Simple Waves and Simple States inR2

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

214, 613]632 Ž1997.

AY975600

Simple Waves and Simple States in R 2 Jordan Tabov Institute of Mathematics, Bulgarian Academy of Sciences, Acad. G. Bonche¨ Str. Block 8, 1113, Sofia, Bulgaria Submitted by William F. Ames Received August 19, 1996

The so-called ‘‘solutions, constructed by means of Riemann invariants’’ for quasi-linear systems of PDEs, play an important role in gas dynamics, since they describe the propagation and the superposition of waves. The existing methods for finding solutions Žsimple waves and simple states . of this type usually deal with systems whose coefficients do not depend on the independent variables, and have a very limited application to non-homogeneous systems. In this paper a method for finding such solutions in the case of general quasi-linear systems of PDEs in R 2 is presented. This method is based on a detailed investigation of the respective overdetermined system and on a special construction, involving Pfaff systems. Q 1997 Academic Press

1. INTRODUCTION The systems of PDEs, describing the propagation and superposition of waves in gas dynamics, are of the form a ijk Ž x, u . ­ i u j s bk Ž x, u . ,

Ž 1.

where uŽ x . s Ž u1 Ž x ., u 2 Ž x ., . . . , u m Ž x .. is the unknown vectorial function of x s Ž x 1, x 2 , . . . , x n . g R n, a ijk Ž x, u. and bk Ž x, u. are given functions, and ­ i s ­r­ x i. The waves themselves correspond to systems of a special type and to their solutions, constructed by means of Riemann invariants, i.e., solutions of the form ui Ž x . s ¨ i Ž R Ž x . . ,

i s 1, 2, . . . , n.

First of all we should mention the classical case, when Ž1. has the form u t q AŽ u . u x s 0

Ž 2.

613 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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with 2 independent variables t and x where A is a n = n matrix. If A has n distinct eigenvalues, this system is called hyperbolic, and the respective eigenvectors are called characteristics. It has n solutions of the form ui Ž t , x . s ¨ i Ž R Ž t , x . . ,

i s 1, 2, . . . , n,

Ž 3.

where ¨ i are functions of a single variables and RŽ t, x . is a suitable function. These solutions are called simple waves; there exists a classical method, the so-called method of characteristics, which is usually applied for finding wave solutions of Ž2.; for more details see Jeffrey w1x and the monographs and the articles quoted there. The results mentioned above are extended to similar hyperbolic systems with more than 2 independent variables ŽBurnat w2x, Peradzynski w3x, and others.. The relations Ž3. imply the following DEFINITION ŽBurnat w2x.. We say that a solution uŽ x . of Ž1. is constructed by means of a Riemann invariant if it is of the form ui Ž x . s ¨ i Ž R Ž x . . ,

i s 1, 2, . . . , n,

where the ¨ i are functions of a single variable and RŽ x . s RŽ x 1 , x 2 , . . . , x n . is a suitable function. More general non-homogeneous quasilinear systems of the form a ikj Ž u . ­ j u i s bk Ž u .

Ž 4.

are treated by Grundland w4, 5, 7x. In this case the solutions, constructed by means of Riemann invariants, are called simple states; in certain cases they correspond to solitary waves Žsee Grundland w5x.. The results obtained by Grundland cover only the case when the respective Riemann invariant is a linear function. Homogeneous quasilinear systems of the form a ikj Ž x, u . ­ j u i s 0

Ž 5.

and their simple wave solutions are considered by P. Kucharczyk, Z. Peradzynski, and E. Zawistowska in the paper w6x from the point of view of the classical method of characteristics; however, the obtained results are related only with the case when in fact the coefficients a ikj in Ž5. do not depend on x. In order to find and to investigate the solutions, constructed by means of Riemann invariants Žsimple waves and simple states in the homogeneous and non-homogeneous case, respectively. for quasi-linear systems of the

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form Ž1., we need a method, more powerful than the method of the characteristics Žwhich is applicable only in the case of homogeneous hyperbolic, or at least non-elliptic, systems with coefficients not depending on x .. In this article we present such a method in the case of two independent variables and two unknown functions. The idea developed here can be used for extensions Žwhich are however more complicated. to more unknown functions. In particular, this method can be used for finding simple waves and simple states.

2. RIEMANN INVARIANTS FOR QUASILINEAR SYSTEMS OF PDES IN R 2 AND THE RESPECTIVE OVERDETERMINED SYSTEMS The main object, which we investigate in this article, is the system

­ 1 u1 s a11 Ž x, u . ­ 2 u1 q a12 Ž x, u . ­ 2 u 2 q b1 Ž x, u . ­ 1 u 2 s a12 Ž x, u . ­ 2 u1 q a22 Ž x, u . ­ 2 u 2 q b 2 Ž x, u .

Ž 6.

and its solutions, constructed by means of Riemann invariants, i.e., its solutions of the form u1 Ž x . s ¨ 1 Ž R Ž x . . ,

u2 Ž x . s ¨ 2 Ž R Ž x . . ,

Ž 7.

where ¨ 1 and ¨ 2 are functions of a single variable and R is a suitable function. In the case when x g R 2 and u g R 2 , which we consider here, this system generalizes both Ž4. and Ž5.. LEMMA 1 ŽGrundland w4x.. The functions u1 Ž x . and u 2 Ž x . can be represented in the form Ž7. if and only if grad u1 and grad u 2 are collinear at each point, i.e., when

­ 1 u1 ­ 2 u 2 s ­ 2 u1 ­ 1 u 2 . The proof is standard. Note that in the conditions of Lemma 1 if u1 is non-degenerated, i.e., if grad u1 / 0, then u1 is a Riemann invariant. Hence we obtain THEOREM 1. Ž u1, u 2 . is a solution of the system Ž6., constructed by means of Riemann in¨ ariants, if and only if Ž u1, u 2 . is a solution of the system

­ 1 u1 s a11 Ž x, u . ­ 2 u1 q a12 Ž x, u . ­ 2 u 2 q b1 Ž x, u . ­ 1 u 2 s a12 Ž x, u . ­ 2 u1 q a22 Ž x, u . ­ 2 u 2 q b 2 Ž x, u . ­1u ­2 u s ­2 u ­1u . 1

2

1

2

Ž 8.

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This system consists of three equations, and since the number of unknown functions equals two, it is overdetermined. But in order to represent it in a form suitable from the point of view of a desired integration, we must resolve it with respect to certain partial derivatives; this requires an analysis of this system as an algebraic system with respect to the partial derivatives of the unknown functions. Substituting ­ 1 u1 and ­ 1 u 2 from the first two equations of Ž8. in the third one and denoting ­ 2 u1 and ­ 2 u 2 by X and Y, respectively, we obtain the equation a12 X 2 y Ž a11 y a22 . XY y a12 Y 2 q b 2 X y b1 Y s 0.

Ž 9.

Let D s Ž a11 y a22 . 2 q 4 a12 a12 . In accordance with the type of the conic section Ž9., we will call the system Ž6. hyperbolic, elliptic, or parabolic pro¨ ided D ) 0, D - 0, or D s 0, respecti¨ ely. This definition is equivalent to the definition given in terms of eigenvalues and eigenvectors of matrices, used in the Introduction of this paper and in many other papers Žsee, e.g., Jeffrey w1x and Grundland w5x quoted above.. Note that the coefficients in Ž6. depend on the point Ž x, u. of the x = u space R 2 = R 2 , and therefore Ž6. can be of different types in different domains. A straightforward analysis of Ž9. leads to several results, formulated below. THEOREM 2. Ži. If the conic section Ž9. is imaginary, then Ž6. has a solution if and only if ­ 2 b1 s ­ 2 b 2 s 0, and its solutions satisfy the equations ­ 2 u1 s ­ 2 u 2 s 0, ­ 2 u1 s b1, and ­ 2 u 2 s b 2 . Žii. If the conic section Ž9. is real, then Ž9. can be resol¨ ed with respect to one of the deri¨ ati¨ es X s ­ 2 u1 and Y s ­ 2 u 2 . Now we will pay special attention on the homogeneous case, when b1 s b 2 s 0, i.e., when Ž6. is of the form

­ 1 u1 s a11 Ž x, u . ­ 2 u1 q a12 Ž x, u . ­ 2 u 2 ­ 1 u 2 s a12 Ž x, u . ­ 2 u1 q a22 Ž x, u . ­ 2 u 2 .

Ž 10 .

Then Ž9. reduces to a12 X 2 y Ž a11 y a22 . XY y a12 Y 2 s 0.

Ž 11 .

In the elliptic case, when D s Ž a11 y a22 . 2 q 4 a12 a12 - 0, the only solution of Ž11. is the trivial one: X s Y s 0. Then the equations of Ž10. imply that all the derivatives ­ i u j equal 0, and hence we obtain

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THEOREM 3. A homogeneous elliptic system Ž10. has only tri¨ ial solutions, constructed by means of Riemann in¨ ariants, i.e., only solutions of the form u1 s const, u 2 s const. ŽNote that in the classical case, when the coefficients a ij do not depend on x, this result is well known.. In the non-elliptic case of Ž10., letting, XrY s K, we reduce Ž11. to a quadratic equation with respect to K, whose roots are K 1, 2 s

1 2 a12

Ž a11 y a22 " 'D . .

Ž 12 .

If D s 0, then K 1 s K 2 . THEOREM 4. If Ž u1, u 2 . is a simple wa¨ e solution, satisfying Ž10., then D G 0 and ­ 2 u1 s K Ž x, u. ­ 2 u 2 , where K Ž x, u. is one of the functions, determined by Ž12.. In view of the above considerations and Theorem 3, the system Ž8. in the non-elliptic homogeneous case reduces to the system

­ 1 u1 s t 1 Ž x, u . ­ 2 u 2 ­ 1 u 2 s t 2 Ž x, u . ­ 2 u 2

Ž 13 .

­ 2 u1 s K Ž x, u . ­ 2 u 2 , where t 1 s a11 K q a12 and t 2 s a12 K q a22 . In the non-homogeneous case in view of Theorem 2 the system Ž8. can be represented in the form

­ 1 u1 s T1 Ž x, u, ­ 2 u 2 . ­ 1 u 2 s T2 Ž x, u, ­ 2 u 2 .

Ž 14 .

­ 2 u s L Ž x, u, ­ 2 u . , 1

2

where LŽ x, u, ­ 2 u 2 . is the solution of the quadratic equation Ž9. with respect to X, T1 s a11 L q a12 ­ 2 u 2 q b1, and T2 s a12 L q a22 ­ 2 u 2 q b 2 . Thus our main purpose in this paper reduces to presenting a method for solving Ž14..

3. THE CORRESPONDING PFAFF SYSTEM The most general and powerful approach to the overdetermined systems of PDEs is based on the Pfaff systems. Having in mind our goals, we will start with some details of the classical relations between the system Ž14.

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and the respective Pfaff system du1 y T1 dx 1 y L dx 2 s 0 du 2 y T2 dx 1 y Ž ­ 2 u 2 . dx 2 s 0. Denoting for symmetry ­ 2 u 2 , u1 , and u 2 by x 3 , x 4 , and x 5, respectively, the last system can be rewritten as

v 1 Ž dx . ' dx 4 y T1 dx 1 y L dx 2 s 0 v 2 Ž dx . ' dx 5 y T2 dx 2 y x 3 dx 2 s 0,

Ž 15 .

where x s Ž x 1, x 2 , . . . , x 5 . g R 5, and T1 , T2 , and L are the functions, described at the end of the previous section. Here v 1 and v 2 are differential forms and dx is a vector field. The classical Pfaff problem in this case is Žroughly speaking. to find three functions Fi Ž x 1 , x 2 , . . . , x 5 ., i s 1, 2, 3, so that the left hand sides of the two equations of Ž15. are linear combinations Žwith variable coefficients. of dF 1 s Ý ­ i F 1 dx i, dF 2 , and dF 3 . It is easy to verify that if F 1 , F 2 , and F 3 satisfy this condition and if det < ­ i Fj < / 0, i, j s 1, 2, 3, then according to the implicit function theorem the system Fi s 0, i s 1, 2, 3, determines three implicit functions x 3 s x 3 Ž x 1, x 2 ., x 4 s x 4 Ž x 1 , x 2 ., and x 5 s x 5 Ž x 1, x 2 ., which satisfy Ž14. with x 3 s ­ 2 u 2 , i.e., it determines a solution of Ž14.. Therefore we can look for solutions of Ž15. instead of solutions of Ž14.. Consider Ž15. as a linear algebraic system with respect to the coordinates of the vector field dx. Since the rank of Ž15. equals 2, it has three linearly independent solutions; denote them by j 1Ž x ., j 2 Ž x ., and j 3 Ž x . and by u Ž x . their linear hull. u Ž x . is a distribution, and its dimension equals 3. It is the linear hull of all vector fields, satisfying Ž15.. Any two linearly independent vector fields h1 g u and h 2 g u determine a subdistribution u 1 of u of dimension 2, which is the linear hull of h1 and h 2 . If wh1 , h 2 x g u 1 , then u 1 is in¨ oluti¨ e, and, according to Frobenius’ theorem, it is completely integrable, i.e., the system of linear first order PDEs

h1F Ž x . s 0,

h 2 F Ž x . s 0,

Ž 16 .

where F Ž x . is the unknown function, has three functionally independent solutions F 1 , F 2 , and F 3 . Comparing Ž15. and Ž16., in view of the fact that h1 and h 2 satisfy Ž14. we conclude that the left hand sides of the equations Ž15. are linear combinations of dF 1 , dF 2 , and dF 3 .

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Thus we proved LEMMA 2. If there exists a 2-dimensional in¨ oluti¨ e subdistribution u 1 s linear hull of h1 and h 2 , then Ž14. has a solution, which is determined by the implicit function theorem from any three functionally independent solutions of the system Ž16.. The converse is also true. LEMMA 3. If Ž14. has a solution Ž u1 s f 1 Ž x 1, x 2 ., u 2 s f 2 Ž x 1 , x 2 .., then there exists a 2-dimensional in¨ oluti¨ e subdistribution u 1 s linear hull of h1 and h 2 . Indeed, let F 1 s x 4 y f 1 Ž x 1 , x 2 ., F 2 s x 5 y f 2 Ž x 1, x 2 ., and F 3 s x 3 y ­ 2 f 2 Ž x 1, x 2 .. Then v 1 Ž dx . s dF 1 . and v 2 Ž dx . s dF 2 ., and therefore the 2-dimensional distribution u 1 , orthogonal to grad Fi , i s 1, 2, 3, possesses the required property. Lemma 2 and Lemma 3 reduce the problem of solving Ž14. to the problem of finding two linearly independent vector fields h1 and h 2 , satisfying the algebraic system Ž15., i.e., v i Žhj . s 0, j s 1, 2, and such that wh1 , h 2 x g linear hull of h1 and h 2 . 4. 2-DIMENSIONAL RESOLVING DISTRIBUTIONS OF PFAFF’S SYSTEM Ž15. In Section 3 the distribution u was characterized as the linear hull of the solutions of the linear system v i Ž j . s 0, i s 1, 2, with respect to j . Now our task is to build a basis for finding all possible involutive 2-dimensional subdistributions u 1 of u . DEFINITION Žsee w8x.. The involutive subdistributions of u are called resol¨ ing distributions for Pfaff ’s system Ž15.. Let the vector fields j 1 , j 2 , and j 3 form a basis of u . THEOREM 5. There exists only one Ž up to a scalar multiple. ¨ ector field h 02 , satisfying the system

v i Ž h . s 0, ­v

2

Ž j j , h . s 0,

i s 1, 2 j s 1, 2, 3.

Ž 17 .

If Ž15. has a 2-dimensional resol¨ ing distribution u 1 , then h 02 g u 1. Proof. In order to satisfy the first two equations of Ž17., h 02 must be of the form h 02 s e1j 1 q e 2j 2 q e 3j 3 .

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Then the last three equations of Ž17. are equivalent respectively to is3

Ý e i ­v 2 Ž j j , j i . s 0,

j s 1, 2, 3.

Ž 18 .

is1

But ­v 2 is antisymmetric, and therefore the rank of this linear homogeneous system equals either 0 or 2. In view of the fact that the expression of v 2 Ž dx . in Ž15. contains the term x 3 dx 2 , the restriction of ­v 2 on u is non-trivial; consequently the above rank cannot be equal to 0, hence it equals 2. Therefore Ž18. has a unique Žup to a scalar multiple. solution Ž e01 , e02 , e03 ., which gives the respective solution

h 02 s e01 j 1 q e02 j 2 q e03 j 3 . Now let u 1 be a 2-dimensional resolving distribution for Ž15.. Since u 1 ; u , then without loss of generality we can assume that j 2 and j 3 form a basis of u 1. And since u 1 is involutive, we have

­v 2 Ž j 2 , j 3 . s 0.

Ž 19 .

We will show that h 02 g u 1 , i.e., that in this case e1 s 0. Indeed, Ž18. reduces to e 2 ­v 2 Ž j 1 , j 2 . q e 3 ­v 2 Ž j 1 , j 3 . s 0 e1 ­v 2 Ž j 2 , j 1 . s 0 e1 ­v 2 Ž j 3 , j 1 . s 0, and the required result follows. THEOREM 6. If the restriction of ­v 1 on u is non-tri¨ ial, then here exists only one Ž up to a scalar multiple. ¨ ector field h 01 , satisfying the system

v i Ž h . s 0, ­v

1

Ž j j , h . s 0,

i s 1, 2 j s 1, 2, 3.

Ž 20 .

If in addition Ž15. has a 2-dimensional resol¨ ing distribution u 1 , then h 01 g u 1. DEFINITION Žsee w9x.. A vector field h 0 , satisfying the system

v i Ž h . s 0, ­v i Ž j j , h . s 0,

i s 1, 2,

is called a characteristic vector field for Ž15..

j s 1, 2, 3

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THEOREM 7. If the solution h 02 of Ž17. satisfies also Ž20., then there exist infinitely many 2-dimensional resol¨ ing distributions for Ž15.. Proof. Note that if h 02 satisfies Ž17. and Ž20., it is in fact characteristic for Ž15.. Let h 0 be a non-zero vector field, which is characteristic for Ž15.. Let Ž c 1 Ž x ., c 2 Ž x ., c 3 Ž x ., c 4 Ž x .. be a set of four functionally independent solutions of the equations h 0 c s 0, and let c 5 be functionally independent with this set. Take y i s c i, i s 1, 2, 3, 4, 5, as new Žin general curvilinear. coordinates. Then

h0 s aŽ y .

­ ­ y5

,

Ž 21 .

where y s Ž y 1 , y 2 , y 3, y 4 , y 5 . g R 5 and aŽ y . is a scalar function. In this new system of coordinates Ž15. has the form

v i Ž dy . ' b1i Ž y . dy 1 q b 2i Ž y . dy 2 q b 3i Ž y . dy 3 q b4i Ž y . dy 4 s 0, i s 1, 2.

Ž 22 .

ŽNote that, since h 0 s aŽ y .Ž ­r­ y 5 . satisfies Ž15. and hence Ž22., the coefficients of dy 5 in Ž22. equal zero.. Since the rank of Ž22. equals 2, we can assume that b11 b 22 y b12 b12 / 0. Then Ž22. can be resolved with respect to dy 1 and dy 2 , and so reduced to the equivalent system

v i Ž dy . ' dy 1 q c 3i Ž y . dy 3 q c 4i Ž y . dy 4 s 0,

i s 1, 2,

Ž 23 .

where the differential forms v 1 and v 2 are linear combinations of the forms v 1 and v 2 . Consequently h 0 is characteristic for Ž23., and hence the rank of the system

v i Ž dy . s 0,

­v i Ž h 0 , dy . s 0,

i s 1, 2,

equals 2. But

­v i Ž h 0 , dy . s a

­ c 3i ­ y5

dy 3 q a

­ c 4i ­ y5

dy 4 ,

i s 1, 2,

which leads to the conclusion that ­ c 3i r­ y 5 s ­ c 4i r­ y 5 s 0, i s 1, 2, i.e., the coefficients of Ž23. do not depend on y 5. Note that the vector fields z 1 s Žyc 31 , c 32 , 1, 0, 0. and z 2 s Žyc41 , c 41 , 0, 1, 0. satisfy Ž23., and consider a vector field h of the form h s pz 1 q q z 2 , where p s pŽ y 1, y 2 , y 3, y 4 . and q s q Ž y 1, y 2 , y 3, y 4 . are

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arbitrary scalar functions, not depending on y 5. Taking into account Ž21., we obtain

h0 , h s y Ž h aŽ y . .

­ ­y

5

sy

Ž h aŽ y . . h0 . aŽ y .

Hence the linear hull of h 0 and h is involutive, and since the pair Ž p, q . can be chosen in infinitely many ways, the required result follows immediately.

5. THE METHOD FOR FINDING SIMPLE WAVES AND SIMPLE STATES IN R 2 The above results can be summarized in the form of the following method. Ž1. The starting point is the system Ž6.. Ž2. Form the system Ž8.. Ž3. Resolve the system Ž8. with respect to the partial derivatives ­ 1 u1, ­ 1 u 2 , and ­ 2 u1 Žor ­ 2 u 2 instead of ­ 2 u1 . to obtain system Ž14.. Ž4. Form Pfaff’s system Ž15. and find the fields j i , i s 1, 2, 3. Ž5. Form the system Ž17. and find its non-trivial solution h 02 ; for more details see the proof of Theorem 5. Ž6. Form the system Ž20. and if its rank equals 4, find its non-trivial solution h 01 ; for more details see the proof of Theorem 5. Ž7. If h 02 and h 01 are linearly independent, solve the linear system of first order PDEs h 0 i F Ž x . s 0, i s 1, 2. If the distribution u 1 spanned by h 01 and h 02 is involutive, then this system has a set of 3 functionally independent solutions Fi , i s 1, 2, 3, and the implicit functions x j s x j Ž x 1, x 2 . determined by the system F Ž x . s const., i s 1, 2, 3 give the solution u1 s x 4 Ž x 1 , x 2 ., u 2 s x 5 Ž x 1 , x 2 . of Ž6.; this solution is a solution, constructed by means of Riemann invariants Žsimple wave or simple state.. If u 1 is not involutive, then Ž14. has no solution. Ž8. In the degenerated cases when either the rank of Ž20. equals 2 or the fields h 01 and h 02 are collinear, take h 0 s h 02 and continue following the way given in the proof of Theorem 7. Ž9. Note that if Ž9. has two solutions, then the stages Ž3. ] Ž8. must be performed twice, once for the one solution and once for the other one. Now we will sketch the details of the application of this method in the case when the given system Ž6. is homogeneous, but its coefficients depend

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on x and u; in this case the method of characteristics cannot be applied. We describe only the non-degenerate case, when the rank of Ž20. equals 2 and the vector fields h 01 and h 02 are not collinear; therefore we perform only the stages Ž1. ] Ž7.. Ž1. The starting point is the system Ž6., which in our case reduces to the system Ž10.:

­ 1 u1 s a11 Ž x, u . ­ 2 u1 q a12 Ž x, u . ­ 2 u 2 ­ 1 u 2 s a12 Ž x, u . ­ 2 u1 q a22 Ž x, u . ­ 2 u 2 . Ž2.

Form the system Ž8.. In the homogeneous case Ž8. reduces to

­ 1 u1 s a11 Ž x, u . ­ 2 u1 q a12 Ž x, u . ­ 2 u 2 ­ 1 u 2 s a12 Ž x, u . ­ 2 u1 q a22 Ž x, u . ­ 2 u 2 ­ 1 u1 ­ 2 u 2 s ­ 2 u1 ­ 1 u 2 . Ž3. Resol¨ e the system Ž8. with respect to the partial deri¨ ati¨ es ­ 1 u1 , ­ 1 u , and ­ 2 u1 Ž or ­ 2 u 2 instead of ­ 2 u1 . to obtain system Ž14.. As it was shown in Section 2, Ž14. in the non-elliptic homogeneous case reduces to the system Ž13., namely to 2

­ 1 u1 s t 1 Ž x, u . ­ 2 u 2 ­ 1 u 2 s t 2 Ž x, u . ­ 2 u 2 ­ 2 u1 s K Ž x, u . ­ 2 u 2 ; here K Ž x, u. equals either K 1 or K 2 , where K 1, 2 s

1 2 a12

Ž a11 y a22 " 'D .

Žthe procedure should be carried out for each of the values K s K 1 and K s K 2 ., t 1 s a11 K q a12 and t 2 s a12 K q a22 . Ž4. Form Pfaff ’s system Ž15. and find the fields j i , i s 1, 2, 3. In our case Ž15. is of the form

v 1 Ž dx . ' dx 4 y t 1 x 3 dx 1 y Kx 3 dx 2 s 0 v 2 Ž dx . ' dx 5 y t 2 x 3 dx 2 y x 3 dx 2 s 0, so we can choose j 1 s Ž0, 0, 1, 0, 0., j 2 s Ž1, 0, 0, t 1 x 3 , t 2 x 3 ., and j 3 s Ž0, 1, 0, kx 3, x 3 ..

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Ž5. Form the system Ž17. and find its non-tri¨ ial solution h 02 ; for more details see the proof of Theorem 5. Taking the solution h 02 to be of the form

h 02 s e1j 1 q e 2j 2 q e 3j 3 in order to satisfy the first two equations of Ž17., we reduce Ž17. to the system is3

Ý

­v 2 Ž j j , j i . e i s 0,

j s 1, 2, 3

Ž 24 .

is1

Žsee the proof of Theorem 5. with unknowns e 1, e 2 , e 3. Since ­v 2 Ž j 1 , j 2 . s yt 2 , ­v 2 Ž j 1 , j 3 . s y1, ­v 2 Ž j 2 , j 3 . s f , ­v 2 Ž j i , j i . s 0, i s 1, 2, 3, where 2

f s x 3 ­ 2 t 2 q Kx 3 ­4 t 2 q Ž x 3 . ­ 5 t 2 , Ž24. becomes yt 2 e 2 y e 3 s 0 t 2 e1 q f e 3 s 0 e1 y f e 2 s 0. The rank of this system equals 2; we need one of its non-zero solutions, and the solution Ž e1 , e 2 , e 3 . s Ž f , 1, yt 2 . is suitable for our purposes. Thus h 02 s fj 1 q j 2 y t 2 j 3 is determined. Ž6. Form the system Ž20. and if its rank equals 4, find its non-tri¨ ial solution h 01 ; for more details see the proof of Theorem 5. Taking the solution h 01 to be of the form

h 01 s e1j 1 q e 2j 2 q e 3j 3 in order to satisfy the first two equations of Ž20., we reduce Ž20. to the system is3

Ý ­v 1 Ž j j , j i . e i s 0,

j s 1, 2, 3

Ž 25.

is1

Žsee the proof of Theorem 5., where e 1, e 2 , e 3 are unknowns. Since ­v 1 Ž j 1 , j 2 . s yt1 , ­v 1 Ž j 1 , j 3 . s yK, ­v 1 Ž j 2 , j 3 . s c , ­v 1 Ž j i , j i . s 0, i s 1, 2, 3, where

c s x 3 ­ 2 t 2 q Kx 3 ­4 t 2 q x 3 ­ 5 t 2 ,

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Ž25. becomes yt1 e 2 y Ke 3 s 0 t1 e1 q c e 3 s 0 Ke1 y c e 2 s 0. We assume that the rank of this system equals 2; we need one of its nonzero solutions, and the solution Ž e1, e 2 , e 3 . s Ž c , K, yt1 . is suitable for our purposes. Thus h 01 s cj 1 q K j 2 y t 1 j 3 is determined. Ž7. Here we assume that h 02 and h 01 are linearly independent. Now it is necessary to solve the linear system of first order PDEs h 0 i F Ž x . s 0, i s 1, 2. We omit the details, because they follow standard classical techniques.

6. GENERALIZATION The above method can be applied also to general non-linear system of the form

­ 1 u i s H i Ž x, u, ­ 2 u1 , ­ 2 u 2 , . . . , ­ 2 u n . i s 1, 2, . . . , n, where x s Ž x 1, x 2 . and u s Ž u1 , u 2 , . . . , u n ., provided the respective system

­ 1 u i s H i Ž x, u, ­ 2 u1 , ­ 2 u 2 , . . . , ­ 2 u n . ­ 1 ui ­ 2 u j s ­ 2 ui ­ 1 u j i , j s 1, 2, . . . , n can be resolved with respect to 2 n y 1 of the partial derivatives ­ i u j, i s 1, 2; j s 1, 2, . . . , n, e.g., if the last system can be represented in the form

­ 1 u i s F i Ž x, u, ­ 2 u n . ­ 2 u j s G i Ž x, u, ­ 2 u n . i s 1, 2, . . . , n, j s 1, 2, . . . , n y 1.

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The respective Pfaff system Žof rank n in R nq 3 . is of the form

v 1 Ž dz . ' du1 y F 1 dx 1 y G1 dx 2 s 0 .. . v ny 1 Ž dz . ' du ny1 y F ny1 dx 1 y G ny1 dx 2 s 0 v n Ž dz . ' du n y F n dx 1 y y dx 2 s 0, where z ' Ž x 1, x 2 , y, u1, u 2 , . . . , u n .. Since it determines a 3-dimensional distribution u , the method for finding all suitable Žif any. integrable 2-dimensional subdistributions described above can be applied.

7. EXAMPLE 1 Here we apply the above method to the problem of the existence of simple states for systems of the form ut q ¨ x s 0 ¨ t s hu x s f ,

Ž 26 .

where h s hŽ t, ¨ . and f s f Ž t, ¨ . are given functions. It is known that in the case when h and f do not depend on t, Ž1. describes different processes in gas dynamics and in metals Žsee w10, 11x.. The particular case when h and F do not depend on t is studied in the author’s paper w12x. For the sake of convenience we will keep the notation of Ž26., so that t, x, u, ¨ correspond to x 1 , x 2 , x 4 , x 5, and will denote u x s ­ ur­ x by y. The steps of our method for the system Ž26. with h and t depending on t and ¨ look as follows. The starting point is the system Ž26.. The system Ž8. in this case is ut q ¨ x s 0 ¨ t q hu x s f u t¨ x s u x¨ t .

Ž 27 .

Resolving the system Ž27. with respect to u t , ¨ t , and ¨ x , we find that it splits into two systems, corresponding to two values 1 and y1 for e in the system u t s ye u x Ž hu x y f .

'

¨ t s yhu x q f ¨ x s e u x Ž hu x y f . ,

'

SIMPLE WAVES AND SIMPLE STATES IN R

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or, using the notations y and T for u x and hu x y f, in the system u t s ye yT

'

¨ t s yT

Ž 28 .

¨ x s e yT .

'

Further we suppose that f / 0. The following Pfaff system in R 5 corresponds to Ž28.,

v 1 Ž dz . ' d¨ q T dt y e'yT dx s 0 v 2 Ž dz . ' du q e'yT dt y y dx s 0, where dz s Ž dt, dx, dy, d¨ , du.. Further, we find the fields

j 1 s Ž 0, 1, 0, e L, y . ,

j 2 s Ž y, e L, 0, 0, 0 . ,

j 3 s Ž 0, 0, 1, 0, 0 . , where L s yT , and then

'

h 01 s e f j 1 q 2 L y j 2 y 2 Ž L2 L¨ q yL t . j 3 h 02 s e f j 1 q 2

L y

j 2 y 2 L2 L¨ j 3 ,

Changing h 01 and h 02 by their linear combinations, we form the system of PDEs, L1F ' e f Fx y y yT T¨ q

'

ž

2 f

Tt Fy q f yT F¨ q e fyFu s 0

/

'

Ž 29 .

L2 F ' f'y Ft q e f'T Fx q y'y Tt Fy s 0, where F s F Ž t, x, y, ¨ , u. is the unknown function of five variables, T Ž t, y, ¨ . s yhŽ t, ¨ . q f Ž t, ¨ ., and e attains the values 1 and y1; this system is equivalent to the system h 0 i F Ž x . s 0, i s 1, 2. In result we obtain the following theorem: THEOREM 8.

Let f / 0 and u 0x Ž h Ž ¨ 0 . u 0x y f Ž ¨ 0 . . ) 0.

If the system Ž29. is in¨ oluti¨ e, i.e., if the commutator w L1 , L2 x belongs to the linear hull of L1 and L2 , and since the conditions of the existence of the implicit function are satisfied, the system Ž29. has 3 functionally independent

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solutions F 1Ž t, x, y, ¨ , u., F 2 Ž t, x, y, ¨ , u., and F 3 Ž t, x, y, ¨ , u.; the system of equations Fi Ž t , x, y, ¨ , u . s Fi Ž t 0 , x 0 , u 0x , ¨ 0 , u 0 . ,

i s 1, 2, 3

determines the implicit functions y s y Ž t, x ., ¨ s ¨ Ž t, x ., and u s uŽ t, x ., and then Ž uŽ t, x ., ¨ Ž t, x .. is the unique solution of the type of simple state for the system Ž26. with initial conditions uŽ t0 , x 0 . s u0 ¨ Ž t0 , x 0 . s ¨ 0

u 0x Ž t 0 , x 0 . s u 0x , corresponding to the selected ¨ alue of 1 of y1 for e . If f / 0 and u 0x Ž h Ž ¨ 0 . u 0x y f Ž ¨ 0 . . - 0, or if the system Ž29. is not in¨ oluti¨ e, then the system Ž26. has no simple state solutions. Maybe the most unexpected conclusion from this result is that the existence of simple states is not related with the type Žhyperbolic or parabolic. of the initial system Ž26.. Another important conclusion is that the natural initial value problem for Ž26. is not the Cauchy problem. Since the problem for solving Ž29. reduces to the problem for solving system of ordinary differential equations, the above Theorem 8 reduces the problem of finding the simple states of Ž26. to solving systems of ordinary differential equations.

8. EXAMPLE 2 D. Kolev in w13x represents an application of a particular case of the above method to systems of the form u1t q u 2x s 0 u 2t y a2 Ž u 2ru1 . u1x q 2 a Ž u 2ru1 . u 2x q h Ž u . s 0

Ž 30 .

u 3x y g Ž u1 . s 0, where u ' Ž u1 , u 2 , u 3 . and the given functions a, g, and h are sufficiently smooth. Such systems are physical models of non-linear electric field oscillations and are investigated by U. Emets and I. Reppa in w14x in the case when a, g, h satisfy certain additional conditions.

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The system analogous to Ž8. in this case consists of the equations of Ž30. and of the following two equations: u1t u 2x y u1x u 2t s 0 u1t u 3x y u1x u 3t s 0.

Ž 31 .

Solving this system with respect to K ' u1t and L s u 2t and denoting u 3t by y, we obtain two solutions: Ži. Žii.

K 1 s L1 s 0, K 2 s ghy Ž ga q y .y2 , L2 s yhy 2 Ž ga q y .y2 .

Omitting the trivial case Ži., we further continue with the investigation of Žii.. The following Pfaff system in R 6 corresponds to the system consisting of Ž30. and Ž31.,

v 1 Ž dz . ' du1 y K dt y K gyy1 dx s 0 v 2 Ž dz . ' du 2 q K ygy1 dt q K dx s 0 v 3 Ž dz . ' du 3 y y dt y g dx s 0, where dz s Ž dt, dx, dy, du1 , du 2 , du 3 .. The last system simplifies to y du1 y K du 3 s 0 g du 2 q K du 3 s 0 du 3 y y dt y g dx s 0. Further, we find the fields

h 01 s Ž yghy1 Ž K y yK y . , K y yK y , 0, 0, 0, 0 . , h 02 s Ž ygK y , yK y , 0, 0, 0, 0 . , h 03 s Ž 0, 1, gK , gyy1 K , yK , g . . Obviously h 01 and h 02 are colinear. Hence the distribution spanned by h 01 , h 02 , and h 03 is 2-dimensional. It is easy to verify that wh 01 , h 03 x is collinear with h 01 , i.e., this distribution is involutive and therefore Žaccording to Frobenius’ theorem. completely integrable.

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In order to integrate it, we have to solve the respective system of PDEs

h 01 F s yg Ft q yFx s 0 h 03 s Fx q gK Fy q gKyy1 Fu1 y K Fu 2 q g Fu 3 s 0. It is equivalent to the system of ODEs dxrdt s yygy1 dyrdt s yyK du1rdt s yK du 2rdt s yKgy1 du 3rdt s yy, which has integrals y s y 0 exp Ž u1 y u10 . ,

u 2 s u 02 y y 0

1

Huu exp Ž s y u . g 1 0

1 0

y1

Ž s . ds.

Using these two integrals we can express y and u 2 in terms of u1 , and then K in terms only of u1 and u 3. Now a third integral can be obtained from the equation 1 1 du1rdu 3 s K Ž u1 , u 3 . yy1 0 exp Ž u 0 y u . .

These integrals reduce our problem to solving a first order ODE with one unknown function.

9. A COMPARISON WITH OTHER METHODS Here we give a brief comparison of our methods with the approaches of Yanenko Žand others, see w15, 16x. and P. Olver and J. Rosenau w17x. The method suggested by Yanenko Žsee w15, 16x. is closely related with Cauchy’s problem for a given overdetermined system and with suitable Cartan prolongations. As a result it deals only with solutions, belonging to the family of solutions of a certain Cauchy problem. The natural initial value problem for the classical wave solutions in the case of quasilinear homogeneous systems with coefficients not depending on the independent variables is Cauchy’s problem. However, the general case of solutions, constructed by means of Riemann invariants Žfor nonhomogeneous systems, or when the coefficients depend on the independent variables., leads to another type of initial value problems, different

SIMPLE WAVES AND SIMPLE STATES IN R

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from Cauchy’s problem. This fact is clear from Theorem 8 for the example in Section 7. Even in the case of the relatively simple system equivalent to the Fermi]Pasta]Ulam equation the search for solutions based on Cartan’s prolongations and Cauchy’s problem leads to obstacles, as it is shown in the recent paper of E. Fackerell, D. Hartley, and R. Tucker w18x. A discussion of the reasons for the ‘‘obstacles’’ and a correct initial value problem are given in the author’s paper w19x. The approach of Olver and Rosenau Žin w17x. using groups of transformations, with respect to which the solutions of the overdetermined system remain invariant, corresponds to S. Lie’s idea for characteristic vector fields of the respective Pfaff system. In the framework of the method represented above, such groups of transformations Žand characteristic vector fields in the sence of S. Lie. arise in the case when h 01 s h 02 . Hence our method includes the determination of the suitable group Žin the context of Olver’s approach., provided such a group exists. It should be noted, however, that the existence of suitable groups is rather an exception.

REFERENCES 1. A. Jeffrey, Equations of evolution and waves, in ‘‘Wave Phenomena: Modern Theory and Applications,’’ pp. 123]152, North-Holland Math. Studies, Vol. 97, Elsevier, New York, 1984. 2. M. Burnat, Riemann invariants, Fluid Dynam. Trans. 4 Ž1969., 17]27. 3. Z. Peradzynski, Riemann invariants for nonplanar k-waves, Bull. Acad. Polon. Sci. Ser. Sci. Tech. 19 Ž1971., 717]732. 4. A. M. Grundland, Riemann invariants for nonhomogeneous systems of quasilinear partial differential equations, Bull. Acad. Polon. Sci. Ser. Sci. Tech. 22 Ž1974., 273]282. 5. A. M. Grundland, Riemann invariants, in ‘‘Wave Phenomena: Modern Theory and Applications,’’ pp. 123]152, North-Holland Math. Studies, Vol. 97, Elsevier, New York, 1984. 6. P. Kucharczyk, Z. Peradzynski, and E. Zawistowska, Unsteady multidimensional isentropic flows described by linear Riemann invariants, Arch. Mech. 25 Ž1973., 319]350. 7. A. M. Grundland, Generalized Riemann invariants, in ‘‘Nonlinear Functional Analysis and Its Applications’’ ŽS. P. Singh, Ed.., pp. 258]276, Reidel, Dordrecht, 1986. 8. J. Tabov, On the extending of the resolving distributions of Pfaff’s systems, Russian Math. Sur¨ eys 29 Ž1974., 243]244. 9. J. Schouten and W. van der Kulk, ‘‘Pfaff’s Problem and Its Generalizations,’’ Clarendon Press, Oxford, 1947. 10. D. Fusco and N. Manganaro, Riemann invariants-like solutions for a class of rate-type materials, Acta Mech. 105 Ž1994., 23]32. 11. R. J. Clifton, High strain rate behavior of metals, Appl. Mech. Re¨ . 43 Ž1990., 9]22. 12. J. Tabov, Riemann invariants for certain systems of PDEs in R 2 , Differential Equations 32, No. 11 Ž1996., 1554]1557.

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13. D. Kolev, Simple states of non-linear electric field model in R 2 , Math. Balkanica, in press. 14. U. Emets and I. Reppa, Non-linear variations of electric field in continuous medium, Appl. Math. Tech. Phys. 3 Ž1969., 55]57. 15. B. L. Rozdestvenskii and N. N. Janenko, ‘‘Systems of Quasilinear Equations and Their Applications to Gas Dynamics,’’ Vol. 55, Amer. Math. Soc., Providence, 1983. 16. A. Sidorov, V. Shapeev, and N. Yanenko, ‘‘Method of Constraint Conditions and Its Applications in Gas Dynamics,’’ Nauka, Novosibirsk, 1984. wIn Russianx 17. P. Olver and P. Rosenau, Group-invariant solutions of differential equations, SIAM J. Appl. Math. 47, No. 2 Ž1987., 263]278. 18. E. Fackerell, D. Hartley, and R. Tucker, An obstruction to the integrability of a class of non-linear wave equation by 1-stable Cartan characteristics, J. Differential Equations 115 Ž1995., 153]165. 19. J. Tabov, Initial value problems for the Fermi]Pasta]Ulam equation, Fundamental and Applied Mathematics, in press. wIn Russianx