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Reduction to linear canonical forms and generation of conservation laws for a class of quasilinear hyperbolic systems
Inr. J. Non-Linear ~Mechuucs. Printed in Great Britain.
Vol. 23. No. I. pp. 25-35.
0020-X62,88 Per&mm
1988
S3.00 + 0.00 Journals Ltd.
REDUCTION TO LINEAR CANONICAL’ FORMS AND GENERATION OF CONSERVATION LAWS FOR A CLASS OF QUASILINEAR HYPERBOLIC SYSTEMS C. CURRY and D. FUSCO* Dipartimento de Matematica, Universitl di Messina, Salita Sperone, contrada Papardo, Vill. S. pgata, 98010, Messina, Italy (Received 27 April 1987; received fir public&on 17 June 1987) Abstract-We consider a quasilinear hyperbolic homogeneous system of two first order equations involving two dependent and two independent variables. For the associated hodograph equations we investigate the reducibility to canonical forms allowing for an explicit integration. Such a kind of requirement, in problems of physical interest, provides a suitable method for characterizing possible material model laws. The theoretical approach shown herein can be relevant for studying the existence of conservation laws to non-homogeneous first order systems and also for describing the evolution of weak shock waves. I. INTRODUCTION
The reduction of the hodograph system to canonical forms which allow closed form integration has been investigated in several physical contexts. Such a method of approach has proved to be a useful tool for determining non-linear functional forms of the constitutive laws. A survey of results can be found in [l]. It must be noted also that the media so characterized are of relevant interest in non-linear wave propagation and especially in wave interactions phenomena; (see [2,3] and the references quoted there). In this paper we aim to develop a general procedure for obtaining explicit solutions to the hodograph system connected to a quasilinear hyperbolic system of first order:
u, + A(U)U, where
us
[Iu v
A(U)
’
=
= 0,
all [ a21
(1.1) al2 a22
1
and x, t are, respectively, space and time coordinates. Moreover, subscripts denote partial derivatives with respect to the indicated variable. Of course the hyperbolicity of (1.1) requires the matrix A to admit two real distinct eigenvalues E. and p (wavespeeds) with corresponding left eigenvectors 1(‘), VP)and right eigenvectors d(l), d@' spanning the Euclidean space E2. First we show that if J. and cc satisfy suitable conditions (which are usually fulfilled by the models of physical interest belonging to the class considered herein) then the problem of determining the solutions to the hodograph system arising from (1.1) is reduced to the integration of a linear second order equation involving a self-adjoint operator. Therefore, by generalizing to the case under investigation the method of approach shown in [l] and [4], a solution is sought in the form of a series whose coefficients are determined by solving recurrence relations. The uniform convergence of this series can be investigated, in principle, after the manner of Bergman [S, 63. Furthermore, in view of obtaining classes of solutions which can be of direct use in physical applications, we deduce the general condition to be satisfied in order that the series terminates. Such a request suggests also a class of canonical forms to which the hodograph system can be reduced. Moreover, through a suitable variable transformation, we establish a link with the canonical solutions already obtained
l Permanent address: Dipartimento di Matematica ed Applicazioni, Universita di Napoli, via Meuocannone 8. 80134 Napoli, Italy.
25
26
C.
CURRY
and D. Fusco
in [7] for a polytropic gas with special values of the adiabatic exponent. It must be pointed out that by requiring the basic model (1.1) to satisfy the conditions which allow the afore-mentioned Bergman-like series to terminate, one can generate possible sets of functional forms to the constitutive laws which are involved in a case of a physical interest. The canonical forms and the classes of explicit solutions to the hodograph system obtained in different physical contexts by several authors Cl-43 emerge from our analysis as particular cases. It is worth noting that the study we carry on herein can be viewed also within a theoretical framework which is different from the one of finding out solutions to the hodograph system arising from (1.1). In fact, in the second part of the paper, we consider also.the quasilinear hyperbolic nonhomogeneous system:
u, + A(U)U,
= B(u, x, t).
(1.2)
Thus we are able to show that, investigating the existence of conservation laws to (1.2), one is led to the integration of a linear system of equations which is similar to the hodograph system connected to the reducible model (1.1). Therefore our analysis fits into the theoretical approach considered in [8,9, lo] for constructing infinitely many conservation laws to a class of quasilinear nondispersive mathematical models. At the end of the paper we illustrate our procedure by means of a physical example. 2. GENERAL
REMARKS
ON THE
HODOGRAPH
SYSTEM
As is well known, the Riemann invariants related to (1.1) are defined by:
r =
q,l(*‘*dU,
s=
s
qJ”‘.dU I
(2.1)
where ~7,and q2 are integrating factors satisfying
(2.2) Thus, via the hodograph transformation x = x(r,s),
t = t(r, s)
(2.3)
under the assumption 8(x, t)/@,s) # 0, the system (1.1) can be reduced to the pair of linear equations:
x, = pt,,
x, = It,.
(2.4)
Whereupon, by eliminating x, the following linear second order equation is obtained:
.A+
Cm - 1. -
p
r
++ I A
-
p
t, = 0.
(2.5)
The classical Riemann method of integration provides the solution of the equation (2.5) in terms of a Riemann function and initial data specified along an initial line in the hodograph plane. However, according to [l 13, let us assume that the characteristic speeds 1 and p satisfy the condition:
(2.6)
27
Quasilinear hyperbolic systems
Hence, writing
g(r,s) = [&dr
+ *ds,
(2.7)
through the variable transformation
(2.8)
t* = exp(g)t
the equation (2.5) takes the form
(2.9) where the differential operator is self-adjoint. We remark that it is quite natural to require the condition (2.6) to hold. In fact, as already noticed in [l 1,123, all the models of physical interest belonging to the class (1.1) satisfy (2.6) without any restriction to the functional forms of the constitutive laws involved there. Now, generalizing to the case of interest the method of approach developed in [l] and L-43,a solution of (2.9) is sought in the form
r*(r, 4 = ; I
z {%h w,w + P.(r,W”(4),
(2.10)
0
where r,, /3” are to be determined, while cb., Y, satisfy the recurrence relations (2.11) Further substitution of (2.10) into (2.9) and use of (2.11) yields the following system of recurrence relations for a, and 8, (2.12)
azr -!
-$&+25I(r,s)or._,
=o
(2.13)
av
=o
(2.14)
-1
-+&+2!&A(r,s)B._r
where A(r,s) = -/
exp(g). Once A(r, s) is specified, the coefficients a,@,s) and /?“(r,s)
can be easily obtained by integrating (2.13) and (2.14). According to [4], the functions O., Y, satisfying (2.11) are given by
s Zr
Q)“(r)= ;
Y,(S) = ;
n
d@o
(2r - 5) --&l;
.
(2.15)
0
(2.16)
28
C. CURROand D. Fusco
where Q*(r) and Y,(S) are arbitrary functions. Alternative expressions for ctr,, Y, are
(2.17)
(2s - <)~2n-1~‘2dC,
n = 1,2,...
(2.18)
with
(2.19)
where K and L are arbitrary. Therefore the equation (2.9) admits the formal series solution (2.10) where Q,, 8, are determined by solving iteratively the recurrence relations (2.13), (2.14) and @#, Y, are given by (2.15), (2.16) [or alternatively by (2.17), (2.18)]. We notice that (2.10) can be considered a Berman-like series whose unif~~ convergence, in principle, can be studied within the theory developed in [S]. Moreover we refer to f6J for additional material on this subject. On account of the complexity of the solution obtained herein, we will deduce in the next section the conditions for which the Bergman series terminates. 3. CANONICAL
FORMS
AND EXPLlCIT
SOLUTIONS
TO THE HODOGRAPH
SYSTEM
The aim of this section is as follows. In order to obtain sets of explicit solutions to the hodograph system which can be of direct use in physical applications, we will require that the series solution (2.10) can be truncated after M+ 1 terms. In the process that requirement will suggest a wide class of canonical forms to which the hodograph system can be reduced. Furthermore we will establish a link with the canonical solutions to the hodograph system obtained in [7] in the case of a polytropic gas with special values of the adiabatic exponent. Let us assume in (2.10) that aO, c1i, . ,. , a, # 0, while a&= 0, k = n + 1,. . . . The same assumption is made, of course for the coefficients &,. Relation (2.12), shows that a0 = so(r). Hence if n = 0 (i.e. the series terminates after the first term), the conditions (2.13) require that A = 0.
(3.1)
This is the case mainly investigated in [43 in connection with non-linear dielectrics. Now let us consider the case n 2 1. In this instance the relations (2.13) specialize to: d2a.
*
+ 22_hcli_r
= 0,
i = 1,2,...,h
(3.21
(3.3) where a0 = so(r) is arbitrary. By defining
PO = A,
P, = A - &(ln
PO),. . . , Pi = A -~[ln(PbP:-‘...P,.,)],
(34J
29
Quasilinear hyperbolic systems it is not difficult to ascertain, after some algebra, that the compatibility determined system (3.2) and (3.3) requires
P,=A-
&
[ln(PeP;- '...P._I)1 = 0.
of the over-
(3.5)
For n = 1, [i.e.; truncation at the second term of the series (2,10)], the relation (3.5) takes the form:
A=
On introduction
&On4.
(3.6)
of w = In A, (3.6) specializes to the Liouville equation W,S=
exp(w).
(3.7)
As is well known (see for instance Cl]), equation (3.7) can be linearized and its solution is given by:
lnA=
expC(Xt(r) - X&))/21
w=2ln
(34
expCXk3ldS+ j ’ evC-X2K)ld5 I SO
where X,(r) and X,(s) are arbitrary functions and 6 is a non-zero constant known as a “Backlund parameter”. For n > 1 it does not seem to be possible to obtain the general expression for A which satisfies condition (3.5). However it is easy to show that if A satisfies the relation
mA =
&
Mm 41,
2 m=n(n
(3.9)
then (3.5) holds. (3.9) is again a Liouville equation which, after integration, provides the following expression for A:
evE&(r) - X2(41
(3.10)
expCXAS)ldS + f ’ expC - X,~~~ldS s 10
d2exp(g) Further substitution of A = drds exp(g) into (3.10) gives rise to the condition which, I [together with (2.6)], must be satisfied by the coefficients of the basic model (1.1) in order that the above results hold. That, of course, will impose suitable restrictions on the functional forms of the constitutive laws which are involved in a system of field equations describing a physical problem*. If A = Ot, then the equation (2.9) specializes to the wave equation t:, = 0
(3.11)
*The functions X,(r) and X,(s) arising from integration of (3.9) will be specified in order to characterize the response functions for the model of interest. *The case A = 0 can be recovered from (3.10) with n = 0.
30
C. CURROand D. Fusco
whose immediate integration yields t*(r,s) = L(r) + M(s)
(3.12)
where L.(r) and M(s) are arbitrary functions. Now we aim to characterize the canonical form taken by (2.9) in connection with the expression of A given by (3.10). To this end we introduce the change of variables
r* = -6 2
s
expW,(OkK
'0
s
s*= d
s
t*=i(r* f
exp(--X,(<))dS, 00
s*)“+‘. (3.13)
By means of (3.13), (2.9) reduces to
(3.14) The equation (3.14) is similar to the one obtained by R. Courant and K. 0. Friedrichs in the hodograph plane for a polytropic gas with special values of the adiabatic exponent 5 7 y given by y = 3, 51 ?.... $ On use of the results deduced in [73 the solution of (3.14) can be written in the form
c=
evk) (r*
+
s*y+
1
(3.15) c=~+&[(r*~~~“+l]+&i[(r*~~:;+l
with arbitrary functions L and M and any constant T. Whence, through (3.13) and (2.8), the corresponding expression for t(r,s) can be obtained. It is easy to see that the canonical forms of the hodograph system and the corresponding classes of mathematical models considered in different physical contexts by several authors Cl-43 (see also references quoted there), can be recovered by means of the method of approach we have already developed. As a very special case corresponding to n = 1 is the class of models characterized in 133 for studying special wave interactions. This class follows from X,(s) = 2&G
X,(r) = 2JKvr, r0
=-s,+-
’ ln[2cofi], 2JKv
6* = $ ,/$,
co, Y,K constants,
so that the general solution of (1.1) has the form
c(r,s) = [l(r) + m(s)]exp(-g) + 4v[L(r) + M(s)] where L(r) and M(s)are arbitrary functions, whereas I(r) and m(s) are determined by I(r) = g,
*dM
m(s) = 7.
However other choices of X,(r), X,(s), give other models for which the series (2.10) truncates at n = 1. Furthermore other values of n, and other choices of X,(r),‘X,(s) give many other classes of models for which the representation (2.10) terminates.
$ The limit case y = - 1 considered in [7] would correspond here to n = - 1 which leads to (3.11).
31
Quasilinear hyperbolic systems 4. CONSERVATION
LAWS
We will show here that the present approach to the integration of the hodograph system can be used also within the relevant theoretical framework for finding conservation laws to non-linear evolution equations. In fact, let us consider the quasilinear hyperbolic non-homogeneous system of first order (1.2):
u, + A(U)U,
= B(U,x,t).
(4.1)
The hyperbolic model (4.1) is not reducible [13]. However, in this case, by means of (2.1), the field Riemann variables r = r(U) and s = s(U) may also be introduced, and they satisfy r, + i.(r,s)r, = p(r, s, x, t)
(4.2)
s, + Ar,sh = b(r, s, x, t)
(4.3)
where p = (1’“’- B)q L, to= (1”’- B)q,. Now we shall deduce the conditions for system (4.1) to admit conservation form
Wr, 4 + Wr, s)
-
at
The compatibility
2X
=
laws of the
q(r, s, x, t).
of (4.2), (4.3) and (4.4) requires the following conditions to hold: G
r
=
l.F,,
Gs = PF,
q = pF, + OF,.
(4.5) (4.6)
The linear system of equations (4.5), which determines F and G is similar to the hodograph system (2.4). Furthermore, from (4.5) by eliminating G the following second order equation is obtained F,s+“IF,2. - p
-F,lb 1 - p
= 0.
(4.7)
Hence the results obtained in [8,9], in connection with the existence of conservation laws to certain classes of non-linear evolution equations, are easily recovered. However it is noteworthy that the analysis developed in [8], [9] deals with models which belong to (4.1) as particular cases, since they do not involve source-like terms. According to [S], we remark that the solution of equation (4.7) will, in general, depend on two arbitrary functions of two variables. Therefore (4.1) admits infinitely many conservation laws of the form (4.4). Obviously the method of approach already shown in the case of equation (2.5) still holds for the integration of (4.7). In particular, if (2.6) is satisfied then, in the case under investigation, we can define &r, s) = ln(i. - P) - g(r, s).
(4.8)
Thus, further investigation of the solution of (4.7) and possible reduction to canonical forms can be worked out as in Sections 2 and 3. Finally, let us assume that E.and p satisfy the conditions Vj_ . d(i)
=
0
=
VP.
d(P)
(4.9)
C. CURRY and D. Fusco
32
In this case the system (4.1) is “completely etceptional”, [13-153. It is easily seen [IO], that conditions (4.9) imply VI a 1"' and Vp a I(*.), sothat 1 = s and p = r. Therefore (4.7) specializes to the conventional wave equation (4.10)
C(r - s)Cl,, = 0
which gives rise to the class of conservation laws already obtained in [lo]. In closing we recall that an exhaustive literature on modern developments of the theory of quasilinear hyperbolic conservative systems which admit conservation law may be found in [16,17]. Remark
Since the solution of (4.4) will depend, in general, on two arbitrary functions it is possible to characterize two independent conservation equations like (4.4). Therefore, it is fairly obvious that a system like (4.1) can be written in infinitely many conservation forms. Such a feature may be relevant for studying shocks. That was already noticed in [lo] for the very special case of 2 x 2 homogeneous and completely exceptional systems. 5. PHYSICAL
EXAMPLE:
NON-LINEAR
DIELECTRICS
Maxwell’s equations in the case of an electromagnetic wave polarized in the x1, x2 plane and propagating along the direction of x3 = x can be written in the form (1.1) with
a= H,
v = E,
all = a2* =
42=
[
jjj
1
1
dD -’
dB -’
0,
v
a2,=
[
do
(5.1)
where E and D = D(E) are, respectively, the x’components of the electric field and electric displacement, while H and B = B(H) are the x*-components of the magnetic field and magnetic flux. Here the characteristic speeds are given by: p = _(B’D’)-“2
). = (B’D’)- I’*,
B’_E
,
D’
-dH’
dD
=dE
(5.2)
so that g = In A”*. Furthermore the Riemann invariants specialize to
s=f(e-h)
r=i(e+h),
(5.3)
where
e =
’ [D’(t)]“* s
H d&
h=
Eo
CB’(Ol I’*d5.
I HO
(5.4)
In connection with the analysis developed through Section 3, let us assume X,(r) = In(2rr + t?),
X*(s)
a, c?,/?, j? constants. Further insertion of (5.5) into (3.10) gives rise to
=
-
In(2bs + fl,
(5.5)
33
Quasilinear hyperbolic systems
where K=
- i(wi
+ Br,) - i(flsij +
hJ.
Several functional forms of i.(r,s) can be chosen in order to satisfy (5.6). In particular it is possible to see that a solution of (5.6) is given by i.(r,s) = [Maze + aJtM[Ma3h
+ ~~]~~~,a,,a,,a,,a,constants,
(5.7)
if M = n + 1 and the constants CX, d, B, p, K are such that 2 ‘1 = -(n + l)2a2a, 6 B = -(n + l)%J2U~ 2 s = -r&n+ l)(a,a, + a,a,) j? = (n +
i j&a,*, - a3a4)
K = adas.
Furthermore,
B=B,-
the integration of (5.7) leads to the following constitutive laws: 1 ((fl + 1)(2n + 3)a,(H - H,) + a:n+J} - H (n + 1)(2n + l)a,
1 {(rl + 1)(2n + 3)U,(S - IS,) + ,F+‘}-%% D = Do - (n + l)(Zn + l)a,
(5.8)
(5.9)
where do, Bo, are constants of integration. It is easy to ascertain that when n = 0, (5.8) and (5.9) reduce to the model laws proposed by Kazakia and Venkataraman in [ 181 for solving a class of non-linear boundary value problems. Taking into account the available parameters occuring in (5.8) and (5.9), the comparison of the general constitutive laws obtained here with other experimentally or theoretically determined material responses can be carried on as in [18]. Of course, other classes of model D-E and B-H laws can be obtained in connection with different possible choices of functional forms of i(r,s), which satisfy (5.6). For instance the case n = 0, i.e. A = 0, leads also to the sets of constitutive equations already characterized in [4]. After (5.5), the variables r* and s* specialize to r* = :[z(r’
- rg) + id(r - r,)],
s* = f [&
- s;, + /T(s - so)].
(5.10)
Whereupon i.(r*,s*) takes the form j. = (r* + s*)2(n+ 1)
(5.11)
Taking into account (5.11) into (3.15), the solution of (2.5) is given by
t(r*,s*)= 5 +
&[(r* :‘;n+l] +&[(r*yg!+l]
(5.12)
34
C. CURRYand D. Fusco
whence further integration of (2.4) provides the following expression for x:
[n
+(-l),+‘
(2n + 2 -j)]{/M(s*)ds*
- IL(r*)dr*}.
(5.13)
j=l.n
As far as the theoretical framework of the conservation laws considered through the Section 4 is concerned, it is immediately seen that in the present case F = r while G = -x and q = 0. 6. CONCLUSIONS
AND REMARKS
In this paper we developed a systematic approach for obtaining general conditions to be satisfied by the coefficients of the basic system (1.1) in order that the corresponding hodograph equations reduce to suitable canonical forms allowing for an explicit integration. Also we provided a suitable mathematical vehicle for characterizing classes of material model laws. The system (1.1) encompasses a number of one-dimensional model of physical interest so that the procedure worked out here can be applied to analyze wave propagation problem arising from different contexts like non-linear elasticity, isentropic fluid-dynamics, dielectrics etc. However, as we already noticed in the introduction, beyond the scope of obtaining explicit solutions to the hodograph equations which can be of direct use in physical applications, the method of approach we developed here can be useful also for investigating the existence of conservation laws to the non-reducible system (1.2). Finally, within the context of the “geometrical shock dynamics” proposed by Whitham [19], we would like to remark that as far as two-dimensional wave propagation on the shock front is concerned, one is led to the following pair of quasilinear equations: d@ 1 dM z+;iT@T=
o
a@ ----=
o.
ag
iaA M aa
(6.1)
In (6.1) the family CL(X, ,x,) = a,t (a0 indisturbed sound speed), describes the successive shock positions whereas b(x,,x,) = const represent the rays (orthogonal trajectories to a = const). Moreover 0 = @(a, /?) is the ray inclination and A = A(M) denotes the ray tube area, with M Mach number. Since the system (6.1) is reducible, then all the results obtained in Sections 2 and 3 can be relevant also for studying the evolution of weak shock waves. Acknowledgements-This
work was supported by the C.N.R. through the G.N.F.M. REFERENCES
1. C. Rogers and W. F. Shadwick, Backlund Trangormations and Their Applicarions. Academic Press, New York (1982). 2. B. Sevmour and E. Varley, Exact solutions describing soliton-like interactions in a nondispersive medium. (SIAM)
J. appi. Math. 4i
804-821
(1982).
_
3. C. Currb and D. Fusco. On a class of auasilinear hyperbolic reducible systems allowing for special wave interactions, Z. angew. Math. Phys. 38,580-594, (198ij. 4. C. Rogers, H. M. Cekirge and A. Askar, Electromagnetic wave propagation in non-linear dielectric media. Acta Mech. 26, 59-73 (1977).
5. S. Bergman, Integral Operators in the Theory of Linear Partial Differential Equations, Vol. 23. Springer, Berlin (1971).
Quasilinear hyperbolic systems
35
6. K. W. Bauer and S. Ruscheweyh, Differential operators for partial differential equations and function theoretic applications. In Lecture Notes in Mothemurics, Vol. 791. Springer, Berlin (1980). 7. R. Courant and K. 0. Friedrichs, Supersonic Plow and Shock Waws. Interscience, New York (1948). 8. K. 0. Friedrichs, Conservation equations and the laws of motion in classical physics. Comm. pure appl. Math. XXXI, 123-131 (1978). 9. Y. Nutku, On a new class of completely integrable non-linear wave equations. I. Infinitely many conservation laws. J. Math. Phys. 26(6), 1237-1242 (1985). 10. G. Boillat and T. Ruggeri, Characteristic shocks: completely and strictly exceptional systems. Boll. U.M.I. 5, 15-A, 197-204 (1978). II. A. Donato and D. Fusco, Some applications of the Riemann method to electromagnetic wave propagation in non-linear media. 2. angew Math. Mech 60, 539-542 (1980). 12. D. Fusco, Generation of solutions and constitutive laws via the hodograph transformation, to appear in the Proceedings of the III Meeting on Wares and Srability, October 1985. Bari. 13. A. Jeffrey, Quasilinear Nyperbolic Systems and Woues. Pitman, London (1976). 14. P. D. Lax, Hyperbolic systems of conservation laws II. Comm. pure appl. Math. 10, 537-566 (1957). 15. C. Boillat, La Propagation des Ondes. Gauthier-Villars, Paris (1965). 16. T. Ruggeri and A. Strumia, Main field and convex covariant density for quasilinear hyperbolic systems. Annls fnsr. PoincarP 34, 65 (1981). 17. T. Ruggeri, Symmetric-hyperbolic system of conservative equations for a viscous heat-conducting fluid. Acra Mech. 47, 167-184 (1983). 18. J. Y. Kazakia and R. Venkataramam, Propagation of electromagnetic waves in a non-linear dielectric slab Z. angew. Math. Phys. 26, 61-76 (1975). 19. G. B. Whitham, Linear and Non-linear Waces. Wiley-Interscience, New York (1974).
Vol. 23. No. I. pp. 25-35.
0020-X62,88 Per&mm
1988
S3.00 + 0.00 Journals Ltd.
REDUCTION TO LINEAR CANONICAL’ FORMS AND GENERATION OF CONSERVATION LAWS FOR A CLASS OF QUASILINEAR HYPERBOLIC SYSTEMS C. CURRY and D. FUSCO* Dipartimento de Matematica, Universitl di Messina, Salita Sperone, contrada Papardo, Vill. S. pgata, 98010, Messina, Italy (Received 27 April 1987; received fir public&on 17 June 1987) Abstract-We consider a quasilinear hyperbolic homogeneous system of two first order equations involving two dependent and two independent variables. For the associated hodograph equations we investigate the reducibility to canonical forms allowing for an explicit integration. Such a kind of requirement, in problems of physical interest, provides a suitable method for characterizing possible material model laws. The theoretical approach shown herein can be relevant for studying the existence of conservation laws to non-homogeneous first order systems and also for describing the evolution of weak shock waves. I. INTRODUCTION
The reduction of the hodograph system to canonical forms which allow closed form integration has been investigated in several physical contexts. Such a method of approach has proved to be a useful tool for determining non-linear functional forms of the constitutive laws. A survey of results can be found in [l]. It must be noted also that the media so characterized are of relevant interest in non-linear wave propagation and especially in wave interactions phenomena; (see [2,3] and the references quoted there). In this paper we aim to develop a general procedure for obtaining explicit solutions to the hodograph system connected to a quasilinear hyperbolic system of first order:
u, + A(U)U, where
us
[Iu v
A(U)
’
=
= 0,
all [ a21
(1.1) al2 a22
1
and x, t are, respectively, space and time coordinates. Moreover, subscripts denote partial derivatives with respect to the indicated variable. Of course the hyperbolicity of (1.1) requires the matrix A to admit two real distinct eigenvalues E. and p (wavespeeds) with corresponding left eigenvectors 1(‘), VP)and right eigenvectors d(l), d@' spanning the Euclidean space E2. First we show that if J. and cc satisfy suitable conditions (which are usually fulfilled by the models of physical interest belonging to the class considered herein) then the problem of determining the solutions to the hodograph system arising from (1.1) is reduced to the integration of a linear second order equation involving a self-adjoint operator. Therefore, by generalizing to the case under investigation the method of approach shown in [l] and [4], a solution is sought in the form of a series whose coefficients are determined by solving recurrence relations. The uniform convergence of this series can be investigated, in principle, after the manner of Bergman [S, 63. Furthermore, in view of obtaining classes of solutions which can be of direct use in physical applications, we deduce the general condition to be satisfied in order that the series terminates. Such a request suggests also a class of canonical forms to which the hodograph system can be reduced. Moreover, through a suitable variable transformation, we establish a link with the canonical solutions already obtained
l Permanent address: Dipartimento di Matematica ed Applicazioni, Universita di Napoli, via Meuocannone 8. 80134 Napoli, Italy.
25
26
C.
CURRY
and D. Fusco
in [7] for a polytropic gas with special values of the adiabatic exponent. It must be pointed out that by requiring the basic model (1.1) to satisfy the conditions which allow the afore-mentioned Bergman-like series to terminate, one can generate possible sets of functional forms to the constitutive laws which are involved in a case of a physical interest. The canonical forms and the classes of explicit solutions to the hodograph system obtained in different physical contexts by several authors Cl-43 emerge from our analysis as particular cases. It is worth noting that the study we carry on herein can be viewed also within a theoretical framework which is different from the one of finding out solutions to the hodograph system arising from (1.1). In fact, in the second part of the paper, we consider also.the quasilinear hyperbolic nonhomogeneous system:
u, + A(U)U,
= B(u, x, t).
(1.2)
Thus we are able to show that, investigating the existence of conservation laws to (1.2), one is led to the integration of a linear system of equations which is similar to the hodograph system connected to the reducible model (1.1). Therefore our analysis fits into the theoretical approach considered in [8,9, lo] for constructing infinitely many conservation laws to a class of quasilinear nondispersive mathematical models. At the end of the paper we illustrate our procedure by means of a physical example. 2. GENERAL
REMARKS
ON THE
HODOGRAPH
SYSTEM
As is well known, the Riemann invariants related to (1.1) are defined by:
r =
q,l(*‘*dU,
s=
s
qJ”‘.dU I
(2.1)
where ~7,and q2 are integrating factors satisfying
(2.2) Thus, via the hodograph transformation x = x(r,s),
t = t(r, s)
(2.3)
under the assumption 8(x, t)/@,s) # 0, the system (1.1) can be reduced to the pair of linear equations:
x, = pt,,
x, = It,.
(2.4)
Whereupon, by eliminating x, the following linear second order equation is obtained:
.A+
Cm - 1. -
p
r
++ I A
-
p
t, = 0.
(2.5)
The classical Riemann method of integration provides the solution of the equation (2.5) in terms of a Riemann function and initial data specified along an initial line in the hodograph plane. However, according to [l 13, let us assume that the characteristic speeds 1 and p satisfy the condition:
(2.6)
27
Quasilinear hyperbolic systems
Hence, writing
g(r,s) = [&dr
+ *ds,
(2.7)
through the variable transformation
(2.8)
t* = exp(g)t
the equation (2.5) takes the form
(2.9) where the differential operator is self-adjoint. We remark that it is quite natural to require the condition (2.6) to hold. In fact, as already noticed in [l 1,123, all the models of physical interest belonging to the class (1.1) satisfy (2.6) without any restriction to the functional forms of the constitutive laws involved there. Now, generalizing to the case of interest the method of approach developed in [l] and L-43,a solution of (2.9) is sought in the form
r*(r, 4 = ; I
z {%h w,w + P.(r,W”(4),
(2.10)
0
where r,, /3” are to be determined, while cb., Y, satisfy the recurrence relations (2.11) Further substitution of (2.10) into (2.9) and use of (2.11) yields the following system of recurrence relations for a, and 8, (2.12)
azr -!
-$&+25I(r,s)or._,
=o
(2.13)
av
=o
(2.14)
-1
-+&+2!&A(r,s)B._r
where A(r,s) = -/
exp(g). Once A(r, s) is specified, the coefficients a,@,s) and /?“(r,s)
can be easily obtained by integrating (2.13) and (2.14). According to [4], the functions O., Y, satisfying (2.11) are given by
s Zr
Q)“(r)= ;
Y,(S) = ;
n
d@o
(2r - 5) --&l;
.
(2.15)
0
(2.16)
28
C. CURROand D. Fusco
where Q*(r) and Y,(S) are arbitrary functions. Alternative expressions for ctr,, Y, are
(2.17)
(2s - <)~2n-1~‘2dC,
n = 1,2,...
(2.18)
with
(2.19)
where K and L are arbitrary. Therefore the equation (2.9) admits the formal series solution (2.10) where Q,, 8, are determined by solving iteratively the recurrence relations (2.13), (2.14) and @#, Y, are given by (2.15), (2.16) [or alternatively by (2.17), (2.18)]. We notice that (2.10) can be considered a Berman-like series whose unif~~ convergence, in principle, can be studied within the theory developed in [S]. Moreover we refer to f6J for additional material on this subject. On account of the complexity of the solution obtained herein, we will deduce in the next section the conditions for which the Bergman series terminates. 3. CANONICAL
FORMS
AND EXPLlCIT
SOLUTIONS
TO THE HODOGRAPH
SYSTEM
The aim of this section is as follows. In order to obtain sets of explicit solutions to the hodograph system which can be of direct use in physical applications, we will require that the series solution (2.10) can be truncated after M+ 1 terms. In the process that requirement will suggest a wide class of canonical forms to which the hodograph system can be reduced. Furthermore we will establish a link with the canonical solutions to the hodograph system obtained in [7] in the case of a polytropic gas with special values of the adiabatic exponent. Let us assume in (2.10) that aO, c1i, . ,. , a, # 0, while a&= 0, k = n + 1,. . . . The same assumption is made, of course for the coefficients &,. Relation (2.12), shows that a0 = so(r). Hence if n = 0 (i.e. the series terminates after the first term), the conditions (2.13) require that A = 0.
(3.1)
This is the case mainly investigated in [43 in connection with non-linear dielectrics. Now let us consider the case n 2 1. In this instance the relations (2.13) specialize to: d2a.
*
+ 22_hcli_r
= 0,
i = 1,2,...,h
(3.21
(3.3) where a0 = so(r) is arbitrary. By defining
PO = A,
P, = A - &(ln
PO),. . . , Pi = A -~[ln(PbP:-‘...P,.,)],
(34J
29
Quasilinear hyperbolic systems it is not difficult to ascertain, after some algebra, that the compatibility determined system (3.2) and (3.3) requires
P,=A-
&
[ln(PeP;- '...P._I)1 = 0.
of the over-
(3.5)
For n = 1, [i.e.; truncation at the second term of the series (2,10)], the relation (3.5) takes the form:
A=
On introduction
&On4.
(3.6)
of w = In A, (3.6) specializes to the Liouville equation W,S=
exp(w).
(3.7)
As is well known (see for instance Cl]), equation (3.7) can be linearized and its solution is given by:
lnA=
expC(Xt(r) - X&))/21
w=2ln
(34
expCXk3ldS+ j ’ evC-X2K)ld5 I SO
where X,(r) and X,(s) are arbitrary functions and 6 is a non-zero constant known as a “Backlund parameter”. For n > 1 it does not seem to be possible to obtain the general expression for A which satisfies condition (3.5). However it is easy to show that if A satisfies the relation
mA =
&
Mm 41,
2 m=n(n
(3.9)
then (3.5) holds. (3.9) is again a Liouville equation which, after integration, provides the following expression for A:
evE&(r) - X2(41
(3.10)
expCXAS)ldS + f ’ expC - X,~~~ldS s 10
d2exp(g) Further substitution of A = drds exp(g) into (3.10) gives rise to the condition which, I [together with (2.6)], must be satisfied by the coefficients of the basic model (1.1) in order that the above results hold. That, of course, will impose suitable restrictions on the functional forms of the constitutive laws which are involved in a system of field equations describing a physical problem*. If A = Ot, then the equation (2.9) specializes to the wave equation t:, = 0
(3.11)
*The functions X,(r) and X,(s) arising from integration of (3.9) will be specified in order to characterize the response functions for the model of interest. *The case A = 0 can be recovered from (3.10) with n = 0.
30
C. CURROand D. Fusco
whose immediate integration yields t*(r,s) = L(r) + M(s)
(3.12)
where L.(r) and M(s) are arbitrary functions. Now we aim to characterize the canonical form taken by (2.9) in connection with the expression of A given by (3.10). To this end we introduce the change of variables
r* = -6 2
s
expW,(OkK
'0
s
s*= d
s
t*=i(r* f
exp(--X,(<))dS, 00
s*)“+‘. (3.13)
By means of (3.13), (2.9) reduces to
(3.14) The equation (3.14) is similar to the one obtained by R. Courant and K. 0. Friedrichs in the hodograph plane for a polytropic gas with special values of the adiabatic exponent 5 7 y given by y = 3, 51 ?.... $ On use of the results deduced in [73 the solution of (3.14) can be written in the form
c=
evk) (r*
+
s*y+
1
(3.15) c=~+&[(r*~~~“+l]+&i[(r*~~:;+l
with arbitrary functions L and M and any constant T. Whence, through (3.13) and (2.8), the corresponding expression for t(r,s) can be obtained. It is easy to see that the canonical forms of the hodograph system and the corresponding classes of mathematical models considered in different physical contexts by several authors Cl-43 (see also references quoted there), can be recovered by means of the method of approach we have already developed. As a very special case corresponding to n = 1 is the class of models characterized in 133 for studying special wave interactions. This class follows from X,(s) = 2&G
X,(r) = 2JKvr, r0
=-s,+-
’ ln[2cofi], 2JKv
6* = $ ,/$,
co, Y,K constants,
so that the general solution of (1.1) has the form
c(r,s) = [l(r) + m(s)]exp(-g) + 4v[L(r) + M(s)] where L(r) and M(s)are arbitrary functions, whereas I(r) and m(s) are determined by I(r) = g,
*dM
m(s) = 7.
However other choices of X,(r), X,(s), give other models for which the series (2.10) truncates at n = 1. Furthermore other values of n, and other choices of X,(r),‘X,(s) give many other classes of models for which the representation (2.10) terminates.
$ The limit case y = - 1 considered in [7] would correspond here to n = - 1 which leads to (3.11).
31
Quasilinear hyperbolic systems 4. CONSERVATION
LAWS
We will show here that the present approach to the integration of the hodograph system can be used also within the relevant theoretical framework for finding conservation laws to non-linear evolution equations. In fact, let us consider the quasilinear hyperbolic non-homogeneous system of first order (1.2):
u, + A(U)U,
= B(U,x,t).
(4.1)
The hyperbolic model (4.1) is not reducible [13]. However, in this case, by means of (2.1), the field Riemann variables r = r(U) and s = s(U) may also be introduced, and they satisfy r, + i.(r,s)r, = p(r, s, x, t)
(4.2)
s, + Ar,sh = b(r, s, x, t)
(4.3)
where p = (1’“’- B)q L, to= (1”’- B)q,. Now we shall deduce the conditions for system (4.1) to admit conservation form
Wr, 4 + Wr, s)
-
at
The compatibility
2X
=
laws of the
q(r, s, x, t).
of (4.2), (4.3) and (4.4) requires the following conditions to hold: G
r
=
l.F,,
Gs = PF,
q = pF, + OF,.
(4.5) (4.6)
The linear system of equations (4.5), which determines F and G is similar to the hodograph system (2.4). Furthermore, from (4.5) by eliminating G the following second order equation is obtained F,s+“IF,2. - p
-F,lb 1 - p
= 0.
(4.7)
Hence the results obtained in [8,9], in connection with the existence of conservation laws to certain classes of non-linear evolution equations, are easily recovered. However it is noteworthy that the analysis developed in [8], [9] deals with models which belong to (4.1) as particular cases, since they do not involve source-like terms. According to [S], we remark that the solution of equation (4.7) will, in general, depend on two arbitrary functions of two variables. Therefore (4.1) admits infinitely many conservation laws of the form (4.4). Obviously the method of approach already shown in the case of equation (2.5) still holds for the integration of (4.7). In particular, if (2.6) is satisfied then, in the case under investigation, we can define &r, s) = ln(i. - P) - g(r, s).
(4.8)
Thus, further investigation of the solution of (4.7) and possible reduction to canonical forms can be worked out as in Sections 2 and 3. Finally, let us assume that E.and p satisfy the conditions Vj_ . d(i)
=
0
=
VP.
d(P)
(4.9)
C. CURRY and D. Fusco
32
In this case the system (4.1) is “completely etceptional”, [13-153. It is easily seen [IO], that conditions (4.9) imply VI a 1"' and Vp a I(*.), sothat 1 = s and p = r. Therefore (4.7) specializes to the conventional wave equation (4.10)
C(r - s)Cl,, = 0
which gives rise to the class of conservation laws already obtained in [lo]. In closing we recall that an exhaustive literature on modern developments of the theory of quasilinear hyperbolic conservative systems which admit conservation law may be found in [16,17]. Remark
Since the solution of (4.4) will depend, in general, on two arbitrary functions it is possible to characterize two independent conservation equations like (4.4). Therefore, it is fairly obvious that a system like (4.1) can be written in infinitely many conservation forms. Such a feature may be relevant for studying shocks. That was already noticed in [lo] for the very special case of 2 x 2 homogeneous and completely exceptional systems. 5. PHYSICAL
EXAMPLE:
NON-LINEAR
DIELECTRICS
Maxwell’s equations in the case of an electromagnetic wave polarized in the x1, x2 plane and propagating along the direction of x3 = x can be written in the form (1.1) with
a= H,
v = E,
all = a2* =
42=
[
jjj
1
1
dD -’
dB -’
0,
v
a2,=
[
do
(5.1)
where E and D = D(E) are, respectively, the x’components of the electric field and electric displacement, while H and B = B(H) are the x*-components of the magnetic field and magnetic flux. Here the characteristic speeds are given by: p = _(B’D’)-“2
). = (B’D’)- I’*,
B’_E
,
D’
-dH’
dD
=dE
(5.2)
so that g = In A”*. Furthermore the Riemann invariants specialize to
s=f(e-h)
r=i(e+h),
(5.3)
where
e =
’ [D’(t)]“* s
H d&
h=
Eo
CB’(Ol I’*d5.
I HO
(5.4)
In connection with the analysis developed through Section 3, let us assume X,(r) = In(2rr + t?),
X*(s)
a, c?,/?, j? constants. Further insertion of (5.5) into (3.10) gives rise to
=
-
In(2bs + fl,
(5.5)
33
Quasilinear hyperbolic systems
where K=
- i(wi
+ Br,) - i(flsij +
hJ.
Several functional forms of i.(r,s) can be chosen in order to satisfy (5.6). In particular it is possible to see that a solution of (5.6) is given by i.(r,s) = [Maze + aJtM[Ma3h
+ ~~]~~~,a,,a,,a,,a,constants,
(5.7)
if M = n + 1 and the constants CX, d, B, p, K are such that 2 ‘1 = -(n + l)2a2a, 6 B = -(n + l)%J2U~ 2 s = -r&n+ l)(a,a, + a,a,) j? = (n +
i j&a,*, - a3a4)
K = adas.
Furthermore,
B=B,-
the integration of (5.7) leads to the following constitutive laws: 1 ((fl + 1)(2n + 3)a,(H - H,) + a:n+J} - H (n + 1)(2n + l)a,
1 {(rl + 1)(2n + 3)U,(S - IS,) + ,F+‘}-%% D = Do - (n + l)(Zn + l)a,
(5.8)
(5.9)
where do, Bo, are constants of integration. It is easy to ascertain that when n = 0, (5.8) and (5.9) reduce to the model laws proposed by Kazakia and Venkataraman in [ 181 for solving a class of non-linear boundary value problems. Taking into account the available parameters occuring in (5.8) and (5.9), the comparison of the general constitutive laws obtained here with other experimentally or theoretically determined material responses can be carried on as in [18]. Of course, other classes of model D-E and B-H laws can be obtained in connection with different possible choices of functional forms of i(r,s), which satisfy (5.6). For instance the case n = 0, i.e. A = 0, leads also to the sets of constitutive equations already characterized in [4]. After (5.5), the variables r* and s* specialize to r* = :[z(r’
- rg) + id(r - r,)],
s* = f [&
- s;, + /T(s - so)].
(5.10)
Whereupon i.(r*,s*) takes the form j. = (r* + s*)2(n+ 1)
(5.11)
Taking into account (5.11) into (3.15), the solution of (2.5) is given by
t(r*,s*)= 5 +
&[(r* :‘;n+l] +&[(r*yg!+l]
(5.12)
34
C. CURRYand D. Fusco
whence further integration of (2.4) provides the following expression for x:
[n
+(-l),+‘
(2n + 2 -j)]{/M(s*)ds*
- IL(r*)dr*}.
(5.13)
j=l.n
As far as the theoretical framework of the conservation laws considered through the Section 4 is concerned, it is immediately seen that in the present case F = r while G = -x and q = 0. 6. CONCLUSIONS
AND REMARKS
In this paper we developed a systematic approach for obtaining general conditions to be satisfied by the coefficients of the basic system (1.1) in order that the corresponding hodograph equations reduce to suitable canonical forms allowing for an explicit integration. Also we provided a suitable mathematical vehicle for characterizing classes of material model laws. The system (1.1) encompasses a number of one-dimensional model of physical interest so that the procedure worked out here can be applied to analyze wave propagation problem arising from different contexts like non-linear elasticity, isentropic fluid-dynamics, dielectrics etc. However, as we already noticed in the introduction, beyond the scope of obtaining explicit solutions to the hodograph equations which can be of direct use in physical applications, the method of approach we developed here can be useful also for investigating the existence of conservation laws to the non-reducible system (1.2). Finally, within the context of the “geometrical shock dynamics” proposed by Whitham [19], we would like to remark that as far as two-dimensional wave propagation on the shock front is concerned, one is led to the following pair of quasilinear equations: d@ 1 dM z+;iT@T=
o
a@ ----=
o.
ag
iaA M aa
(6.1)
In (6.1) the family CL(X, ,x,) = a,t (a0 indisturbed sound speed), describes the successive shock positions whereas b(x,,x,) = const represent the rays (orthogonal trajectories to a = const). Moreover 0 = @(a, /?) is the ray inclination and A = A(M) denotes the ray tube area, with M Mach number. Since the system (6.1) is reducible, then all the results obtained in Sections 2 and 3 can be relevant also for studying the evolution of weak shock waves. Acknowledgements-This
work was supported by the C.N.R. through the G.N.F.M. REFERENCES
1. C. Rogers and W. F. Shadwick, Backlund Trangormations and Their Applicarions. Academic Press, New York (1982). 2. B. Sevmour and E. Varley, Exact solutions describing soliton-like interactions in a nondispersive medium. (SIAM)
J. appi. Math. 4i
804-821
(1982).
_
3. C. Currb and D. Fusco. On a class of auasilinear hyperbolic reducible systems allowing for special wave interactions, Z. angew. Math. Phys. 38,580-594, (198ij. 4. C. Rogers, H. M. Cekirge and A. Askar, Electromagnetic wave propagation in non-linear dielectric media. Acta Mech. 26, 59-73 (1977).
5. S. Bergman, Integral Operators in the Theory of Linear Partial Differential Equations, Vol. 23. Springer, Berlin (1971).
Quasilinear hyperbolic systems
35
6. K. W. Bauer and S. Ruscheweyh, Differential operators for partial differential equations and function theoretic applications. In Lecture Notes in Mothemurics, Vol. 791. Springer, Berlin (1980). 7. R. Courant and K. 0. Friedrichs, Supersonic Plow and Shock Waws. Interscience, New York (1948). 8. K. 0. Friedrichs, Conservation equations and the laws of motion in classical physics. Comm. pure appl. Math. XXXI, 123-131 (1978). 9. Y. Nutku, On a new class of completely integrable non-linear wave equations. I. Infinitely many conservation laws. J. Math. Phys. 26(6), 1237-1242 (1985). 10. G. Boillat and T. Ruggeri, Characteristic shocks: completely and strictly exceptional systems. Boll. U.M.I. 5, 15-A, 197-204 (1978). II. A. Donato and D. Fusco, Some applications of the Riemann method to electromagnetic wave propagation in non-linear media. 2. angew Math. Mech 60, 539-542 (1980). 12. D. Fusco, Generation of solutions and constitutive laws via the hodograph transformation, to appear in the Proceedings of the III Meeting on Wares and Srability, October 1985. Bari. 13. A. Jeffrey, Quasilinear Nyperbolic Systems and Woues. Pitman, London (1976). 14. P. D. Lax, Hyperbolic systems of conservation laws II. Comm. pure appl. Math. 10, 537-566 (1957). 15. C. Boillat, La Propagation des Ondes. Gauthier-Villars, Paris (1965). 16. T. Ruggeri and A. Strumia, Main field and convex covariant density for quasilinear hyperbolic systems. Annls fnsr. PoincarP 34, 65 (1981). 17. T. Ruggeri, Symmetric-hyperbolic system of conservative equations for a viscous heat-conducting fluid. Acra Mech. 47, 167-184 (1983). 18. J. Y. Kazakia and R. Venkataramam, Propagation of electromagnetic waves in a non-linear dielectric slab Z. angew. Math. Phys. 26, 61-76 (1975). 19. G. B. Whitham, Linear and Non-linear Waces. Wiley-Interscience, New York (1974).