no global solution and a zero pressure gas dynamics model which is linearly degenerative but for which the Riemann problem also does not have global solutions. In both cases singular solutions appear, called singular shock waves. We prove that these solutions are solutions in the sense of 1-strong association. After computing all projectable symmetry groups of these systems we show that they are 1-strongly associated, hence transform existing solutions (given in [8] and [13]) into other solutions.

2000 Mathematics Subject Classification: Primary 35L65; Secondary 35D99, 46F30, 58D19.

Key words and phrases: symmetry group, infinitesimal generator, conservation law, Riemann problem, singular shock wave, solution in the sense of association.

Work supported by MNZŽS, project 1835.

1. Introduction

1.1. Symmetry groups of differential equations. Let S be a system of differential equations:

Δ_{ν}(x, u) = 0, 1 ⩽ ν ⩽ l.

Denote by X = ℝ^{p} and U = ℝ^{q} the spaces of independent and dependent variables with coordinates x = (x_{1}, x_{2}, …, x_{p}) and u = (u^{1}, u^{2}, …, u^{q}) respectively. Also, denote by M an open subset of X × U. Identify a function u = f(x) with its graph Γ_{f} = {(x, f(x)) : x ∈ Ω} ⊂ X × U, where Ω ⊂ X is the domain of f. Let G be a local group of transformations acting on M. The transform of Γ_{f} by g ∈ G is defined by g · Γ_{f} = {(x̃, ũ) = g · (x, u) : (x, u) ∈ Γ_{f}}. In local coordinates this action is given by g · f = (Φ_{g} ∘ (id_{X} × f)) ∘ (Ξ_{g} ∘ (id_{X} × f))^{−1}, where Ξ_{g} and Φ_{g} are smooth function on M, and id_{X} is the identity mapping on X. Supposing that Ξ_{g} does not depend on the dependent variables we get a projectable action of g on f, i.e.,

(1)  g · f = (Φ_{g} ∘ (id_{X} × f)) ∘ Ξ_{g}^{−1}.

Definition 1.1. The symmetry group of the system S is a local transformation group G acting on the space of independent and dependent variables with the property that whenever u = f(x) is a solution of the system and g · f is defined, g ∈ G, then u = g · f is also a solution of the system.

The n-th prolonged or n-jet space X × U^{(n)} is a space which represents all independent variables, dependent variables and all different partial derivatives of dependent variables up to the order n. For the construction of the n-th prolonged space we refer to [16]. Write M^{(n)} for a subset of n-jet space X×U^{(n)}. An arbitrary point in U^{(n)} will be denoted by u^{(n)} and its components by u_{J}^{α}, where 1 ⩽ α ⩽ q, while J runs over the set of all unordered multi-indices J = (j_{1}, …, j_{k}), 1 ⩽ j_{k} ⩽ p, 0 ⩽ k ⩽ n.

The n-th prolongation of a function f : X → U, denoted by pr^{(n)} f, is a function from X to U^{(n)}, which maps x into (∂_{J}f^{α}(x))_{α,J}, 1 ⩽ α ⩽ q, 0 ⩽ |J| ⩽ n.

The n-th prolongation of a group G which acts on M ⊂ X × U, pr^{(n)} G, is again a local group of transformations which acts on M^{(n)} such that it transforms the derivatives of a smooth function u = f(x) into the corresponding derivatives of the transformed function ũ = f̃(x̃). For the precise definition see [16].

The n-th prolongation of a vector field v on M, pr^{(n)} v, is a vector field on the n-jet space M^{(n)} with the following property:

pr^{(n)} v|_{(x,u^{(n)})} = \frac{d}{dη} |_{η=0} pr^{(n)}(exp(ηv))(x, u^{(n)}),

where exp(ηv) is the corresponding local one-parameter group generated by v. If

v = ∑_{i=1}^{p} ξ^{i}(x, u)\frac{∂}{∂x_{i}} + ∑_{α=1}^{q} ϕ_{α}(x, u)\frac{∂}{∂u^{α}}

then we calculate the n-th prolongation of v using the formula:

(2)  pr^{(n)} v = v + ∑_{α=1}^{q} ∑_{J} ϕ_{α}^{J}(x, u^{(n)})\frac{∂}{∂u_{J}^{α}} ,

where the coefficients ϕ_{α}^{J}(x, u^{(n)}) are given by

(3)  ϕ_{α}^{J}(x, u^{(n)}) = D_{J}(ϕ_{α} − ∑_{i=1}^{p} ξ^{i}u_{i}^{α}) + ∑_{i=1}^{p} ξ^{i}u_{J,i}^{α},

u_{i}^{α} = ∂u^{α}/∂x^{i}, u_{J,i}^{α} = ∂u_{J}^{α}/∂x^{i} and D_{J} denotes a total differential.

Then the infinitesimal criterion for a system of differential equations reads:

Theorem 1.2. Let

(4)  Δ_{ν}(x, u^{(n)}) = 0,  ν = 1, …, l

be a system of differential equations of a maximal rank (meaning that the corresponding Jacobian matrix J_{Δ}(x, u^{(n)}) = (∂Δ_{ν}/∂x^{i}, ∂Δ_{ν}/∂u_{J}^{α}) is of rank l on the set of all solutions of S, S_{Δ}). If G is a local transformation group acting on M ⊂ X×U and

(5) pr^{(n)} v(Δ_{ν}(x, u^{(n)})) = 0, ν = 1, …, l, whenever Δ(x, u^{(n)}) = 0,

for every infinitesimal generator v of G then G is a symmetry group of (4).

The condition (5) from this theorem will also be necessary if we additionaly suppose that the system (4) is locally solvable, i.e., at each point (x_{0}, u_{0}^{(n)}) ∈ S_{Δ} there exists a smooth solution u = f(x) of the system, defined in a neighborhood of x_{0}, which has the prescribed “initial conditions” u_{0}^{(n)} = pr^{(n)} f(x_{0}). (We say that a system of differential equations is nondegenerate if at every point of the solution set it is both locally solvable and of maximal rank.)

For later use, we mention here a result which is a consequence of the maximal rank condition (imposed on the system (4) in the above theorem). Namely, under the conditions of Theorem 1.2, the infinitesimal criterion (5) can be replaced by the equivalent condition

(6)  pr^{(n)} v(Δ_{ν}(x, u^{(n)})) = ∑_{μ=1}^{l} Q_{νμ}Δ_{μ}(x, u^{(n)}), ν = 1, …, l,

for functions Q_{νμ}, μ, ν = 1, …, l to be determined.

We finish this short introduction into symmetry groups of differential equations by a description of a procedure for calculating symmetry groups of a given system S. The procedure consists of the following steps:

1) Write the vector field (i.e., infinitesimal generator) in the most general form:

v(x, u) = ∑_{i=1}^{p} ξ_{i}(x, u)\frac{∂}{∂x_{i}} + ∑_{α=1}^{q} ϕ_{α}(x, u)\frac{∂}{∂u^{α}} ,

where ξ_{i} and ϕ_{α} are functions which should be calculated.

2) According to (2) and (3) calculate the corresponding prolongation of v.

3) Then apply the infinitesimal criterion (5) and equate pr^{(n)} v(Δ_{ν}(x, u^{(n)})) with zero. Since those equations must hold on S_{Δ}, eliminate the dependence of derivatives of u by the equations from the system. After that we have the equations which have to be satisfied with respect to x, u and the remaining partial derivatives of u.

4) After solving these equations we obtain a certain number of partial differential equations for ξ_{i} and ϕ_{α}.

5) Compute the ξ_{i} and ϕ_{α} from them, thereby computing vector fields v which generate a Lie algebra of infinitesimal symmetries.

6) At the end find the corresponding one-parameter symmetry groups as the flows of the infinitesimal generators calculated in the previous step.

1.2. Symmetry groups of weak solutions. Next, we look for the symmetries which transform weak solutions of the system of PDEs

(7)  Δ_{ν}(x, u^{(n)}) = 0,  1 ⩽ ν ⩽ l,

into other weak solutions, mainly associated solutions to (7) into other associated solutions to (7) (hence the system (7) should be replaced by Δ_{ν}(x, u^{(n)}) ≈ 0). Such a symmetry group is called symmetry group in the sense of association or associated symmetry group for short. The symmetry groups we are interested in are projectable in order to avoid the problem of inverting Colombeau functions. Thus unless explicitly stated otherwise, all symmetry groups are assumed to be projectable. Beside this, we need some more assumptions on G (cf. [5]). We suppose that a local transformation group G is slowly increasing, uniformly for x in compact sets, and analogously for the mapping u^{(n)} ↦ Δ(x, u^{(n)}).

Look at the system

(8)  Δ_{ν}(x, u^{(n)}) ≈ 0,  ν = 1, …, l.

We recall the following definitions from [5]:

Definition 1.3. u = (u^{1}, …, u^{q}) ∈ G(Ω)^{q} is a solution of (8) and also associated solution to (7) if u = (u^{1}, …, u^{q}) has a representative (u_{ε}^{1}, …, u_{ε}^{q})_{ε} ∈ E_{M}(Ω)^{q} such that for each test function φ ∈ D(Ω)

(9)  ∫ Δ_{ν}(x, pr^{(n)} u_{ε}(x))φ(x) dx → 0 as ε → 0, 1 ⩽ ν ⩽ l.

The set of all associated solutions to (7) is denoted by A_{Δ} and moreover, the set of all u ∈ (G_{∞})^{q} which satisfies (9) with AB_{Δ}. The symmetry group G of (7) is called A-symmetry group if for every u ∈ A and every g_{η} ∈ G it follows that g_{η}u ∈ A, whenever g_{η}u is defined. Beside solution in the sense of association we can also define a solution in the sense of strong association.

Definition 1.4. Let k ∈ ℕ_{0}. Then u = (u^{1}, …, u^{q}) ∈ A_{Δ}(Ω) (resp. u ∈

AB_{Δ}(Ω)) is called k-strongly associated or ≈^{k}-associated solution to the system (7) if there exists a representative (u_{ϵ}^{1}, …, u_{ϵ}^{q})_{ε} ∈ E_{M}(Ω)^{q} such that for each B ⊆ C_{c}^{∞}(Ω) which is bounded in C_{c}^{k}(Ω) we have

lim_{ϵ→0} sup_{φ∈B} | ∫ Δ_{ν}(x, pr^{(n)} u_{ε}(x))φ(x) dx| = 0, 1 ⩽ ν ⩽ l.

The space of all k-strongly associated solutions to (7) is denoted by AS_{Δ}^{k}. Also, ABS_{Δ}^{k} := AS_{Δ}^{k} ∩ G_{∞}. The main role in the calculation of associated, resp. k-strongly associated symmetry groups is played by the theorem which is based on the factorization property of the system, derived in [6]. In matrix form this property is given by

(10) Δ(Ξ_{η}(x, u(x)), pr^{(n)}(g_{η}u)(Ξ_{η}(x, u(x))))=Q(η, x, pr^{(n)} u(x))Δ(x, pr^{(n)} u(x)),

where Q : W → ℝ^{l^{2}} and W is an open subset of (−η_{0}, η_{0})×M^{(n)} with {0}×M^{(n)} ⊆ W.

Theorem 1.5. Let G be a slowly increasing symmetry group of the system (7) which admits a global factorization of the form (10). Then

(i) if Q depends only on η, x and u then G is also an ABS_{Δ}-symmetry group of (7);

(ii) if Q depends only on η and x then G is also an AS_{Δ}^{k}-symmetry group of (7), for each k > 0. Moreover, G is in this case an associated symmetry group of (7) as well.

1.3. Systems of conservation laws. We look at a system of conservation laws in one space dimension:

u_{t}^{1} + (f_{1}(u^{1}, …, u^{n}))_{x} = 0
⋮
u_{t}^{n} + (f_{n}(u^{1}, …, u^{n}))_{x} = 0,

or written in a shorter (matrix) form:

(11)  u_{t} + (f(u))_{x} = 0,

where t > 0, x ∈ ℝ, u = (u^{1}, …, u^{n}) is the conserved density and f(u) = (f_{1}, …, f_{n}) is the flux. Differentiating (11) we obtain a quasilinear system

(12)  u_{t} + A(u)u_{x} = 0,

where A(u) = Df(u) is the Jacobian matrix of f. The systems (11) and (12) are equivalent for all smooth solutions u. Otherwise, if u has a jump, the left hand side of (12) contains a product of a discontinuous function with a distributional derivative, while (11) is still well defined in the distributional sense.

The eigenvalues of the matrix A(u) determine the system of conservation laws in the following way:

Definition 1.6. The system of conservation laws is hyperbolic, resp. strictly hyperbolic, if all eigenvalues of the matrix A(u) are real, resp. real and different.

Suppose that the system (11) is strictly hyperbolic and denote by λ_{1}(u), …, λ_{n}(u) the eigenvalues of A(u) with λ_{1}(u) < ⋯ < λ_{n}(u). Next, denote by l_{1}, …, l_{n} and r_{1}, …, r_{n} the corresponding left and right eigenvectors. The eigenvalue λ_{i} of A is also called the i-th characteristic speed and the pair (λ_{i}, r_{i}) the i-th characteristic field of (11).

Definition 1.7. The i-th characteristic field of the system (11) is called genuinely nonlinear if Dλ_{i}(u)·r_{i}(u) ≠ 0, for all u. The i-th characteristic field is called linearly degenerate if Dλ_{i}(u) · r_{i}(u) = 0, for all u.

If the solution of (11) is a piecewise smooth function u = u(t, x) having a discontinuity across a line x = γ(t) with u^{±} = lim_{x→γ(t)^{±}} u(t, x), then it satisfies (12) outside the γ, while along the line of discontinuities the Rankine–Hugoniot conditions holds: (u^{+} − u^{−})γ̇ = f(u^{+}) − f(u^{−}).

In order to have a unique solution we must require some additional conditions, known as entropy conditions. One of the most useful is the Lax condition, which says that a shock connecting the states u^{−} and u^{+}, travelling with speed γ̇ = λ_{i}(u^{−}, u^{+}) (λ_{i}(u^{−}, u^{+}) is an eigenvalue of the averaged matrix A(u^{−}, u^{+}), cf. [3]) is admissible if

(13)  λ_{i}(u^{−}) ⩾ λ_{i}(u^{−}, u^{+}) ⩾ λ_{i}(u^{+}).

Now we define two types of curves: fix a state u_{0} ∈ ℝ^{n}. Let r_{i}(u) be the i-th eigenvector of A(u). The i-th rarefaction curve through u_{0} is the integral curve of the vector field r_{i} and is denoted by σ ↦ R_{i}(σ)(u_{0}). The i-th shock curve through u_{0} is the curve of states u which can be connected to the right of u_{0} by an i-shock, satisfying the Rankine–Hugoniot conditions. It is denoted by σ ↦ S_{i}(σ)(u_{0}). The i-th rarefaction and shock curve are tangent to the r_{i}(u) at u_{0}.

Next we study the Riemann problem

(14)  \begin{aligned}u_{t} + f(u)_{x} & = 0 \\ u(0, x) & = \begin{cases}u^{−}, & x < 0, \\ u^{+}, & x > 0. \end{cases} \end{aligned}

Under the assumption that the system is strictly hyperbolic with smooth coefficients and each i-th characteristic field is either genuinely nonlinear or linearly degenerate, there exist three special cases:

(1) Centered rarefaction waves: the i-th characteristic field is genuinely nonlinear and u^{+} lies on the positive i-rarefaction curve through u^{−}, i.e., u^{+} = R_{i}(σ)(u^{−}) for some σ > 0. Then the solution of (14) is the centered rarefaction wave:

u(t, x) = \begin{cases}u^{−}, & x < tλ_{i}(u^{−}), \\ R_{i}(s)(u^{−}), & x = tλ_{i}(s), s ∈ [0, σ] \\ u^{+}, & x > tλ_{i}(u^{+}). \end{cases}

(2) Shocks: again the i-th characteristic field is genuinely nonlinear, but this time the state u^{+} lies on the i-th shock curve through u^{−}, i.e., u^{+} = S_{i}(σ)(u^{−}). Denote the Rankine–Hugoniot speed of the shock λ_{i}(u^{−}, u^{+}) by λ. Then the solution of (14) is the shock

(15)  u(t, x) = \begin{cases}u^{−}, & x < λt, \\ u^{+}, & x > λt. \end{cases}

(3) Contact discontinuities: the i-th characteristic field is linearly degenerate and u^{+} lies on the i-th rarefaction curve through u^{−}, i.e., u^{+} = R_{i}(σ)(u^{−}) for some σ. Then the function (15) is again a solution, but this time called the contact discontinuity.

The parameter σ, for which u^{+} = R_{i}(σ)(u^{−}) or u^{+} = S_{i}(σ)(u^{−}), is called the wave strength. Therefore, if u^{+} lies on the rarefaction or shock curve the solution of the Riemann problem (14) is one of the elementary waves- a centered rarefaction, a shock or a contact discontinuity. Otherwise, for u^{+} sufficiently close to u^{−}, the Riemann problem (14) can be decomposed in n auxiliary Riemann problems, which can be solved by an elementary wave. Piecing together those solutions we obtain a solution of the initial Riemann problem (14).

2. Symmetry Groups of the Systems of Conservation Laws

After we gave the brief overview of the notation and results from symmetry group analysis (classical and in the generalized setting) and conservation laws, we turn our attention to concrete systems of conservation laws. As we have just seen, a system of conservation laws is a system of first order partial differential equations. We introduced symmetry groups of systems of differential equations as local transformation groups which act on the space of independent and dependent variables, transforming the solution of the system to other solutions. Also, we defined associated and k-strongly associated symmetry groups. The aim of this section is to verify the results given in the introduction in two examples of conservation laws.

2.1. A model system of two strictly hyperbolic laws. The first system we consider is

\begin{aligned}u_{t} + (u^{2} − v)_{x} & = 0 \\ v_{t} + (\frac{1}{3}u^{3} − u)_{x} & = 0 \end{aligned} ; (16)

with initial conditions:

(17)  u(x, 0) = \{ \begin{matrix}u_{0}, & x < 0 \\ u_{1}, & x > 0 \end{matrix}  v(x, 0) = \{ \begin{matrix}v_{0}, & x < 0 \\ v_{1}, & x > 0. \end{matrix}

A motivation for studying this system arises from some physical models like a model for a nonlinear elastic system or a model for the evolution of ion-acoustic waves. We start by calculating symmetry groups of the system (16), using the procedure described in the introduction.

1) Since v denotes one of the dependent variables, denote the infinitesimal generator by w:

w = ξ(x, t, u, v)∂_{x} + τ(x, t, u, v)∂_{t} + ϕ(x, t, u, v)∂_{u} + ψ(x, t, u, v)∂_{v}.

2) From (2) we get the first prolongation of this vector field

pr^{(1)} w = w + ϕ^{x}∂_{u_{x}} + ϕ^{t}∂_{u_{t}} + ψ^{x}∂_{v_{x}} + ψ^{t}∂_{v_{t}},

and by (3) we calculate

(18) \begin{matrix}ϕ^{x} = & ϕ_{x} + & ϕ_{u}u_{x} + & ϕ_{v}v_{x} − & ξ_{x}u_{x} − & ξ_{u}u_{x}^{2} − & ξ_{v}u_{x}v_{x} − & τ_{x}u_{t} − & τ_{u}u_{x}u_{t} − & τ_{v}u_{t}v_{x} \\ ϕ^{t} = & ϕ_{t} + & ϕ_{u}u_{t} + & ϕ_{v}v_{t} − & ξ_{t}u_{x} − & ξ_{u}u_{x}u_{t} − & ξ_{v}u_{x}v_{t} − & τ_{t}u_{t} − & τ_{u}u_{t}^{2} − & τ_{v}u_{t}v_{t} \\ ψ^{x} = & ψ_{x} + & ψ_{u}u_{x} + & ψ_{v}v_{x} − & ξ_{x}v_{x} − & ξ_{u}u_{x}v_{x} − & ξ_{v}v_{x}^{2} − & τ_{x}v_{t} − & τ_{u}u_{x}v_{t} − & τ_{v}v_{x}v_{t} \\ ψ^{t} = & ψ_{t} + & ψ_{u}u_{t} + & ψ_{v}v_{t} − & ξ_{t}v_{x} − & ξ_{u}u_{t}v_{x} − & ξ_{v}v_{x}v_{t} − & τ_{t}v_{t} − & τ_{u}u_{t}v_{t} − & τ_{v}v_{t}^{2} \end{matrix}

3) Now we have

Δ_{1}(x, t, u, v, u_{x}, v_{x}, u_{t}, v_{t}) = u_{t} + 2uu_{x} − v_{x}
Δ_{2}(x, t, u, v, u_{x}, v_{x}, u_{t}, v_{t}) = v_{t} + u^{2}u_{x} − u_{x},

therefore we need to solve the system

pr^{(1)} w(Δ_{1}) = ϕ^{t} + 2ϕu_{x} + 2ϕ^{x}u − ψ^{x} = 0
pr^{(1)} w(Δ_{2}) = ψ^{t} + 2ϕuu_{x} + ϕ^{x}u^{2} − ϕ^{x} = 0,

whenever u_{t} = −2uu_{x} + v_{x} and v_{t} = −u^{2}u_{x} + u_{x}. Inserting (18) in these equations and replacing u_{t} by −2uu_{x} + v_{x} and v_{t} by −u^{2}u_{x} + u_{x}, whenever they appear, we arrive at

ϕ_{t} + ϕ_{u}(−2uu_{x}+v_{x}) + ϕ_{v}(−u^{2}u_{x}+u_{x}) − ξ_{t}u_{x} − ξ_{u}u_{x}(−2uu_{x}+v_{x})
− ξ_{v}u_{x}(−u^{2}u_{x}+u_{x}) − τ_{t}(−2uu_{x}+v_{x}) − τ_{u}(−2uu_{x}+v_{x})^{2}
− τ_{v}(−2uu_{x}+v_{x})(−u^{2}u_{x}+u_{x}) + 2ϕu_{x} + 2u[ϕ_{x} + ϕ_{u}u_{x} + ϕ_{v}v_{x}
− ξ_{x}u_{x} − ξ_{u}u_{x}^{2} − ξ_{v}u_{x}v_{x} − τ_{x}(−2uu_{x}+v_{x}) − τ_{u}u_{x}(−2uu_{x}+v_{x})
− τ_{v}(−2uu_{x}+v_{x})v_{x}] − [ψ_{x} + ψ_{u}u_{x} + ψ_{v}v_{x} − ξ_{x}v_{x} − ξ_{u}u_{x}v_{x}
− ξ_{v}v_{x}^{2} − τ_{x}(−u^{2}u_{x}+u_{x}) − τ_{u}u_{x}(−u^{2}u_{x}+u_{x}) − τ_{v}v_{x}(−u^{2}u_{x}+u_{x})] = 0

ψ_{t} + ψ_{u}(−2uu_{x} + v_{x}) + ψ_{v}(−u^{2}u_{x} + u_{x}) − ξ_{t}v_{x} − ξ_{u}(−2uu_{x} + v_{x})v_{x}

− ξ_{v}v_{x}(−u^{2}u_{x} + u_{x}) − τ_{t}(−u^{2}u_{x} + u_{x}) − τ_{u}(−2uu_{x} + v_{x})(−u^{2}u_{x} + u_{x})

− τ_{v}(−u^{2}u_{x} + u_{x})^{2} + 2ϕuu_{x} + (u^{2} − 1)[ϕ_{x} + ϕ_{u}u_{x} + ϕ_{v}v_{x} − ξ_{x}u_{x} − ξ_{u}u_{x}^{2}

− ξ_{v}u_{x}v_{x} − τ_{x}(−2uu_{x} + v_{x}) − τ_{u}u_{x}(−2uu_{x} + v_{x}) − τ_{v}(−2uu_{x} + v_{x})v_{x}] = 0.

4) Apparently, solving this system is quite complicated. Hence, we are going to look only for projectable symmetry groups. So, assume that ξ and τ only depend on x and t. Then we have

ϕ_{t} − 2ϕ_{u}uu_{x} + ϕ_{u}v_{x} − ϕ_{v}u^{2}u_{x}ϕ_{v}u_{x} − ξ_{t}u_{x} + 2τ_{t}uu_{x} − τ_{t}v_{x}
+ 2ϕu_{x} + 2ϕ_{x}u + 2ϕ_{u}uu_{x} + 2ϕ_{v}uv_{x} − 2ξ_{x}uu_{x} + 4τ_{x}u^{2}u_{x}
− 2τ_{x}uv_{x} − ψ_{x} − ψ_{u}u_{x} − ψ_{v}v_{x} + ξ_{x}v_{x} − τ_{x}u^{2}u_{x} + τ_{x}u_{x} = 0
ψ_{t} − 2ψ_{u}uu_{x} + ψ_{u}v_{x} − ψ_{v}u^{2}u_{x} + ψ_{v}u_{x} − ξ_{t}v_{x} + τ_{t}u^{2}u_{x} − τ_{t}u_{x}
+ 2ϕuu_{x} + ϕ_{x}u^{2} + ϕ_{u}u^{2}u_{x} + ϕ_{v}u^{2}v_{x} − ξ_{x}u^{2}u_{x} + 2τ_{x}u^{3}u_{x}
− τ_{x}u^{2}v_{x} − ϕ_{x} − ϕ_{u}u_{x} − ϕ_{v}v_{x} + ξ_{x}u_{x} − 2τ_{x}uu_{x} + τ_{x}v_{x} = 0.

These equations are in fact polynomials of free variables x, t, u, v, u_{x} and v_{x}. The solution will be found by looking at their coefficients on the left and right hand side of the equations. Since the functions ϕ, ψ and their derivatives depend on x, t, u and v we equate the coefficients of 1, u_{x} and v_{x} to 0. Then we arrive at the following equivalent system:

ϕ_{t} − ψ_{x} + 2uϕ_{x} = 0
2ϕ + ϕ_{v} − ψ_{u} − ξ_{t} + τ_{x} + u(2τ_{t} − 2ξ_{x}) + u^{2}(3τ_{x} − ϕ_{v}) = 0
ϕ_{u} − ψ_{v} + ξ_{x} − τ_{t} + u(2ϕ_{v} − 2τ_{x}) = 0
ψ_{t} − ϕ_{x} + u^{2}ϕ_{x} = 0
ψ_{v} − ϕ_{u} − τ_{t} + ξ_{x} + u(2ϕ − 2ψ_{u} − 2τ_{x}) + u^{2}(ϕ_{u} − ψ_{v} − ξ_{x} + τ_{t}) + 2u^{3}τ_{x} = 0
ψ_{u} − ϕ_{v} − ξ_{t} + τ_{x} + u^{2}(ϕ_{v} − τ_{t}) = 0

5) The general solution of this system is:

ξ(x, t) = c_{1}x + c_{3}t + c_{4}, ϕ(x, t, u, v) = c_{3},
τ(x, t) = c_{1}t + c_{2}  ψ(x, t, u, v) = c_{3}u + c_{5}

c_{1} − c_{5} are arbitrary constants. The linearly independent infinitesimal generators of projectable symmetry groups are:

w_{1} = x∂_{x} + t∂_{t}, w_{2} = ∂_{t}, w_{3} = t∂_{x} + ∂_{u} + u∂_{v}, w_{4} = ∂_{x}, w_{5} = ∂_{v}

6) It remains to compute the corresponding one-parameter symmetry groups. Hence, the one-parameter group G_{1}, generated by the vector field w_{1}, is a solution to the system of ODEs

ẋ(η) = x(η), ṫ(η) = t(η),

with initial data x(0) = x and t(0) = t, i.e., g_{η} : (x, t, u, v) → (e^{η}x, e^{η}t, u, v) . Since G_{1} is a symmetry group, (1) implies that if u and v are solutions of (16) so are the functions

ũ : (x, t) → u (e^{−η}x, e^{−η}t)
ṽ: (x, t) → v (e^{−η}x, e^{−η}t) .

We repeat the same procedure for the remaining symmetry groups and calculate that G_{2}, G_{4} and G_{5} are translations of t, x and v, respectively. Finally, the action of G_{3} is given by g_{η} : (x, t, u, v) → (x + ηt, t, u + η, v + ηu + η^{2}/2) , and the functions

ũ : (x, t) → u(x − ηt, t) + η
ṽ: (x, t) → v(x − ηt, t) + ηu(x − ηt, t) + 3η^{2}/2

are solutions of the system whenever u and v are.

Therefore, we calculated all projectable symmetry groups of the system (16) and all transformed solutions. As we saw, the calculation of non-projectable symmetry groups is rather complicated on the one hand, and on the other, as was mentioned in the introduction, it is enough to study projectable groups if the solution is in G, D′ or if it is a solution in the sense of association. We recall from [8] that the system (16) has a solution in the sense of association: the Jacobian matrix of (16) is

A = Df = [\begin{matrix}2u & −1 \\ u^{2} − 1 & 0 \end{matrix}] ,

the eigenvalues are λ_{1}(u, v) = u−1 and λ_{2}(u, v) = u+1, and the corresponding right eigenvectors are r_{1}(u, v) = [1 u + 1]^{T} and r_{2}(u, v) = [1 u − 1]^{T}. Since Dλ_{i}(u, v) · r_{i}(u, v) > 0, i = 1, 2, it follows that both characteristic fields are genuinely nonlinear and hence the solution consists only of centered rarefaction waves and shocks.

The rarefaction curves are calculated as the integral curves of the vector fields r_{1} and r_{2}:

R_{1} = \{(u, v) : v = \frac{1}{2}u^{2} + u + c_{1}\}

R_{2} = \{(u, v) : v = \frac{1}{2}u^{2} − u + c_{2}\},

and the shocks are found from the Rankine–Hugoniot equations:

(19)  v − v_{0} = (u − u_{0})(\frac{u + u_{0}}{2} ∓ \sqrt{1 − \frac{(u − u_{0})^{2}}{12}} ), for |u − u_{0}| ⩽ 12.

The corresponding shock speeds are

γ̇ = u_{0} + \frac{u − u_{0}}{2} ± \sqrt{1 − \frac{(u − u_{0})^{2}}{12}} ,

where the sign − refers to the speed of 1-shock, and + to the speed of 2-shock. The Riemann problem (16)–(17) has a classical solution for each (u, v) lying in the area bounded by

J(u, v) = \{(u, v) : v = \frac{1}{2}u^{2} + u + \frac{9}{2} + v_{0} − \frac{1}{2}u_{0}^{2} − u_{0} ∧ u ⩾ u_{0} − 3\}

J_{1}(u, v) = {(u, v) : (u, v) satisfies (19) ∧ u ⩽ u_{0} − 3}
J_{2}(u, v) = \{(u, v) : v = \frac{1}{2}u^{2} − u − \frac{9}{2} + v_{0} − \frac{1}{2}u_{0}^{2} + u_{0} ∧ u ⩾ u_{0} − 3\}

The remaining (u, v) are in the exterior of this area, which we denote by Q and which is divided by the curves

D(u, v) = {(u, v) : v = v_{0} + u^{2} + (1 − u_{0})u − u_{0} ∧ u ⩽ u_{0} − 3}
E(u, v) = {(u, v) : v = v_{0} + (u − u_{0})(u_{0} − 1) ∧ u ⩽ u_{0} − 3}

into three open regions. In each of them the solution consists of a singular shock wave which is given by:

(20)  \begin{aligned}U(x, t) & = G(x − ct) + s_{1}(t)(α_{0}d^{−}(x − ct) + α_{1}d^{+}(x − ct)) \\ V (x, t) & = H(x − ct) + s_{2}(t)(β_{0}D^{−}(x − ct) + β_{1}D^{+}(x − ct)), \end{aligned}

where G(x − ct) and H(x − ct) are generalized step functions (cf. [13, Def. 1(a)]), D(x − ct) = β_{0}D^{−}(x − ct) + β_{1}D^{+}(x − ct) is an Sδ-function with value (β_{0}, β_{1}), β_{0} + β_{1} = 1 (cf. [13, Def. 1(b)]), d(x − ct) = α_{0}d^{−}(x − ct) + α_{1}d^{+}(x − ct) is an 3′SD-function with value (α_{0}, α_{1}) (cf. [13, Def. 3 with Ex. (ii)]), such that D(x−ct) and d(x − ct) are compatible and

−c[G] + [G^{2}] − [H] = 0, (21)

s_{2}(t) = s_{1}^{2}(t)(α_{0}^{2} + α_{1}^{2}), (22)

s_{2}(t) = σ_{1}t, σ_{1} = c[H] − \frac{1}{3}[G^{3}] + [G], σ_{1} > 0, (23)

cs_{2}(t) = s_{1}^{2}(t)(α_{0}u_{0} + α_{1}u_{1}). (24)

The function s_{2}(t) is called the strength of the singular shock wave and is the most important part of the solution which has to be uniquely determined. α_{0} and α_{1} can be chosen such that α_{0}^{2} + α_{1}^{2} = 1, hence the condition (22) becomes s_{2}(t) = s_{1}^{2}(t), and the condition (24) becomes α_{0}^{2}u_{0} + α_{1}^{2}u_{1} = c.

We are going to show that this is a solution in the sense of 1-strong association.

Theorem 2.1. The solution (20) of the system (16) is a 1-strongly associated solution to (16).

Proof. In order to show this we use Definition 1.4. Let B be a bounded subset of C_{c}^{1}(ℝ × [0, ∞)). This means that there exists K ⋐ ℝ × [0, ∞), with the property supp φ ⊆ K for each φ ∈ B and with sup_{(x,t)∈K}{|∂^{α}φ(x, t)| : φ ∈ B, |α| ⩽ 1} < ∞ . It suffices to show that there exist representatives U_{ε} and V_{ε} of the solution (20) such that

(25)  lim_{ε→0} sup_{φ∈B} | ∫_{ℝ×[0,∞)} ((U_{ε})_{t}(x, t) + (U_{ε}^{2} − V_{ε})_{x}(x, t))φ(x, t) dx dt| = 0,

and

(26) lim_{ε→0} sup_{φ∈B} | ∫_{ℝ×[0,∞)} ((V_{ε})_{t}(x, t) + (\frac{1}{3}U_{ε}^{3} − U_{ε})_{x}(x, t))φ(x, t) dx dt| = 0.

40 KONJIK

Look first at (25). Let φ ∈ B. Then

∫ℝ×[0,∞) [(U_{ε}(x, t))_{t} + (U_{ε}^{2}(x, t) − V_{ε}(x, t))_{x}]φ(x, t) dx dt
= ∫_{ℝ×[0,∞)}[{G_{ε}(x − ct) + s_{1}(t)(α_{0}d^{−}(x − ct) + α_{1}d^{+}(x − ct))}_{t}

+ {G_{ε}^{2}(x − ct) + 2s_{1}(t)(α_{0}u_{0}d^{−}(x − ct) + α_{1}u_{1}d^{+}(x − ct)) + s_{1}^{2}(t)(α_{0}^{2}(d^{−})^{2}(x − ct) + α_{1}^{2}(d^{+})^{2}(x − ct)) − H_{ε}(x − ct)

− s_{2}(t)(β_{0}D^{−}(x − ct) + β_{1}D^{+}(x − ct))}_{x}]φ(x, t) dx dt
= ∫_{ℝ×[0,∞)}[ − c∂_{x}G_{ε}(x − ct) + s′_{1}(t)(α_{0}d^{−}(x − ct) + α_{1}d^{+}(x − ct))

− cs_{1}(t)∂_{x}(α_{0}d^{−}(x − ct) + α_{1}d^{+}(x − ct)) + ∂_{x}G_{ε}(x − ct) + s_{1}(t)∂_{x}(α_{0}u_{0}d^{−}(x − ct) + α_{1}u_{1}d^{+}(x − ct)) + s_{1}^{2}(t)∂_{x}(α_{0}^{2}(d^{−})^{2}(x − ct) + α_{1}^{2}(d^{+})^{2}(x − ct)) − ∂_{x}H_{ε}(x−ct) − s_{2}(t)∂_{x}(β_{0}D^{−}(x−ct) + β_{1}D^{+}(x−ct))]φ(x, t) dx dt

= ∫_{ℝ×[0,∞)} \underbrace{s′_{1}(t)(α_{0}d^{−}(x − ct) + α_{1}d^{+}(x − ct))}_{(1)} φ(x, t) dx dt
− ∫_{ℝ×[0,∞)}[ − \underbrace{cG_{ε}(x − ct)}_{(2)} − \underbrace{cs_{1}(t)(α_{0}d^{−}(x − ct) + α_{1}d^{+}(x − ct))}_{(3)}

+ \underbrace{G_{ε}^{2}(x − ct)}_{(4)} + \underbrace{s_{1}(t)(α_{0}u_{0}d^{−}(x − ct) + α_{1}u_{1}d^{+}(x − ct))}_{(5)}

+ \underbrace{s_{1}^{2}(t)(α_{0}^{2}(d^{−})^{2}(x − ct) + α_{1}^{2}(d^{+})^{2}(x − ct))}_{(6)}

− \underbrace{H_{ε}(x − ct)}_{(7)} − \underbrace{s_{2}(t)(β_{0}D^{−}(x − ct) + β_{1}D^{+}(x − ct))}_{(8)} ]φ_{x}(x, t) dx dt

= (∗) For the first member of this sum we have

∫_{ℝ×[0,∞)} s′_{1}(t)(α_{0}d^{−}(x − ct) + α_{1}d^{+}(x − ct))φ(x, t) dx dt
= ∫_{0}^{∞} ∫_{ℝ} s′_{1}(t)(α_{0}( − \frac{1}{2ε}ϕ(\frac{−x + ct − 4ε}{ε}) + \frac{1}{2ε}ϕ(\frac{−x + ct − 6ε}{ε}))^{1/2}

+ α_{1}(\frac{1}{2ε}ϕ(\frac{x − ct − 4ε}{ε}) − \frac{1}{2ε}ϕ(\frac{x − ct − 6ε}{ε}))^{1/2})φ(x, t) dx dt

Here we used the 3′SD-function from [13]. Introducing suitable substitutions we obtain

∫_{0}^{∞} s′_{1}(t) ∫_{−1}^{∞} α_{0}\sqrt{ε}(\frac{1}{2}ϕ(z))^{1/2}φ(−εz − 4ε + ct, t) dz dt

− ∫_{0}^{∞} s′_{1}(t) ∫_{−1}^{∞} α_{0}\sqrt{ε}(\frac{1}{2}ϕ(z))^{1/2}φ(−εz − 6ε + ct, t) dz dt

+ ∫_{0}^{∞} s′_{1}(t) ∫_{−1}^{∞} α_{1}\sqrt{ε}(\frac{1}{2}ϕ(z))^{1/2}φ(εz − 4ε + ct, t) dz dt

− ∫_{0}^{∞} s′_{1}(t) ∫_{−1}^{∞} α_{1}\sqrt{ε}(\frac{1}{2}ϕ(z))^{1/2}φ(εz − 6ε + ct, t) dz dt.

Applying the Lebesgue dominated convergence theorem two times successively to the corresponding sequences we conclude that this term tends to 0 as ε → 0. A similar argument shows that each of the terms in the sum with the functions d^{±} or (d^{±})^{3}, i.e., (3) and (5), also goes to 0 as ε → 0. So, look now at (2). We have

− ∫_{ℝ×[0,∞)} − cG_{ε}(x − ct)φ_{x}(x, t) dx dt

= ∫_{0}^{∞} ∫_{−∞}^{ct−ε} cu_{0}φ_{x}(x, t) dx dt + ∫_{0}^{∞} ∫_{ct+ε}^{∞} cu_{1}φ_{x}(x, t) dx dt

= cu_{0} ∫_{0}^{∞} φ(ct − ε, t) dt − cu_{1} ∫_{0}^{∞} φ(ct + ε, t) dt

⟶^{ε→0} (cu_{0} − cu_{1}) ∫_{0}^{∞} φ(ct, t) dt = −c[G] ∫_{0}^{∞} φ(ct, t) dt,

where we again applied the Lebesgue dominated convergence theorem. For (8) we obtain

− ∫_{ℝ×[0,∞)} −s_{2}(t)(β_{0}D^{−}(x − ct) + β_{1}D^{+}(x − ct))φ_{x}(x, t) dx dt

= ∫_{0}^{∞} ∫_{−∞}^{ct−ε} s_{2}(t)\frac{β_{0}}{ε} ϕ(\frac{x − ct + 2ε}{ε})φ_{x}(x, t) dx dt

+ ∫_{0}^{∞} ∫_{ct+ε}^{∞} s_{2}(t)\frac{β_{1}}{ε} ϕ(\frac{x − ct − 2ε}{ε})φ_{x}(x, t) dx dt

= ∫_{0}^{∞} s_{2}(t) ∫_{−∞}^{1} β_{0}ϕ(z)φ_{x}(εz − 2ε + ct, t) dz dt

+ ∫_{0}^{∞} s_{2}(t) ∫_{−1}^{∞} β_{1}ϕ(z)φ_{x}(εz + 2ε + ct, t) dz dt

⟶^{ε→0} ∫_{0}^{∞} s_{2}(t)β_{0}φ_{x}(ct, t) ∫_{−∞}^{1} ϕ(z) dz dt + ∫_{0}^{∞} s_{2}(t)β_{1}φ_{x}(ct, t) ∫_{−1}^{∞} ϕ(z) dz dt

= (β_{0} + β_{1}) ∫_{0}^{∞} s_{2}(t)φ_{x}(ct, t) dt = ∫_{0}^{∞} s_{2}(t)φ_{x}(ct, t) dt.

Repeating this for the remaining terms yields that (4) tends to [G^{2}] ∫_{0}^{∞} φ(ct, t) dt,

(6) to (α_{0}^{2}+α_{1}^{2}) ∫_{0}^{∞} s_{1}^{2}(t)φ_{x}(ct, t) dt, and (7) to −[H] ∫_{0}^{∞} φ(ct, t) dt, as ε → 0. Therefore,

(∗) ⟶^{ε→0} − c[G] ∫_{0}^{∞}φ(ct, t) dt + [G^{2}] ∫_{0}^{∞}φ(ct, t) dt + (α_{0}^{2} + α_{1}^{2}) ∫_{0}^{∞}s_{1}^{2}(t)φ(ct, t) dt
− [H]∫_{0}^{∞}φ(ct, t) dt − ∫_{0}^{∞}s_{2}(t)φ(ct, t) dt = 0,

by (21)–(24) and (25) is satisfied. Similarly we conclude that it is also true for (26). Hence, the solution (20) is a 1-strongly associated solution to (16). □

The projectable symmetry groups calculated at the beginning of this section transform 1-strongly associated solutions to (16) to other 1-strongly associated solutions, as shown by the following

Theorem 2.2. The symmetry groups G_{1}–G_{5} of the system (16) are AS_{Δ}^{1}- symmetry groups.

Proof. By Theorem 1.5 it suffices to show that G_{1}-G_{5} are slowly increasing and have a factorization (10) such that Q depends only on η and x. First we consider G_{1}. The action of G_{1} is given by g_{η} : (x, t, u, v) → (e^{η}x, e^{η}t, u, v) . Since Φ is the identity it follows that the map (u, v) ↦ Φ_{g}(x, t, u, v) is slowly increasing, uniformly for x and t in compact sets. It is easy to see that this is also true for the remaining groups. Next, for G_{1} we have

Δ_{1}(e^{−η}x, e^{−η}t, pr^{(1)} u(e^{−η}x, e^{−η}t), pr^{(1)} v(e^{−η}x, e^{−η}t)) = e^{−η}u_{t} + 2e^{−η}uu_{x} − e^{−η}v_{x}
= e^{−η}Δ_{1}
Δ_{2}(e^{−η}x, e^{−η}t, pr^{(1)} u(e^{−η}x, e^{−η}t), pr^{(1)} v(e^{−η}x, e^{−η}t)) = e^{−η}v_{t} + e^{−η}u^{2}u_{x} − e^{−η}u_{x}
= e^{−η}Δ_{2},

where Δ_{1} and Δ_{2} denote the first, respectively the second equation of the system (16). The matrix form of this factorization is given by

[Δ̃_{1} ; Δ̃_{2}] = [\begin{matrix}e^{−η} & 0 \\ 0 & e^{−η} \end{matrix}] · [\begin{matrix}Δ_{1} \\ Δ_{2} \end{matrix}] .

Therefore, the matrix Q depends only on η and Theorem 1.5 provides that the G_{1} is AS_{Δ}^{1}-symmetry group. The factorizations for G_{2}–G_{5} are

G_{2}, G_{4}, G_{5} :  [Δ̃_{1} ; Δ̃_{2}] = [\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}] · [\begin{matrix}Δ_{1} \\ Δ_{2} \end{matrix}] ,

G_{3} :  [Δ̃_{1} ; Δ̃_{2}] = [\begin{matrix}1 & 0 \\ η & 1 \end{matrix}] · [\begin{matrix}Δ_{1} \\ Δ_{2} \end{matrix}] ,

hence again we conclude that these four symmetry groups are also AS_{Δ}^{1}-symmetry groups. □

2.2. Zero pressure gas dynamics model. The next system of conservation laws we consider is given by

\begin{aligned}u_{t} + (uv)_{x} & = 0 \\ (uv)_{t} + (uv^{2})_{x} & = 0 \end{aligned} ; (27)

with the same initial conditions (17) as in the previous case. This Riemann problem is a zero pressure gas dynamics model, where u is a density, hence nonnegative, and v is a velocity.

The quasilinear form of (27) is obtained by differentiating:

\begin{aligned}u_{t} + u_{x}v + uv_{x} & = 0 \\ u_{t}v + uv_{t} + u_{x}v^{2} + 2uvv_{x} & = 0. \end{aligned} ; (28)

If we compute u_{t} from the first equation of (28) and insert into the second one we arrive to the following system:

\begin{aligned}u_{t} + u_{x}v + uv_{x} & = 0 \\ u(v_{t} + vv_{x}) & = 0. \end{aligned} ; (29)

From the second equation it can be seen that one possible solution is u = 0, i.e., vacuum state. Therefore, we consider the other possibility v_{t} + vv_{x} = 0, looking at the system

\begin{aligned}u_{t} + u_{x}v + uv_{x} & = 0 \\ v_{t} + vv_{x} & = 0. \end{aligned} ; (30)

The eigenvalues of this system are λ_{1}(u, v) = λ_{2}(u, v) = v, thus the system is weakly hyperbolic. The corresponding right eigenvector is r(u, v) = [0 1]^{T}, so both characteristic fields are linearly degenerative. Hence, only contact discontinuities can appear as a solution and we calculate them: let x = ct be a curve of discontinuities of the system (29). Along this curve the Rankine–Hugoniot conditions must hold:

c[u] = [uv], c[uv] = [uv^{2}].

Equating c from these equations yields [u][uv^{2}] = [uv]^{2}, so u_{0}u_{1}(v_{1} − v_{0})^{2} = 0. Therefore, there exist three solutions: u_{0} = 0, u_{1} = 0 and v_{0} = v_{1}. For u_{0} = 0 we calculate c = v_{1}, λ_{i}(u_{0}, v_{0}) = v_{0} and λ_{i}(u_{1}, v_{1}) = v_{1}, i = 1, 2 and similarly for u_{1} = 0. In the third case c = v_{0} = v_{1} and also λ_{i}(u_{0}, v_{0}) = λ_{i}(u_{1}, v_{1}) = v_{0} = v_{1}, thus the Lax entropy condition (13) is satisfied. Hence the initial conditions (u_{0}, v_{0}) and (u_{1}, v_{1}) can be connected by contact discontinuity only when v_{0} = v_{1}. Finally, combining contact discontinuities and vacuum states we obtain the classical solution of the Riemann problem (27) when v_{0} < v_{1}:

(u, v)(x, t) = \{ \begin{matrix}(u_{0}, v_{0}), & x/t < v_{0} \\ (0, v(x, t)), & v_{0} ⩽ x/t ⩽ v_{1} \\ (u_{1}, v_{1}), & x/t > v_{1}. \end{matrix}

In the case when v_{0} > v_{1} this solution is not uniquely defined and certain nonregularities appear, which is studied in detail in [13]. In that case the solution of (27) is again a singular shock wave

\begin{aligned}U(x, t) & = G(x − ct) + s_{1}(t)(α_{0}D^{−} + α_{1}D^{+}) + s_{2}(t)(β_{0}d^{−} + β_{1}d^{+}) \\ V (x, t) & = H(x − ct) + s_{3}(t)(γ_{0}d^{−} + γ_{1}d^{+}), \end{aligned} ; (31)

where G and H are generalized step functions, D and d are compatible Sδ-and 3SD-functions (cf. [13, Def. 3 with Ex. (i)]) and

s_{1}(t) = σ_{1}t, σ_{1} = c[G] − [GH], σ_{1} > 0, (32)

α_{0} = \frac{v_{1} − c}{v_{1} − v_{0}} , α_{1} = \frac{c − v_{0}}{v_{1} − v_{0}} , (33)

σ_{1}(α_{0}v_{0} + α_{1}v_{1}) = σ_{1}c = c[GH] − [GH^{2}], (34)

−s_{2}(t)s_{3}^{2}(t) = s_{1}(t), (35)

α_{0}(v_{0}^{2} − cv_{0}) + α_{1}(v_{1}^{2} − cv_{1}) = β_{0}γ_{0}^{2} + β_{1}γ_{1}^{2}. (36)

This time the function s_{1}(t) denotes the strength of the singular shock wave.

As for the first system (16) we are going to show that this solution is also a 1-strongly associated solution to (27).

Theorem 2.3. The solution (31) of the system (27) is a 1-strongly associated solution to (27).

Proof. The proof is similar to that for the system (16), so we have to show that there exist representatives U_{ε} and V_{ε} of the solutions U and V defined in (31), such that for arbitrary set B ⊆ C_{c}^{∞}(ℝ × [0, ∞)) bounded in C_{c}^{1}(ℝ × [0, ∞)) the following holds:

(37)  lim_{ε→0} sup_{φ∈B} | ∫_{ℝ×[0,∞)} ((U_{ε})_{t}(x, t) + (U_{ε}V_{ε})_{x}(x, t))φ(x, t) dx dt| = 0,

and for the second equation:

(38)  lim_{ε→0} sup_{φ∈B} | ∫_{ℝ×[0,∞)} ((U_{ε}V_{ε})_{t}(x, t) + (U_{ε}V_{ε}^{2})_{x}(x, t))φ(x, t) dx dt| = 0.

Let B be a bounded subset of C_{c}^{1}(ℝ × [0, ∞)). First we prove (37). Let φ ∈ B. Then

∫ℝ×[0,∞) [(U_{ε}(x, t))_{t} + (U_{ε}(x, t)V_{ε}(x, t))_{x}]φ(x, t) dx dt
= ∫_{ℝ×[0,∞)}[{G_{ε}(x − ct) + s_{1}(t)(α_{0}D^{−}(x − ct) + α_{1}D^{+}(x − ct))
+ s_{2}(t)(β_{0}d^{−}(x − ct) + β_{1}d^{+}(x − ct))}_{t} + {G_{ε}(x − ct)H_{ε}(x − ct)
+ s_{1}(t)s_{3}(t)(α_{0}D^{−}(x−ct)+α_{1}D^{+}(x−ct))(γ_{0}d^{−}(x−ct)+γ_{1}d^{+}(x−ct))
+ s_{2}(t)s_{3}(t)(β_{0}d^{−}(x−ct)+β_{1}d^{+}(x−ct))(γ_{0}d^{−}(x−ct)+γ_{1}d^{+}(x−ct))
+ s_{1}(t)(α_{0}v_{0}D^{−}(x − ct) + α_{1}v_{1}D^{+}(x − ct))
+ s_{2}(t)(β_{0}v_{0}d^{−}(x − ct) + β_{1}v_{1}d^{+}(x − ct))
+ s_{3}(t)(γ_{0}u_{0}d^{−}(x − ct) + γ_{1}u_{1}d^{+}(x − ct))}_{x}]φ(x, t) dx dt

= ∫_{ℝ×[0,∞)}[ − c∂_{x}G_{ε}(x − ct) + s′_{1}(t)(α_{0}D^{−}(x − ct) + α_{1}D^{+}(x − ct))

− cs_{1}(t)∂_{x}(α_{0}D^{−}(x − ct) + α_{1}D^{+}(x − ct))

+ s′_{2}(t)(β_{0}d^{−}(x − ct) + β_{1}d^{+}(x − ct))

− cs_{2}(t)∂_{x}(β_{0}d^{−}(x − ct) + β_{1}d^{+}(x − ct)) + ∂_{x}(G_{ε}(x − ct)H_{ε}(x − ct))

+ s_{1}(t)s_{3}(t)∂_{x}((α_{0}D^{−}(x − ct) + α_{1}D^{+}(x − ct))

× (γ_{0}d^{−}(x − ct) + γ_{1}d^{+}(x − ct)))

+ s_{2}(t)s_{3}(t)∂_{x}((β_{0}d^{−}(x − ct) + β_{1}d^{+}(x − ct))

× (γ_{0}d^{−}(x − ct) + γ_{1}d^{+}(x − ct)))

+ s_{1}(t)∂_{x}(α_{0}v_{0}D^{−}(x − ct) + α_{1}v_{1}D^{+}(x − ct))

+ s_{2}(t)∂_{x}(β_{0}v_{0}d^{−}(x − ct) + β_{1}v_{1}d^{+}(x − ct))

+ s_{3}(t)∂_{x}(γ_{0}u_{0}d^{−}(x − ct) + γ_{1}u_{1}d^{+}(x − ct))]φ(x, t) dx dt

= ∫_{ℝ×[0,∞)}[ \underbrace{s′_{1}(t)(α_{0}D^{−}(x − ct) + α_{1}D^{+}(x − ct))}_{(1)}

+ \underbrace{s′_{2}(t)(β_{0}d^{−}(x − ct) + β_{1}d^{+}(x − ct))}_{(2)} ]φ(x, t) dx dt

− ∫_{ℝ×[0,∞)}[ − \underbrace{cG_{ε}(x − ct)}_{(3)} − \underbrace{cs_{1}(t)(α_{0}D^{−}(x − ct) + α_{1}D^{+}(x − ct))}_{(4)}

− \underbrace{cs_{2}(t)(β_{0}d^{−}(x − ct) + β_{1}d^{+}(x − ct))}_{(5)} + \underbrace{G_{ε}(x − ct)H_{ε}(x − ct)}_{(6)}

+ \underbrace{s_{1}(t)s_{3}(t)(α_{0}D^{−}(x−ct)+α_{1}D^{+}(x−ct))(γ_{0}d^{−}(x−ct)+γ_{1}d^{+}(x−ct))}_{(7)}

+ \underbrace{s_{2}(t)s_{3}(t)(β_{0}d^{−}(x−ct)+β_{1}d^{+}(x−ct))(γ_{0}d^{−}(x−ct)+γ_{1}d^{+}(x−ct))}_{(8)}

+ \underbrace{s_{1}(t)(α_{0}v_{0}D^{−}(x − ct) + α_{1}v_{1}D^{+}(x − ct))}_{(9)}

+ \underbrace{s_{2}(t)(β_{0}v_{0}d^{−}(x − ct) + β_{1}v_{1}d^{+}(x − ct))}_{(10)}

+ \underbrace{s_{3}(t)(γ_{0}u_{0}d^{−}(x − ct) + γ_{1}u_{1}d^{+}(x − ct))}_{(11)} ]φ_{x}(x, t) dx dt = (∗)

Consider now each of the terms in the last sum, like in the proof of Theorem 2.1. For (1) we have:

∫ℝ×[0,∞)s′_{1}(t)(α_{0}D^{−}(x − ct) + α_{1}D^{+}(x − ct))φ(x, t) dx dt
= ∫_{0}^{∞} ∫_{ℝ}s′_{1}(t)(\frac{α_{0}}{ε} ϕ(\frac{x − ct + 2ε}{ε}) + \frac{α_{1}}{ε} ϕ(\frac{x − ct − 2ε}{ε}))φ(x, t) dx dt

Next, split up this integral as a sum of two integrals and take suitable substitutions. This yields

∫_{0}^{∞} ∫_{−∞}^{1}s′_{1}(t)α_{0}ϕ(z)φ(εz − 2ε + ct, t) dz dt
+ ∫_{0}^{∞} ∫_{−1}^{∞}s′_{1}(t)α_{1}ϕ(z)φ(εz + 2ε + ct, t) dz dt

Here we apply the Lebesgue dominated convergence theorem, first to the sequences ( ∫_{−∞}^{1} φ(εz−2ε+ct, t) dz)_{ε} and ( ∫_{−1}^{∞} φ(εz+2ε+ct, t) dz)_{ε}, and then to the sequences (φ(εz − 2ε + ct, t))_{ε} and (φ(εz + 2ε + ct, t))_{ε}. Then the last integral

⟶^{ε→0} ∫_{0}^{∞} s′_{1}(t)α_{0}φ(ct, t) ∫_{−∞}^{1} ϕ(z) dz dt + ∫_{0}^{∞} s′_{1}(t)α_{1}φ(ct, t) ∫_{−1}^{∞} ϕ(z) dz dt

= (α_{0} + α_{1}) ∫_{0}^{∞} s′_{1}(t)φ(ct, t) dt = ∫_{0}^{∞} s′_{1}(t)φ(ct, t) dt,

since by definition of Sδ-functions, ∫ ϕ(z) dz = 1 on the domain of ϕ (and that is the interval [−1, 1]) and α_{0} + α_{1} = 1 by assumption. It is obvious that a procedure similar to this one and those from the proof of Theorem 2.1 is repeated for each of the remaining 9 terms (some of them are explicitly calculated in the proof of Theorem 2.1). The only difference is that for the solution of the system (31) we used 3SD-functions instead of 3′SD-functions from the solution of (16). This implies that all terms with d^{±} or (d^{±})^{2} tend to 0 as ε → 0.

Joining all together we have

(∗) ⟶^{ε→0} ∫_{0}^{∞} s′_{1}(t)φ(ct, t) dt − c[G] ∫_{0}^{∞} φ(ct, t) dt + ∫_{0}^{∞} cs_{1}(t)φ_{x}(ct, t) dt
+[GH] ∫_{0}^{∞} φ(ct, t) dt − ∫_{0}^{∞}(α_{0}v_{0} + α_{1}v_{1})s_{1}(t)φ_{x}(ct, t) dt = 0,

since from (32) s′_{1}(t) = c[G]−[GH] and α_{0}v_{0}+α_{1}v_{1} = c from (34) and the condition α_{0} + α_{1} = 1. Thus (31) is a 1-strongly associated solution to the first equation of (27). Analogously, it can be seen that the same holds also in the case of the second equation, which proves the claim. □

The next task is to calculate symmetry groups of (27). Let us recall that (27), (28) and (29) are equivalent systems for all smooth solutions. Also, these systems are equivalent in the Colombeau algebra, since the elements of this algebra are equivalence classes of nets of smooth functions. Therefore we look for the symmetry groups of the quasilinear system (29). We start with the following

Theorem 2.4. Let Δ(x, u^{(n)}) = 0 and Δ_{i}(x, u^{(n)}), i = 1, …, k, be nondegenerate differential equations on M ⊂ X × U such that Δ can be written as a product

(39)  Δ = ∏_{i=1}^{k} Δ_{i}.

If we denote the corresponding algebras of infinitesimal generators of symmetries of Δ and Δ_{i} by 𝔤 and 𝔤_{i} respectively, i = 1, …, k, then

(40)  ⋂_{i=1}^{k} 𝔤_{i} ⊆ 𝔤.

Proof. Let v ∈ ⋂_{i=1}^{k} 𝔤_{i}. Then v ∈ 𝔤_{i}, for each i = 1, …, k, i.e., v is a generator of a local one-parameter symmetry group of each equation

Δ_{i}(x, u^{(n)}) = 0, i = 1, …, k.

By the infinitesimal criterion and (6) we may write

(41)  pr^{(n)} v(Δ_{i}(x, u^{(n)})) = Q_{i} · Δ_{i}(x, u^{(n)}), i = 1, …, l,

with well-defined functions Q_{i}, i = 1, …, l. Since pr^{(n)} v is a vector field on the n-jet space M^{(n)}, the Leibniz rule for the product derivative yields

pr^{(n)} v(Δ) = pr^{(n)} v( ∏_{i=1}^{k} Δ_{i}) = ∑_{i=1}^{k} Δ_{1} · . . . · pr^{(n)} v(Δ_{i}) · . . . · Δ_{k}

= ∑_{i=1}^{k} Δ_{1} · . . . · Q_{i}Δ_{i} · . . . · Δ_{k} = Q · Δ,

where Q = Q_{1} + ⋯ + Q_{l}. Another application of Theorem 1.2 provides that v ∈ 𝔤, which proves the claim. □

According to this theorem, the intersection of the symmetry groups of the system (30) with the symmetry groups of u = 0 will provide symmetry groups of the system (29). (It should be noticed that due to the inclusion in (40) not all symmetry groups of (29) will be obtained. However, Theorem 2.4 is of great help, since a direct computation of the symmetry groups of (29) is a very difficult task.) The symmetry groups of u = 0 can easily be calculated. Namely, by the infinitesimal criterion (5) it follows that the infinitesimal generators of u = 0 are obtained as solutions of v(u) = ϕ = 0, whenever u = 0 (we assumed here that v = ξ(x, t, u, v)∂_{x} + τ(x, t, u, v)∂_{t} + ϕ(x, t, u, v)∂_{u} + ψ(x, t, u, v)∂_{v}).

Now we follow the procedure for calculating symmetry groups of the system (30).

1) w = ξ(x, t, u, v)∂_{x} + τ(x, t, u, v)∂_{t} + ϕ(x, t, u, v)∂_{u} + ψ(x, t, u, v)∂_{v}.

2) The first prolongation is given by pr^{(1)}w = w+ϕ^{x}∂_{u_{x}} +ϕ^{t}∂_{u_{t}} +ψ^{x}∂_{v_{x}} +ψ^{t}∂_{v_{t}}, where ϕ^{x}, ϕ^{t}, ψ^{x} and ψ^{t} are the same as in (18).

3) Since the equations of this systems are

Δ_{1}(x, t, u, v, u_{x}, v_{x}, u_{t}, v_{t}) = u_{t} + u_{x}v + uv_{x}
Δ_{2}(x, t, u, v, u_{x}, v_{x}, u_{t}, v_{t}) = v_{t} + vv_{x},

we have to solve

pr^{(1)}w(Δ_{1}) = ϕ^{t} + vϕ^{x} + ϕu_{x} + ϕv_{x} + uψ^{x} = 0
pr^{(1)}w(Δ_{2}) = ψ^{t} + ψv_{x} + vψ^{x} = 0.

Again we look only for the projectable symmetry groups. Inserting (18), having in mind that the partial derivatives of ξ and τ with respect to u and v vanish, and then substituting u_{t} by −u_{x}v − uv_{x} and v_{t} by −vv_{x}, we obtain

ϕ_{t} − ϕ_{u}vu_{x} − ϕ_{u}uv_{x} − ϕ_{v}vv_{x} − ξ_{t}u_{x} + τ_{t}vu_{x} + τ_{t}uv_{x}
+ ϕ_{x}v + ϕ_{u}vu_{x} + ϕ_{v}vv_{x} − ξ_{x}vu_{x} + τ_{x}v^{2}u_{x} + τ_{x}uvv_{x}
+ ψu_{x} + ϕv_{x} + ψ_{x}u + ψ_{u}uu_{x} + ψ_{v}uv_{x} − ξ_{x}uv_{x} − τ_{x}uvv_{x} = 0
ψ_{t} − ψ_{u}vu_{x} − ψ_{u}uv_{x} − ψ_{v}vv_{x} − ξ_{t}v_{x}
+ τ_{t}vv_{x} + ψv_{x} + ψ_{x}v + ψ_{u}vu_{x} + ψ_{v}vv_{x} − ξ_{x}vv_{x} + τ_{x}v^{2}v_{x} = 0.

4) Coefficients of 1, u_{x} and v_{x} equating with 0 yield the following equations

ϕ_{t} + vϕ_{x} + uψ_{x} = 0
−ξ_{t} + τ_{t}v − ξ_{x}v + τ_{x}v^{2} + ψ + uψ = 0
−ϕ_{u}u + τ_{t}u + τ_{x}uv + ϕ + ψ_{v}u − ξ_{x}u + τ_{x}uv = 0
ψ_{t} + ψ_{x}v = 0
0 = 0
−ψ_{u}u − ξ_{t} + τ_{t}v + ψ − ξ_{x}v + τ_{x}uv = 0.

5) The solution is

ξ(x, t) = c_{1} + c_{3}t + (c_{2} + c_{8}t)x + c_{5}x^{2}
τ(x, t) = c_{6} + (c_{7} + c_{8}t)t + (c_{4} + c_{5}t)x
ϕ(x, t, u, v) = uα(x, t, v)
ψ(x, t, u, v) = c_{3}+c_{2}v+2c_{5}xv+c_{8}(tv+x)−v(c_{7}+2c_{8}t+c_{5}x)−v^{2}(c_{4}+c_{5}t),

where α is a function which depends on x, t and v and satisfies the equation

(42)  \frac{c_{8} + c_{5}v + α_{t} + vα_{x}}{v} = 0.

The eight constants c_{1} − c_{8} generate eight linearly independent infinitesimal generators of one-parameter projectable symmetry groups, while α(x, t, v) generates an infinite-dimensional group. From (42) we see that α(x, t, v) must depend on constants c_{5} and c_{8}. It is also clear that the function α is not uniquely determined. Hence, in order to calculate the infinitesimal generators of the projectable symmetry groups we choose one possibility for α:

α(x, t, v) = −c_{5}x − c_{8}t + β(v).

Now we can write all infinitesimal generators:

w_{1} = ∂_{x}, w_{2} = x∂_{x} + v∂_{v}, w_{3} = t∂_{x} + ∂_{v}, w_{4} = x∂_{t} − v^{2}∂_{v},

w_{5} = x^{2}∂_{x} + xt∂_{t} − xu∂_{u} + (xv − tv^{2})∂_{v}, w_{6} = ∂_{t}, w_{7} = t∂_{t} − v∂_{v},

w_{8} = xt∂_{x} + t^{2}∂_{t} − tu∂_{u} + (x − tv)∂_{v}, w_{β} = uβ(v)∂_{u}.

6) The one-parameter transformation groups generated by the vector fields w_{1} − w_{8} and w_{β} are:

(43)  \begin{aligned}G_{1} : (x, t, u, v) & → (x + η, t, u, v) \\ G_{2} : (x, t, u, v) & → (e^{η}x, t, u, e^{η}v) \\ G_{3} : (x, t, u, v) & → (x + ηt, t, u, v + η) \\ G_{4} : (x, t, u, v) & → (x, t + ηx, u, \frac{v}{1 + ηv} ) \\ G_{5} : (x, t, u, v) & → (\frac{x}{1 − ηx}, \frac{t}{1 − ηx}, (1 − ηx)u, \frac{v}{1 − η(x − tv)}) \\ G_{6} : (x, t, u, v) & → (x, t + η, u, v) \\ G_{7} : (x, t, u, v) & → (x, e^{η}t, u, e^{−η}v) \\ G_{8} : (x, t, u, v) & → (\frac{x}{1 − ηt}, \frac{t}{1 − ηt}, (1 − ηt)u, ηx + (1 − ηt)v) \\ G_{β} : (x, t, u, v) & → (x, t, e^{ηβ(v)}u, v) \end{aligned}

Since each of the groups in (43) is a symmetry group of the system (30), from (1) it follows that if u and v are solutions so are the functions

(1)  ũ : (x, t) → u(x − η, t),  ṽ: (x, t) → v(x − η, t)
(2)  ũ : (x, t) → u(e^{−η}x, t),  ṽ: (x, t) → e^{η}v(e^{−η}x, t)
(3)  ũ : (x, t) → u(x − ηt, t),  ṽ: (x, t) → v(x − ηt, t) + η
(4)  ũ : (x, t) → u(x, t − ηx),  ṽ: (x, t) → \frac{v(x, t − ηx)}{1 + ηv(x, t − ηx)}

(5)  ũ : (x, t) → \frac{u (\frac{x}{1+ηx}, \frac{t}{1+ηx})}{1 + ηx},  ṽ: (x, t) → \frac{(1 + ηx)v (\frac{x}{1+ηx}, \frac{t}{1+ηx})}{1 + ηtv (\frac{x}{1+ηx}, \frac{t}{1+ηx})}
(6)  ũ : (x, t) → u(x, t − η),  ṽ: (x, t) → v(x, t − η)
(7)  ũ : (x, t) → u(x, e^{−η}t),  ṽ: (x, t) → e^{−η}v(x, e^{−η}t)

(8)  ũ : (x, t) → \frac{u (\frac{x}{1+ηt}, \frac{t}{1+ηt})}{1 + ηt},  ṽ: (x, t) → \frac{ηx + v (\frac{x}{1+ηt}, \frac{t}{1+ηt})}{1 + ηt}
(β)  ũ : (x, t) → e^{ηβ(v)}u(x, t),  ṽ: (x, t) → v(x, t)

From the remark given after Theorem 2.4 it follows that all calculated symmetry groups are also symmetry groups of the system (29): for infinitesimal generators w_{5}, w_{8} and w_{β} the coefficients of ∂_{u} are \frac{1}{1+ηx}u, \frac{1}{1+ηt}u and e^{ηβ(v)}u respectively, hence they vanish when u = 0, while for the rest ϕ = 0.

The matrix factorizations of (27) with respect to symmetry groups in (43) are:

G_{1}, G_{6} :  [Δ̃_{1} ; Δ̃_{2}] = [\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}] · [\begin{matrix}Δ_{1} \\ Δ_{2} \end{matrix}]

G_{2} :  [Δ̃_{1} ; Δ̃_{2}] = [\begin{matrix}1 & 0 \\ 0 & e^{η} \end{matrix}] · [\begin{matrix}Δ_{1} \\ Δ_{2} \end{matrix}]

G_{3} :  [Δ̃_{1} ; Δ̃_{2}] = [\begin{matrix}1 & 0 \\ η & 1 \end{matrix}] · [\begin{matrix}Δ_{1} \\ Δ_{2} \end{matrix}]

G_{4} :  [Δ̃_{1} ; Δ̃_{2}] = [\begin{matrix}\frac{1+2ηv}{(1+ηv)^{2}} & \frac{−η}{(1+ηv)^{2}} \\ \frac{2ηv^{2}}{(1+ηv)^{3}} & \frac{1−ηv}{(1+ηv)^{3}} \end{matrix} ] · [\begin{matrix}Δ_{1} \\ Δ_{2} \end{matrix}]

G_{5} :  [Δ̃_{1} ; Δ̃_{2}] = [\begin{matrix}\frac{1+2ηtv}{(1+ηx)^{2}(1+ηtv)} & \frac{−ηt}{(1+ηx)^{2}(1+ηtv)} \\ \frac{2ηtv^{2}}{(1+ηx)(1+ηtv)^{3}} & \frac{1−ηtv}{(1+ηx)(1+ηtv)^{3}} \end{matrix}] · [\begin{matrix}Δ_{1} \\ Δ_{2} \end{matrix}]

G_{7} :  [Δ̃_{1} ; Δ̃_{2}] = [\begin{matrix}e^{−η} & 0 \\ 0 & e^{−2η} \end{matrix}] · [\begin{matrix}Δ_{1} \\ Δ_{2} \end{matrix}]

G_{8} :  [Δ̃_{1} ; Δ̃_{2}] = [\frac{1}{(1+ηt)^{3}}  0_{\frac{ηx}{(1+ηt)^{4}}  \frac{1}{(1+ηt)^{4}}} ] · [\begin{matrix}Δ_{1} \\ Δ_{2} \end{matrix}]

G_{β} :  [Δ̃_{1} ; Δ̃_{2}] = [\begin{matrix}(1 − ηvβ′(v))e^{ηβ(v)} & ηβ′(v)e^{ηβ(v)} \\ −ηuv^{2}β′(v)e^{ηβ(v)} & (1 + ηuvβ′(v))e^{ηβ(v)} \end{matrix}] · [\begin{matrix}Δ_{1} \\ Δ_{2} \end{matrix}] .

Therefore, the matrix of factorization Q depends only on x, t and η for all groups except for G_{4}, G_{5} and G_{β}. In these three cases the factor Q depends also on v. Beside that, from the transformed solutions we see that the groups G_{1}–G_{3} and G_{6}–G_{8} are slowly increasing, uniformly for x and t in compact sets. Thereby we have proved the next

Theorem 2.5. The symmetry groups G_{1}, G_{2}, G_{3}, G_{6}, G_{7} and G_{8} of the system (27) are 1-strongly associated symmetry groups, i.e., transform 1-strongly associated solutions to (27) to other 1-strongly associated solutions.

The remaining three groups G_{4}, G_{5} and G_{β} are not AS_{Δ}^{1}-groups for two reasons: first the condition that the map (u, v) ↦ Φ_{g}(x, t, u, v) is slowly increasing, uniformly for x and t in compact sets, does not hold globally. Second, the solution (31) does not belong to the algebra G_{∞}, which is necessary by Theorem 1.5.

Under certain assumptions on the solution (u, v) defined in (31), and also on the parameter η, these problems can be avoided. Namely, if we assume that the function v is nonnegative and η ⩾ 0 (then instead of a group we consider a semigroup) the symmetry groups G_{4} and G_{5} become slowly increasing, while for G_{8} it should be supposed that β(v) is a function of L^{∞}-log-type. The second condition from Theorem 1.5 (i) would be fulfilled if we assume that the solution (u, v) belongs to the algebra G_{∞}.

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Poljoprivredni fakultet (Received 05 07 2004) Trg Dositeja Obradovića 8 (Revised 06 01 2005) 21000 Novi Sad Serbia and Montenegro

kinjoki@neobee.net