The Diffiety Institute Preprint Series

Preprint DIPS–03 March 21, 2002

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

The Diffiety Institute

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

Conservation laws for multidimensional systems and related linear algebra problems

by

Sergei Igonin

Conservation laws for multidimensional systems and related linear algebra problems

SERGEI IGONIN

Abstract. We consider multidimensional systems of PDEs of generalized evolution form with t-derivatives of arbitrary order on the left-hand side and with the right-hand side dependent on lower order t-derivatives and arbitrary space derivatives. For such systems we find an explicit necessary condition for existence of higher conservation laws in terms of the system’s symbol. For systems that violate this condition we give an effective upper bound on the order of conservation laws. Using this result, we completely describe conservation laws for viscous transonic equations, for the Brusselator model, and the Belousov-Zhabotinskii system. To achieve this, we determine the conditions for a quadratic matrix A with entries from an arbitrary field to be similar (conjugate) to its transpose A^{t} or to the matrix −A^{t}, which is of independent interest.

Mathematics Subject Classification (2000): Primary 37K05, 37K10; Secondary 15A24, 76H05, 35K57.

1. Introduction

It is well known that conservation laws are of fundamental importance for clarifying the structure of PDEs. In particular, a common feature of soliton equations is to have conservation laws of arbitrarily high order. Existence of higher order conservation laws imposes very strong conditions on a system of PDEs. Explicit formulation of these conditions would help to classify integrable systems of a given type.

The straightforward study of the conserved current condition is hampered by the fact that one is interested in equivalence classes of conserved currents modulo trivial ones. Therefore, it is convenient to switch from a conserved current to its characteristic, which is the same for equivalent currents and satisfies the equation adjoint to the linearization of the initial system [1, 2, 8, 12].

Thus a part of the problem is to determine conditions for the adjoint linearized equation to have higher order solutions χ. In the present article we perform the first natural step in this direction. For determined, possibly multidimensional, systems of PDEs we find the conditions imposed on the symbol of the system by the fact that some higher order vector-functions satisfy the adjoint linearized equation modulo lower order terms.

Key words and phrases. Multidimensional systems, conservation laws, adjoint equation, characteristic polynomial, invariant factors, viscous transonic equations, Brusselator.

Revised version, minor mistakes corrected.

More precisely, we consider systems of generalized evolution form

\frac{∂^{h}u^{i}}{∂t^{h}} = F^{i}(x_{j}, t, \frac{∂^{s}u^{a}}{∂t^{s}} , \frac{∂^{i_{1}+···+i_{n}}u^{b}}{∂x_{1}^{i_{1}} … ∂x_{n}^{i_{n}}}), (1)

i, a, b = 1, …, m, j = 1, …, n, s = 0, …, h − 1, i_{1} + ⋯ + i_{n} ≤ N,

with t-derivatives of fixed order h > 0 on the left-hand side and with the right-hand side dependent on lower order t-derivatives and arbitrary space derivatives.

To any m-component vector-function χ of the variables x_{j}, t, u^{i} and their derivatives we associate its symbol with respect to the space variables x_{j}, which is an m×m matrix, whose entries are homogeneous polynomials in n variables of degree equal to the order o(χ) of χ with respect to x_{j}.

Let A be the symbol of the right-hand side of (1) and S be the symbol of the characteristic χ of a conservation law for (1). It turns out that if o(χ) > O, where O ≤ N is some constant associated to (1), then the adjoint linearized equation implies the matrix equation

(2)  SA = (−1)^{N+h}A^{t}S.

Here and below A^{t} is the transpose of A.

A linear algebra problem arises naturally: for what matrices A does there exist a nonzero matrix S such that (2) holds? In addition, since for known integrable systems there are normally higher conservation laws with nonsingular S, one is also interested for which A the matrix S can be taken nonsingular, i.e., when the matrices A and ±A^{t} are similar (conjugate).

In solving these problems there is a difference for the cases n = 1 and n > 1. If n = 1, one can switch from homogeneous polynomial in one variable matrices A and S to the corresponding matrices of coefficients and, allowing the coefficients to be complex, make use of the Jordan normal form [3]. While if n > 1 then the entries of the matrices belong to the field of rational functions in several variables, which is essentially not algebraicly closed, hence the Jordan normal form is not generally applicable. Using more sophisticated algebraic technique, we prove the following effective criteria.

Theorem. For any m × m matrix A with the entries from an arbitrary field F and the characteristic polynomial d(λ) ∈ F [λ] we have the following.

(1) The matrices A and A^{t} are always similar.

(2) A nonzero m × m matrix S such that SA = −A^{t}S exists if and only if the polynomials d(λ) and d(−λ) have a common divisor of positive degree.

(3) The matrices A and −A^{t} are similar if and only if all the invariant factors d_{i}(λ) of A (certain divisors of the characteristic polynomial [4]) satisfy d_{i}(−λ) = (−1)^{deg d_{i}(λ)}d_{i}(λ). In particular, d(λ) = (−1)^{deg d(λ)}d(λ), which in the case char F ≠ 2 implies tr A = 0 and, if m is odd, det A = 0.

Statements 2 and 3 of the theorem give a necessary condition for existence of higher conservation laws for systems (1) with odd N + h. In particular, a scalar equation (m = 1) of the form (1) with odd N + h can not have conservation laws of order greater than O. For different ways to write system (1) in the generalized evolution form the symbols A and the resulting conditions are generally different. In order to have higher conservation laws a system of PDEs must satisfy all conditions obtained from various ways to write it in the generalized evolution form.

Let us discuss the previous research on this theme. It seems that only evolution systems (h = 1) in one space variable (n = 1) were studied in this respect. For such systems equation (2) was obtained by a similar technique in [6] and rediscovered in [3]. In [6] it is noticed that SA = −A^{t}S for nonsingular S implies det(A + λI) = (−1)^{m} det(A − λI), which is a weaker version of our Statement 3. Here and below I is the unity matrix. In [3] Statement 2 is proved for complex matrices, and the corresponding necessary condition for existence of conservation laws of order greater than the order of the evolution system is formulated. Even in this simplest case our result is stronger, since the upper bound O is normally much smaller than the order of the system (see the examples in Section 7).

The paper is organized as follows. In Section 2 the method of characteristics of conservation laws is recalled. We specify the method for systems of generalized evolution form in Section 3 and derive equation (2) in Section 4. The above theorem on quadratic matrices is proved in Section 5. In Section 6 we explicitly formulate the obtained necessary conditions for existence of higher conservation laws. Finally, Section 7 contains some mathematical physics equations of the form (1), which violate these conditions and, therefore, do not have conservation laws of order greater than O. This result allows us to describe all conservation laws for two basic equations in the theory of viscous transonic gas flows (see, for example, [5, 7, 9, 11] and references therein) and for two popular reaction-diffusion systems: the Brusselator model and the Belousov-Zhabotinskii system [10, Section 15.4].

2. Characteristics of conservation laws

This is a brief review of the method of characteristic for computation of conservation laws. We refer to [1, 2, 8, 12] for more details.

Consider a system E of differential equations

(3)  F_{s}(y_{i}, v^{j}, …, v_{I}^{k}, …) = 0, s = 1, …, p,

with independent variables y_{1}, …, y_{a}, unknown functions v^{1}, …, v^{b}, and

v_{I}^{k} = \frac{∂^{|I|}v^{k}}{∂y_{1}^{i_{1}} … ∂y_{a}^{i_{a}}} , I = (i_{1}, …, i_{a}) ∈ ℤ_{+}^{a},

being their derivatives. Here and below ℤ_{+} is the set of nonnegative integers and |I| = i_{1} + ⋯ + i_{a}.

Let F be the algebra of smooth functions of the variables y_{i}, u^{j}, and u_{I}^{j}. Although the whole set of the variables is infinite, each function is supposed to depend only on a finite subset. Denote by F_{E} the quotient algebra with respect to the ideal I generated by the left-hand sides of equations (3) and their differential consequences D_{y_{i_{1}}} … D_{y_{i_{k}}} (F_{s}) ∈ F. Here

D_{y_{i}} = \frac{∂}{∂y_{i}} + ∑_{j,I} u_{I+1_{i}}^{j} \frac{∂}{∂u_{I}^{j}}

is the total derivative with respect to y_{i}, where 1_{i} is the multi-index with 1 at the i-th place, the other indices of 1_{i} being zero. For two equivalent functions f_{1}, f_{2} ∈ F, f_{1} − f_{2} ∈ I, one has f_{1}(y_{i}, v^{j}(y_{i})) = f_{2}(y_{i}, v^{j}(y_{i})) for any local solution v^{j}(y_{i}) to (3).

By definition, the ideal I is invariant under the action of D_{y_{i}}, which, therefore, defines a derivation D̄_{y_{i}} of F_{E}. A conserved current for (3) is an a-tuple J = (J_{1}, …, J_{a}), where J_{k} ∈ F_{E}, that satisfies the equation

(4)  ∑_{i=1}^{a} D̄_{y_{i}}(J_{i}) = 0.

A conserved current is called trivial, if it has the form

J_{k} = ∑_{l<k} D̄_{l}(L_{lk}) − ∑_{k<l} D̄_{l}(L_{kl})

for some functions L_{kl} ∈ F_{E}, 1 ≤ k < l ≤ a. Two conserved currents are said to be equivalent if they differ by a trivial one. Conservation laws are defined to be the equivalent classes of conserved currents.

Let J̃_{k} ∈ F be such that J_{k} = J̃_{k} + I. Identity (4) means that

(5)  ∑_{i=1}^{a} D_{y_{i}}(J̃_{i}) = ∑_{s,I} g_{I}^{s}D_{y}^{I}(F_{s})

for some functions g_{I}^{s} ∈ F, only a finite number of which is nonzero. Here and in what follows for each multi-index I = (i_{1}, …, i_{a}) we denote D_{y}^{I} = D_{y_{1}}^{i_{1}} … D_{y_{a}}^{i_{a}}. Consider the functions

(6)  χ̃_{s} = ∑_{I}(−1)^{|I|}D_{y}^{I}(g_{I}^{s}), s = 1, …, p.

Generally speaking, representation (5) and functions χ̃_{s} are not uniquely defined by the conserved current J. Assume that system (3) is non-overdetermined and nondegenerated [1, 8], then the corresponding elements χ_{s} = χ̃_{s} + I of F_{E} are well-defined by J and are all zero if and only if J is trivial. The p-tuple χ = (χ_{1}, …, χ_{p}) is the same for equivalent conserved currents and is called the characteristic (or generating function [1, 12]) of the corresponding conservation law.

In addition, χ satisfies the adjoint linearized equation

(7)  K(χ) = 0,

where K is the b × p matrix differential operator with the entries

(8)  [K]_{ij} = ∑_{I}(−1)^{|I|}D̄_{y}^{I} ∘ \frac{∂F_{j}}{∂v_{I}^{i}} .

The homological interpretation of these concepts can be found in [1, 2, 12].

3. Formulas for systems of generalized evolution form

Consider a system E of m partial differential equations in n+1 independent variables t, x_{1}, …, x_{n} and m unknown functions u^{1}, …, u^{m} of the form

(9)  \frac{∂^{h}u^{i}}{∂t^{h}} = F^{i}, i = 1, …, m,

where F^{i} are smooth functions of the variables x_{j}, t, u^{a} and the following derivatives

(10)  \frac{∂^{i_{1}+···+i_{n}}u^{b}}{∂x_{1}^{i_{1}} … ∂x_{n}^{i_{n}}} , i_{1}, …, i_{n} ∈ ℤ_{+}, \frac{∂^{s}u^{a}}{∂t^{s}} , s < h.

For each multi-index I = (i_{1}, …, i_{n}) ∈ ℤ_{+}^{n} and integer s ≥ 0 we denote

(11)  u_{s,I}^{i} = \frac{∂^{|I|+s}u^{i}}{∂t^{s}∂x_{1}^{i_{1}} … ∂x_{n}^{i_{n}}} .

Let us describe the algebra F_{E} for this system. By induction on s, it follows from (9) that the derivatives u_{s,I}^{i} with s ≥ h are expressed in terms of

(12)  t, x_{j}, u_{s,I}^{i}, s < h.

Therefore, for each function of t, x_{j}, u^{i} and arbitrary derivatives (11) there is a unique equivalent modulo (9) function of variables (12). Thus we can identify F_{E} with the algebra of smooth functions of variables (12). Below all functions are supposed to be from F_{E}.

The restrictions D̄_{x_{i}}, D̄_{t}: F_{E} → F_{E} of the total derivatives are written in coordinates (12) as follows

D̄_{x_{i}} = \frac{∂}{∂x_{i}} + ∑_{a,I,s} u_{s,I+1_{i}}^{a} \frac{∂}{∂u_{s,I}^{a}} ,

D̄_{t} = \frac{∂}{∂t} + ∑_{s<h−1} u_{s+1,I}^{a} \frac{∂}{∂u_{s,I}^{a}} + ∑ D_{x}^{I}(F^{a})\frac{∂}{∂u_{h−1,I}^{a}} . (13)

The equation D̄_{t}J_{0}+∑_{i=1}^{n} D̄_{x_{i}}J_{i} = 0 for a conserved current J = (J_{0}, J_{1}, …, J_{n}) implies the identity

D_{t}J_{0} + ∑_{i=1}^{n} D_{x_{i}}J_{i} = ∑_{i,I} \frac{∂J_{0}}{∂u_{h−1,I}^{i}} D_{x}^{I}(u_{h,0}^{i} − F^{i}),

which is the specification of (5) for system (9). According to general formula (6), the characteristic χ = (χ_{1}, …, χ_{m}) is computed as follows

(14)  χ_{i} = ∑_{I}(−1)^{|I|}D̄_{x}^{I}(\frac{∂J_{0}}{∂u_{h−1,I}^{i}} ).

From (7) and (8) we see that the characteristic regarded as a column vector satisfies the equation

(15)  (−)^{h}D̄_{t}^{h}(χ) = L(χ),

where L is the m × m matrix differential operator with the entries

(16)  [L]_{ij} = ∑_{I}(−1)^{|I|}D̄_{x}^{I} ∘ \frac{∂F^{j}}{∂u_{0,I}^{i}} + ∑_{s=0}^{h−1}(−1)^{s}D̄_{t}^{s} ∘ \frac{∂F^{j}}{∂u_{s,0}^{i}} .

4. Solving the adjoint equation for the highest order terms

For a (vector-)function f the maximal integer k such that ∂f/∂u_{s,I}^{i} ≠ 0 for some 0 ≤ s < h, 1 ≤ i ≤ m, |I| = k is called the order of f and denoted by o(f). If ∂f/∂u_{s,I}^{i} = 0 for all s, i, I then we set o(f) = −1. The maximal integer s < h such that ∂f/∂u_{s,I}^{i} ≠ 0 for some 1 ≤ i ≤ m, |I| = o(f) is denoted by t(f). The order of the characteristic of a conservation law for (9) is called the order of the conservation law.

Consider the ring F[q_{1}, …, q_{n}] of polynomials in n variables with F as the ring of coefficients. For each multi-index I = (i_{1}, …, i_{n}) denote by q^{I} the monomial q_{1}^{i_{1}} … q_{n}^{i_{n}} ∈ F[q_{1}, …, q_{n}]. For any k-component vector-function χ = (χ_{1}, …, χ_{k}) and two integers a, b ≥ 0 let S_{χ}^{a,b} be the k × m matrix with the entries

[S_{χ}^{a,b}]_{ij} = ∑_{|I|=a} \frac{∂χ_{i}}{∂u_{b,I}^{j}} q^{I} ∈ F[q_{1}, …, q_{n}].

We call the nonzero matrix S_{χ}^{o(χ),t(χ)} the symbol of χ and denote it by S_{χ}.

Let A be the symbol of the right-hand side (F^{1}, …, F^{m}) of (9) and set N = o(F^{1}, …, F^{m}) > 0. By assumption (10), one has t(F^{1}, …, F^{m}) = 0. Therefore, by definition,

(17)  [A]_{ij} = ∑_{|I|=N} \frac{∂F^{i}}{∂u_{0,I}^{j}} q^{I}.

Applying the Leibniz rule, differential operators (16) can be uniquely rewritten in the usual form

[L]_{ij} = ∑_{|I|≤N} f_{I}^{ij}D̄_{x}^{I} + ∑_{s<h} g_{s}^{ij}D̄_{t}^{s}.

In particular, from definition (16) one has

(18)  f_{I}^{ij} = (−1)^{N} \frac{∂F^{j}}{∂u_{0,I}^{i}} ∀ I : |I| = N.

We set

(19)  O = max_{i,j,I,s}{−1, o(f_{I}^{ij}) − N, o(g_{s}^{ij}) − N}.

From definition (16) it follows that O ≤ N.

Theorem 1. Let χ be the characteristic of a conservation law for (9). If o(χ) > O then we have

(20)  S_{χ}A = (−1)^{N+h}A^{t}S_{χ}.

Proof. Set a = N + o(χ), b = t(χ). Equation (15) implies, in particular,

(21)  (−1)^{h}S_{D̄_{t}^{h}(χ)}^{a,b} = S_{L(χ)}^{a,b}.

By formula (13) and assumption (10), for any vector-function χ with o(χ) ≥ 0 the vector-function D̄_{t}^{h}(χ) does not depend on the coordinates u_{s,I}^{j} with |I| > a or |I| = a, s > b. Moreover, by formula (17), it is easily seen that

(22)  S_{D̄_{t}^{h}(χ)}^{a,b} = S_{χ}A.

In the case o(χ) > O only the part ∑_{|I|=N} f_{I}^{ij}D̄_{x}^{I} of L contributes to S_{L(χ)}^{a,b}, since D̄_{t}^{s}(χ) for s < h does not depend on u_{p,I}^{j} with |I| = a, p ≥ b and the number a is greater than the order of any coefficient f_{I}^{ij} or g_{s}^{ij}. Therefore, taking into account formulas (18) and (17), we obtain

S_{L(χ)}^{a,b} = (−1)^{N}A^{t}S_{χ}.

Combining this with (22) and (21), one gets (20). □

In the next section we study the conditions imposed on A by equation (20).

5. Linear algebra problems

For a ring R we denote by M_{k}(R) the ring of k × k matrices with entries from R. Consider an arbitrary field F and denote by F̃ its algebraic closure, i.e., the minimal algebraicly closed extention of F.

Theorem 2. Let A ∈ M_{k}(F) and let d(λ) = det(A−λI) be the characteristic polynomial of A. A nonzero matrix S ∈ M_{k}(F) such that

(23)  SA = −A^{t}S

exists if and only if the polynomials d(λ) and d(−λ) have a common divisor of positive degree. Equivalently, there are two roots λ_{1}, λ_{2} ∈ F̃ of d(λ) such that λ_{1} + λ_{2} = 0.

Proof. The polynomials d(λ) and d(−λ) have a common divisor of positive degree if and only if they have a common root λ in F̃, i.e., both λ and −λ are roots of d(λ).

If (23) holds then for any similar to A matrix A′ = CAC^{−1}, C ∈ M_{k}(F), one has S′A′ = −A′^{t}S′ with S′ = C^{−1}^{t}SC^{−1}. Let us regard A as a matrix from M_{k}(F̃) ⊃ M_{k}(F). Then we can assume A to be in the Jordan normal form. For such A one can easily show that the linear map

M_{k}(F̃) → M_{k}(F̃), S ↦ SA + A^{t}S,

has a nontrivial kernel if and only if there are two eigenvalues λ_{1}, λ_{2} ∈ F̃ of A such that λ_{1} + λ_{2} = 0 (see [3] for F = ℂ).

It remains to prove that if (23) holds for some S ∈ M_{k}(F̃) then there is nonzero S′ ∈ M_{k}(F) such that S′A = −A^{t}S′. Consider a (possibly infinite) basis {a_{i}} of F̃ regarded as a vector space over F. One has S = ∑_{i} a_{i}S_{i}, where S_{i} ∈ M_{k}(F). Since A ∈ M_{k}(F), equation (23) implies S_{i}A = −A^{t}S_{i}. □

Recall some criteria for two matrices A, B ∈ M_{k}(F) to be similar (see, for example, [4, Chapter 13]). Consider the ring F [λ] of polynomials in one variable. A matrix C ∈ M_{k}(F [λ]) is said to be unimodular if det C is nonzero and belongs to F. For any matrix A ∈ M_{k}(F) the matrix A − λI ∈ M_{k}(F [λ]) admits a canonical decomposition

(24)  A − λI = C_{1}DC_{2}, C_{1}, D, C_{2} ∈ M_{k}(F [λ]),

such that C_{1}, C_{2} are unimodular, while D is diagonal. Moreover, the polynomials d_{i} = [D]_{ii} have the leading coefficient 1, and d_{i+1} is divisible by d_{i} for each i = 1, …, k − 1. Then the k polynomials d_{i} are defined uniquely by A and are called the invariant factors of A. Note that there is a simple procedure to compute the invariant factors [4, Chapter Proposition 1 ([4, Chapter 13]). Two matrices A, B ∈ M_{k}(F) are similar if and only if they have the same invariant factors.

We call a polynomial d(λ) = ∑_{i} a_{i}λ^{i} ∈ F [λ] skew if d(−λ) = (−1)^{deg d(λ)}d(λ), i.e., for all i ≡ deg d(λ) + 1 mod 2 one has a_{i} = 0.

Theorem 3. For any A ∈ M_{k}(F) we have the following.

(1) The matrices A and A^{t} are similar.

(2) The matrices A and −A^{t} are similar if and only if each invariant factor of A is skew. In this case the characteristic polynomial is also skew. In particular, in the case char F ≠ 2 we have tr A = 0 and, if k is odd, det A = 0.

Remark. Note that in the case char F = 2 the second statement of this theorem as well as Theorem 2 are trivial.

Proof. Consider canonical decomposition (24) for A. Taking the transpose, we obtain

(25)  A^{t} − λI = C_{2}^{t}DC_{1}^{t},

which is a canonical decomposition for A^{t}, since C_{1}^{t}, C_{2}^{t} are clearly unimodular. Therefore, the invariant factors of A^{t} are the same, which, by Proposition 1, implies that A and A^{t} are similar.

Multiplying (25) by −1 and substituting −λ in place of λ, we obtain

(26)  −A^{t} − λI = −C_{2}^{t}(−λ)D(−λ)C_{1}^{t}(−λ).

Denote C′_{1} = −C_{2}^{t}(−λ), C′_{2} = T C_{1}^{t}(−λ), and D′ = D(−λ)T, where T ∈ M_{k}(F) is the diagonal matrix with the entries [T]_{ii} = (−1)^{deg[D]_{ii}}. From (26) we obtain the canonical decomposition −A^{t} − λI = C′_{1}D′C′_{2} for −A^{t}. According to Proposition 1, A and −A^{t} are similar if and only if D′ = D, which says that all the invariant factors [D]_{ii} of A are skew. In this case the characteristic polynomial is also skew, since from (24) it is evidently equal to the product of the invariant factors multiplied by (−1)^{k}. □

6. Necessary conditions for existence of higher conservation laws

According to Theorem 1, a necessary condition for existence of conservation laws for (9) of order greater than O is that there is a nonzero m × m matrix S_{χ} with entries from F[q_{1}, …, q_{n}] such that (20) holds. Let us treat A and S_{χ} as matrices with entries from the field F of rational functions in n variables q_{1}, …, q_{n}. Then Theorem 2 implies the following.

Theorem 4. If N+h is odd and system (9) possesses a conservation law of order greater than O, then the characteristic polynomial d(λ) = det(A−λI) and the polynomial d(−λ) have a common divisor of positive degree. Equivalently, there are eigenvalues λ_{1}, λ_{2} of A (possibly in the algebraic closure of F) such that λ_{1} + λ_{2} = 0.

Remark. Evidently, introducing the new dependent variables

u^{i,s} = \frac{∂^{s}u^{i}}{∂t^{s}} , i = 1, …, m, s = 0, …, h − 1,

we can rewrite (9) in the usual evolution form. But if h > 1 then the symbol of the right-hand side of the obtained evolution system has zero determinant and, therefore, automatically satisfies the condition in Theorem 4, even if the initial system does not meet this condition. Therefore, it is essential to consider the generalized evolution form.

Analyzing examples of known soliton equations, we can conjecture that for (9) to be integrable there must exist higher order conservation laws with nonsingular matrix S_{χ}. Therefore, it is worth formulating a necessary condition for existence of such nonsingular conservation laws. According to Theorem 3, we obtain the following criterion.

Theorem 5. If (9) has a nonsingular conservation law of order greater than O then all the invariant factors of the matrix A are skew. In this case its characteristic polynomial is also skew. In particular, tr A = 0 and, if m is odd, det A = 0.

If m = 1 then A is a nonzero 1 × 1 matrix, and Theorem 4 implies the following.

Theorem 6. A scalar equation (m = 1) of the form (9) with odd N + h can not have conservation laws of order greater than O.

Remark. To obtain stronger conditions, it is sometimes useful to write a system of PDEs in several ways in the form (9). For example, for a scalar evolution equation u_{t} = u_{xxx} + u_{yy} + f(u, u_{x}, u_{y}, u_{xx}) the condition is empty, since the sum of h = 1 and N = 3 is even. But rewriting the equation in the generalized evolution form with respect to y as follows u_{yy} = −u_{xxx} + u_{t} − f(u, u_{x}, u_{y}, u_{xx}), we see, according to Theorem 6, that there are no higher conservation laws.

7. Examples

7.1. The viscous transonic equation. The nonlinear viscous transonic equation

(27)  u_{tt} = −u_{xxx} + u_{x}u_{xx} − \frac{α}{t}u_{t}

describes the asymptotic form of a gas flow in the sonic region (see [7, 9] and references therein). The following conserved currents for (27) were found in [7]

(28)  (u_{t}t^{α}, u_{xx}t^{α} − \frac{u_{x}^{2}}{2} t^{α}), (u_{t}t + (α − 1)u, u_{xx}t − \frac{u_{x}^{2}}{2} t).

All other conserved currents mentioned in [7] are trivial.

Let us show that (28) span the whole space of conservation laws for (27). We have

L = D̄_{x}^{3} + D̄_{x}^{2} ∘ u_{x} − D̄_{x} ∘ u_{xx} + D̄_{t} ∘ \frac{α}{t} = D̄_{x}^{3} + u_{x}D̄_{x}^{2} + u_{xx}D̄_{x} + \frac{α}{t}D̄_{t} − \frac{α}{t^{2}} .

According to (19), one has O = −1. By Theorem 6, since N + h = 5 is odd, the characteristic χ of any conservation law is a function of x, t only. Equation (15) reads

χ_{tt} = χ_{xxx} + u_{x}χ_{xx} + u_{xx}χ_{x} + \frac{α}{t}χ_{t} − \frac{α}{t^{2}} χ.

This implies χ_{x} = 0 and

(29)  χ_{tt} = \frac{α}{t}χ_{t} − \frac{α}{t^{2}} χ.

The characteristics t^{α}, t of conserved currents (28) span the space of solutions to (29). Therefore, (28) span the space of conservation laws.

7.2. Another equation for viscous transonic flows. The equation

(30)  v_{yy} = 2v_{xt} + v_{x}v_{xx} − v_{zz} − μv_{xxx},

where μ is a nonzero real constant, models nonstationary transonic flows around a thin body with effects of viscosity and heat conductivity when the velocity of the gas is close to the local speed of sound, see [5, 11] and references therein.

For this equation we obtain

L = μD̄_{x}^{3} + 2D̄_{x}D̄_{t} − D̄_{z}^{2} + v_{x}D̄_{x}^{2} + v_{xx}D̄_{x}.

By definition (19), we have O = −1. According to Theorem 6, since N + h = 5 is odd, the characteristic χ of any conservation law is a function of x, y, z, t only. Equation (15) implies

(31)  χ_{x} = 0, χ_{yy} + χ_{zz} = 0.

Each function χ satisfying (31) is indeed the characteristic of the conserved current

(32)  D̄_{y}(χv_{y} − χ_{y}v) + D̄_{z}(χv_{z} − χ_{z}v) + D̄_{x}(μχv_{xx} − \frac{1}{2}χv_{x}^{2} − 2χv_{t}) = 0.

Therefore, any conserved current for (30) is equivalent to a conserved current of the form (32).

7.3. The Brusselator model. The Brusselator model governing certain chemical reactions is the following multidimensional system

(33)  \begin{aligned}v_{t} & = ∑_{i=1}^{n} \frac{∂^{2}v}{∂x_{i}^{2}} + v^{2}w − (b + 1)v + a, \\ w_{t} & = c ∑_{i=1}^{n} \frac{∂^{2}w}{∂x_{i}^{2}} − v^{2}w + bv, \end{aligned}

where a, b, and c ≠ 0 are real parameters [10, Section 15.4]. By definition (16) we have

L = ( \begin{matrix}∑_{i} D̄_{x_{i}}^{2} + 2vw − (b + 1) & −2vw + b \\ v^{2} & c ∑_{i} D̄_{x_{i}}^{2} − v^{2} \end{matrix} ) .

The symbol A is diagonal with [A]_{11} = ∑_{i} q_{i}^{2} and [A]_{22} = c ∑_{i} q_{i}^{2}. By definition (19), we have O = −1. By Theorem 4, since [A]_{11} ≠ 0, [A]_{22} ≠ 0, and [A]_{11} + [A]_{22} ≠ 0, the characteristic χ = (χ_{1}, χ_{2}) of any conservation law is a function of x_{i}, t only. For such χ equation (15) reads

(34)  \begin{aligned}−\frac{∂χ_{1}}{∂t} & = ∑_{i} \frac{∂^{2}χ_{1}}{∂x_{i}^{2}} + (2vw − (b + 1))χ_{1} − (2vw − b)χ_{2}, \\ −\frac{∂χ_{2}}{∂t} & = v^{2}χ_{1} + c ∑_{i} \frac{∂^{2}χ_{2}}{∂x_{i}^{2}} − v^{2}χ_{2}. \end{aligned}

Evidently, (34) implies χ_{1} = χ_{2}. Then (34) becomes

(35)  \begin{aligned}−\frac{∂χ_{1}}{∂t} & = ∑_{i} \frac{∂^{2}χ_{1}}{∂x_{i}^{2}} − χ_{1}, \\ −\frac{∂χ_{1}}{∂t} & = c ∑_{i} \frac{∂^{2}χ_{1}}{∂x_{i}^{2}} . \end{aligned}

Clearly, for c = 1 the only solution to (35) is χ_{1} = 0, while for c ≠ 1 the general solution is

(36)  χ_{1} = χ_{2} = G(x_{i}) exp(\frac{ct}{c − 1}),

where G(x_{i}) is an arbitrary solution to the equation

(37)  G + (c − 1) ∑_{i} \frac{∂^{2}G}{∂x_{i}^{2}} = 0.

Each solution (36) is indeed the characteristic of the conserved current

D̄_{t}(G exp(\frac{ct}{c − 1})(v + w))+

+ ∑_{i=1}^{n} D̄_{x_{i}}(exp(\frac{ct}{c − 1})(\frac{∂G}{∂x_{i}} (v + cw) − G(v_{x_{i}} + cw_{x_{i}}) − \frac{a}{n} ∫ G dx_{i}))= 0.

Thus if c ≠ 1 then these conserved currents span the space of conservation laws; while if c = 1 then there are no nontrivial conservation laws for (33).

7.4. The Belousov-Zhabotinskii system. This system describes certain chemical reactions and reads [10, Section 15.4]

\begin{aligned}v_{t} & = ∑_{i} \frac{∂^{2}v}{∂x_{i}^{2}} + v(1 − v − rw) + Lrw, \\ w_{t} & = ∑_{i} \frac{∂^{2}w}{∂x_{i}^{2}} − bvw − Mw. \end{aligned}

Here r, L, b, M are real constants. Evidently, Theorem 4 implies that this system does not possess conservation laws of nonnegative order. Similarly to the above examples, analysis of equation (15) for characteristics of order −1 shows that in the nonlinear case b ≠ 0 there are no nontrivial conservation laws at all.

Acknowledgements

The author is grateful to Prof. I. S. Krasil′shchik for very stimulating attention to this work and many helpful remarks, to Prof. A. L. Onishchik for attracting author’s attention to the used algebraic technique, and to Prof. R. Martini and Prof. V. V. Sokolov for useful discussions.

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. Edited and with a preface by Krasil′shchik and Vinogradov. Amer. Math. Soc., Providence, RI, 1999.

[2] R. L. Bryant and P. A. Griffiths. Characteristic cohomology of differential systems. I. General theory. J. Amer. Math. Soc. 8 (1995), 507–596.

[3] K. Foltinek. Conservation laws of evolution equations: generic non-existence. J. Math. Anal. Appl. 235 (1999), 356–379.

[4] A. Kurosh. Higher algebra. Mir Publishers, Moscow, 1972.

[5] N. Larkin. On one problem of transonic gas dynamics. Mat. Contemp. 15 (1998), 169–186.

[6] A. V. Mikhailov, A. B. Shabat and R. I. Yamilov. A symmetric approach to the classification of nonlinear equations. Complete lists of integrable systems. Russian Math. Surveys 42 (1987), 1–63.

[7] J. Mukherjee and A. Roy Choudhury. On the Lie symmetry, conservation laws and Painlevé analysis for the viscous transonic equation. Ann. Physik (7) 46 (1989), 6–92.

[8] P. J. Olver. Applications of Lie Groups to Differential Equations. Springer-Verlag, New York, 1993.

[9] O. S. Ryzhov. Viscous transonic flows. Annual review of fluid mechanics, Vol. 10, 65–92. Annual Reviews, Palo Alto, Calif., 1978.

[10] M. E. Taylor. Partial differential equations. III. Nonlinear equations. Springer-Verlag, New York, 1996.

[11] P. A. Vel′misov, S. V. Fal′kovič. On the theory of transonic flows of a viscous gas. (Russian) Izv. Vysš. Učebn. Zaved. Matematika 1974, no. 5, 52–61.

[12] A. M. Vinogradov. The C-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory. II. The nonlinear theory. J. Math. Anal. Appl. 100 (1984), 1–129.

Independent University of Moscow; University of Twente, the Netherlands

E-mail address: igonin@mccme.ru