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A reduction method to quasilinear hyperbolic systems of multicomponent field PDEs with application to wave interaction
International Journal of Non-Linear Mechanics 37 (2002) 281}295
A reduction method to quasilinear hyperbolic systems of multicomponent "eld PDEs with application to wave interaction Carmela Curro`*, Domenico Fusco Dipartimento di Matematica, Universita di Messina, Salita Sperone 31, 98166 Messina, Italy Received 1 July 2000; accepted 12 October 2000 Dedicated to the memory of Professor A. Donato, our unforgettable friend and colleague
Abstract A reduction method is worked out for determining a class of exact solutions with inherent wave features to quasilinear hyperbolic homogeneous systems of N'2 "rst-order autonomous PDEs. A crucial point of the present approach is that in the process the original set of "eld equations induces the hyperbolicity of an auxiliary 2;2 subsystem and connection between the respective characteristic velocities can be established. The integration of this auxiliary subsystem via the hodograph method and through the use of the Riemann invariants provides the searched solutions to the full governing system. These solutions also represent invariant solutions associated with groups of translation of space/time coordinates and involving arbitrary functions that can be used for studying non-linear wave interaction. Within such a theoretical framework the two-dimensional motion of an adiabatic #uid is considered. For appropriate model pressure}entropy}density laws, we determine a solution to the governing system of equations which describes in the 2#1 space two non-linear waves which were initiated as plane waves, interact strongly on colliding but emerge with una!ected pro"le from the interaction region. These model material laws include the classical pressure}entropy}density law which is usually adopted for a polytropic #uid. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Hyperbolic systems; Double wave solutions; Reduction methods; Non linear wave interactions; Model constitutive laws
1. Introduction Over the last decades, the development of reduction techniques was proved to be a useful tool for determining exact solutions to non-linear system of
* Corresponding author. Tel.: #39-090-676-5067; fax: #39090-3502. E-mail addresses: [email protected] (C. Curro`), [email protected] (D. Fusco).
PDEs. In general, these procedures are based either upon group methods (classical or non-classical) or upon `ad hoca and direct approaches which permit to look for solutions of special form to a given governing model. A large body of literature is available on this subject so that any list of references is very far from being exhaustive. However, the guidelines along which the di!erent investigations were carried on can be found in [1}19]. As far as the application of reduction techniques to solving problems of physical interest is
0020-7462/02/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 0 ) 0 0 1 1 2 - 8
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concerned, it is worth noticing that often the reduction process also provides a mathematical vehicle for characterizing special classes of material response functions (model constitutive laws) which permit to obtain in a closed form the solution of classes of non-linear initial and/or boundary value problems connected with the existence of relevant wave pro"les. On that concern recent contributions within the framework of non-linear wave propagation can be found in [20}22]. In this paper we consider a quasi-linear hyperbolic system of "rst-order PDEs of the form AA
U "0, x A
(1)
where x ("0,2, m) denote the independent A variables; and U"U(x )3R, is a column vector A representing the "eld while AA are N-order square matrices. Moreover, a subscript means for partial derivative with respect to the indicated variable. We assume throughout the system (1) to be hyperbolic [23,24] with respect to the time coordinate x "t, that is for each unit space vector n"(n ), i"1,2, m, the equation G K det(!A#A )"0, A " AGn L L G G
(2)
has N real roots and the matrix A has a comG L plete set of left and right eigenvectors. It is well known that if N"2 and m"1, under the assumption of strict hyperbolicity the classical hodograph transformation interchanging the role of dependent and independent variables reduces the system (1) to a linear form for which the solutions can be searched by means of well-established techniques [25]. In many cases of physical interest further use of reduction approaches permits to characterize classes of 2;2 models allowing for special wave interactions [26}28]. Of course, when N'2 and/or m'1 the hodograph transformation is no longer valid and classes of exact solutions to (1) can be obtained by means of suitable group reductions. However, for investigating the evolution of non-linear wave processes and mainly wave interaction, solutions to the model in point are
usually found by means of asymptotic or approximate methods [29}31]. Here, our main aim is to outline a framework for obtaining exact solutions to (1) which also have distinguishing wave features as those exhibited by the solutions of the 2;2 systems given in terms of the classical Riemann invariants. Owing to the invariance under the action of the translation group, in Section 2 "rst we reduce the system (1) to an autonomous, homogeneous and hyperbolic system of PDEs involving two independent variables. Furthermore to the latter system a class of exact solutions is found through the integration of an auxiliary 2;2 hyperbolic model via a hodograph transformation whence solutions in a closed form to the original system of "eld equations (1) are obtained. In the process we show the connections between the characteristic velocities of the system (1) and the di!erent hyperbolic models which arise from the reduction. The material response functions which are involved in a given set of governing equations are required to adopt special form (model constitutive laws) in order to make consistent the approach in point. This aspect is illustrated in Section 3 where two-dimensional motion of an adiabatic #uid is considered. Although in the present theoretical setting several cases can be studied, later our attention is focused on the existence of solutions which describe special wave interaction in the #uid #ow.These solutions are consistent with model constitutive laws which include, as a particular case, the classical pressure}density relation characterizing a polytropic #uid [25].
2. Reduction procedure The autonomous system (1) is invariant under the action of the one-parameter () in"nitesimal translation group x "x , A A
(3)
x "x # , O, N N N
(4)
where are constants and x is a chosen space or N A time coordinate. The invariant solutions of (1)
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where, in line with (8), we put
corresponding to (3) and (4) are of the form U"U(x , ), " x N N A N$A
(5)
and they are de"ned by the system AA
U U #AI "0, x A
u , I " u u
(6)
BI BM
, u
, AI I U"
CI CM
(11)
with BI "[B ], CI "[C ], JK JK (7)
Because of the hyperbolicity of the system (1) the matrix A is non-singular whence if x "x , the A system (6) also results to be hyperbolic. Alternatively, if the time coordinate x is included in the similarity variable then the system (6) is hyperbolic with respect to the direction provided that det(AI )O0 which is tantamount to require that none of the eigenvalues of ( AN) with respect N$ N to A coincides with . In the following we will assume the latter case to hold. Owing to the invariance, the characteristic surfaces of the system (1) corresponding to the class of solutions (5) transform into characteristic curves of the system (6), (see [2]). Of course, the characteristic velocities of the latter system are obtained by those of (1) evaluated at the solutions of (6). In view of obtaining solutions to (1) in a closed form by means of hodograph-like techniques we search for the following class of solutions to (6): u "u (x , ), u "u (x , ), A A u "u (u , u ), h"3,2, N. FY FY
V"
AA I U"
where AI " ( AN). N N$A
283
(8)
Insertion of (8) into (6) yields the overdetermined system V V BI #CI "0, x A
(9)
V V BM #CM "0, x A
(10)
BM "[B ], CM "[C ], IYK IYK
(12)
u FY , B "AA #AA JK JFY u JK K u FY , C "AI #AI JK JK JFY u K
(13)
u FY , B "AA #AA IYK IYK IYFY u K u FY , C "AI #AI IYK IYK IYFY u K l, m"1, 2; k, h"3,2, N.
(14)
Next, we assume that the two equations (9) are independent while we require the remaining N!2 equations (10) to be identically satis"ed by any solution V"V(x , ) of the system (9). Hence the A following compatibility conditions must be ful"lled: ¸BI "BM , ¸CI "CM ,
(15)
¸"[¸ ] h"3,2, N, l"1, 2, FYJ
(16)
where the matrix ¸ is a Lagrange multiplier. The overdetermined set of 4(N!2) conditions (15) characterizes the N!2 functions u "u (u , u ), FY FY along with the 2(N!2) components of the matrix ¸ in order that any solution V"V(x , ) of (9) A provides through (8) a solution of the original system (1).
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We prove the following. Proposition. The hyperbolicity of the system (6) induces the hyperbolicity of the system (9) with respect to the -direction. Proof. Owing to the assumed hyperbolicity, at any solution of the system (6) the characteristic equation det(AA !AI )"0
(17)
yields N real eigenvalues '(U) (I"1,2, N) of AA to which there correspond N left and N right eigenvectors spanning the Euclidean space E,. Thus, bearing in mind the consistency conditions (15), the use of a well-known property of linear algebra shows that at the class of solutions (8), Eq. (17) specializes to det(AA !AI )"det(BI !CI ) det(Q!P),
(18)
Q"[Q ], Q "AA !¸ AA , FYIY FYIY FYIY FYJ JIY
(19)
P"[P ], P "AI !¸ AI , FYIY FYIY FYIY FYJ JIY
(20)
k, h"3,2, N, whereupon two of the characteristic speeds, say and , of the spectrum of 's are the eigenvalues of BI with respect to the matrix CI . Of course, and are selected among the 's by the relations (8) and by the pair of equations (9) which have been assumed to be independent. Let lH (" , ), be a left eigenvector of AA : lH(AA !AI )"0.
(21)
If we split lH into two parts lH,[lK H, lM H], lK H3R, lM H3R,\ a right multiplying of (21) by I U yields (lK H#lM H¸) (BI !CI )"0,
(22)
whereupon l H"lK H#l H¸ is the left eigenvector of the matrix BI corresponding to the eigenvalue . Since the left eigenvectors lH and lH of AA are linearly independent, the system (9) results to be hyperbolic.
Remark. According to the above proposition we notice that if dI H(" , ), is a right eigenvector of BI (BI !CI )dI H"0,
(23)
then dH"(I U)dI H
(24)
is the right eigenvector of AA associated with the eigenvalue : (AA !AI )dH"0,
(25)
whence ) dH"I ) dI H,
"
u
u
2
u ,
(26)
so that the genuine non-linearity or the exceptionality of an eigenvalue is preserved by the reduction process which was developed hitherto. Since we aim at constructing exact solutions to (1) which exhibit a wave behaviour, we require (9) to be strictly hyperbolic that is tantamount to assume the matrix BI to admit two real and distinct eigenvalues and , whereupon via the hodo graph transformation it is possible to determine the solutions of (9) by means of the Riemann invariants [24,25]:
r" l H dU, s" l H dU.
(27)
Therefore, the insertion of V(r, s) into the relations (8) gives rise, through (3)}(5), to a class of solutions to (1) which describe in the x -space wave processes A in terms of the wave fronts associated with the characteristic velocities and , respectively. On this concern in Section 3, our attention will be focused on investigating special wave interaction processes for the system of balance equations of adiabatic two-dimensional #uid dynamics.
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Setting "x !t, is a constant, in the present case, (6) specializes to
3. Adiabatic two-dimensional 6uid The adiabatic two-dimensional motion of a #uid is described by the following set of PDEs: u v #u #v # # "0, t x x x x
(28)
u u u 1 p 1 p S #u #v # # "0, t x x x S x (29) v v v 1 p 1 p S #u #v # # "0, x x x S x t (30) S S S #u #v "0, x x t
(31)
where " (x , x , t) is the mass density, S"S(x , x , t) is the entropy density, while u"u(x , x , t) and v"v(x , x , t) are the compo nents of the #uid velocity along the axes x and x , respectively; moreover, p"p( , S) denotes the pressure and the condition p/ '0 is required to hold in order the governing model in point to be hyperbolic [23,24]. The system of equations (28)}(31) can be cast in the form (1) with
U"
u v
, A"I,
285
(32)
U U (A!I) #A "0. x
(34)
Hereafter it is assumed that none of the eigenvalues of the matrix A coincides with , whereupon (34) can also be written in the form U U #A "0, x
(35)
where A"(A!I)\A. In passing we notice that the solutions U( , x ), we are considering herein include the travelling wave solutions of the governing system of a onedimensional #uid motion in the x direction. The eigenvalues of the matrix A are given by
v(u!)$(p/ (v#(u!)!p/ ) " , ! (u!)!p/ (36)
v " u!
(double).
(37)
Since the system (34) as the system of equations (28)}(31) can be written in conservative form, the eigenvalue turns out to be exceptional [32]. In view of the approach which was outlined in Section 2 we search for solutions of the following form to the system (35):
S
u
/N A" M /M 0
A"
0
0
0
0 /N M /1 , v 0
0
0
0
u
v
0
0
0
v
0
0
/N M /M 0
0
v
0
0
/N M /1 v
u
u"U( , S), v"V( , S).
.
(38)
By assuming " ( , x ) and S"S( , x ) to be any solution of the 2;2 system V V #D "0, x
(39)
where (33)
V"
S
, D"
H 0
H
V U
\J
(40)
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where
and
V p (U!)V b " , b "! ,
U V #a , H "! a #a U V H "! a #a #a , S S
(41)
V , c "! (U!)
(U!) a "! ,
(42)
V p p a "! , "(U!)! . (U!) S
(43)
In the present case the compatibility conditions (15) give rise to the overdetermined set of equations
U U V U #a #(a !b ) V !b !b "0,
a
(44)
a
u"U( , S)"#(G cos M,
(50)
v"V( , S)"(G sin M,
(51)
where 1 p d #F(S),
(52)
M( , S)" (45)
(49)
We notice that the Lagrange multiplier matrix ¸ which is involved in the relations (15) here specializes to BM because in the system (40) BI is the unit matrix. A direct inspection shows that the integration of the set of conditions (44)}(47) yields
G( , S)"!2
U V U a ! !b S U!
1 p c "! . (U!) S
V U V #a V #(a !c ) !c "0,
(48)
1 p c "! , (U!)
V (U!)V , a " , a "!
a
1 p V p b " , b " , S
K(S)G
1 2G
!
G
G 2G # d .
(53)
F(S) and K(S) are arbitrary functions, the condition (/ ) ( G)'0 is required to hold and the following equation must be ful"lled:
U V U # a !b !b #a "0, S
(46)
V U V V V #a !c #a "0, S S
(47)
M M tan M G M tan M M p # # S 2G S G S 1 G p 1 p ! ! "0. 2 G S G S
(54)
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Taking into account (50) and (51), it is easily seen that the matrix D admits the eigenvalues (M/ )#sin M cos M "H " , cos M#( /2G)(G/ )
whose integration yields the general solution x !H (r, s) "w (r, s),
(63)
x !r "w (r, s),
(64)
where
U "tan M " V!
(55)
to which there correspond the left eigenvectors
H , H !tan M
l H " 0
1 , tan M!H
H w (r, s)"(r!H ) !¹ !¹ s
(56)
w (r, s)"(H !r) !¹ H !(H !r)¹ !¹ s s
and the right eigenvectors
d H "
1 0
,
H d H " , tan M!H
, S
(58)
whereupon M"M(p)
(59)
and Eq. (54) results to be identically satis"ed. Owing to the condition (58) we have &l H so that the Riemann variables (27) in the present case are given by r"tan M, s"S.
(60)
Thus through the standard hodograph transformation [25]: " (r, s), x "x (r, s),
(61)
the system (39) can be reduced to the linear system x x "r , "H (r, s) , r r s s
(H ) ds, ! r¹ ds# ¹ s s
(57)
so that (39) is strictly hyperbolic. The eigenvalues and result from the characteristic velocities (36) and (37), respectively. Since (37) is exceptional, in line with the remark of Section 2, here the consistency of the full reduction process requires that ) d H "0, "
(H ) ds, (65) !H ¹ ds# ¹ s s
l H " 1
287
d¹ (r)" , dr
d¹ (s)" , ds
(66)
(67)
¹ "¹ (r), ¹ "¹ (s),
(68)
while
1 (r, s)" exp r!H (r, s)
dr . r!H (r, s)
(69)
The functions (x , ) and S(x , ) which are de "ned by (63) and (64) along with the relations (50) and (51) provide an exact solution of the system (28)}(31). It is worth noticing that the arbitrariness of the functions ¹ (s), ¹ (r) makes the class of solutions in point #exible to "t with initial and/or boundary conditions. That is relevant to our later analysis of wave interaction. The form of the response function p"p( , S) which allows the reduction approach developed hitherto to hold is selected by the consistency of the overdetermined set of conditions (52), (53) and (59). Several cases of model pressure}density}entropy law are considered below: (i) Assume
(62)
p( , S)"p (S) #p (S).
(70)
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288
(i1) (1. Substitution of (70) into (52) and (53), taking into account that M"M(p), gives rise to
2 G( , S)" p (S) \#F(S), 1!
p "p "const.,
M(p)"
1 F(S)"0, K(S)"k G 2
1# ln p (S), 1!
2 G( , S)" p (S) \, 1! 1 M(p)"G 2
(71)
1# 1# #F (p!p )\ $ arctan 1! !1
k Garctan
k "const.,
(76)
(1!)#(1!)F (p!p )\ , (!1)
(77)
where p (S) is arbitrary and the following condition must be ful"lled:
1# 2 ln 1! 1!
!1 # ln p!p #k
(72)
or, alternatively, to p "p "const., F(S)"F [p (S)] '0, K(S)"k "const., F "const., 2 G( , S)" p (S) \#F(S), 1!
(73)
p!p (
(!1)F #1
\ .
Of course, relation (70) with '1 includes as a very special case the p! law where is given by "(2n#1)/(2n!1), n3N. These values of the adiabatic exponent were considered in [25] for obtaining explicit integration of the hodograph equations associated with the governing system of an isentropic one-dimensional gas dynamics. (i3) "1. Here we have p"p "const.,
M(p)"
k Garctan
1# 1 #F (p!p )\ $ 1! 2
1# ln 1!
((1!)#F (1!)(p!p )\ #(1! (74) ((1!)#F (1!)(p!p )\ !(1!
with p (S) arbitrary. (i2) '1. Here the consistency of (52), (53) and (59) yields p "p "const., F(S)"F [p (S)] , F "const.,
F(S)"!p (S)[2 ln p #F ], K(S)"k "const., F "const., G( , S)"!2p (S)ln #F(S),
(78)
M(p)"k G(1#F #2 ln(p!p ) $arctan(1#F #2 ln(p!p ),
(79)
where p (S) is arbitrary and the following condition must be ful"lled:
K(S)"k "const., (75)
p!p ((exp(!1!F ).
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(i4) "!1 (Von-KaH rmaH n}Tsien law) In the present case we have p"p "const., F F(S)"! , K(S)"k "const., p (S) F "const., p (S) G( , S)"! #F(S),
(80)
F . M(p)"k $arctan p!p
(81)
As well known [33}35], the relation (70) with "!1 makes the system of equations (28)}(31) completely exceptional and so the subsystem (9) does. As far as the double sign in the M(p) expression is concerned, a direct inspection shows that only the positive case must be considered, otherwise the eigenvalue does not result to be exceptional. (ii) By assuming M(p)"$G p,
289
graph system (62) provides an exact solution of a corresponding initial and/or boundary value problem associated with the original set of the governing equations (28)}(31). Within such a context, here we study simple wave interaction in the ( , x )-plane by means of the general solutions (63) and (64) and later we consider the corresponding wave processes which are described by the full set of equations governing the #uid motion. Along the lines of the approach worked out in [26] and later in a more systematic way in [27], it is appropriate for studying pulse interaction to proceed in terms of the characteristic curves CH , CH de"ned, respectively, by ( , x )"const. and ( , x )"const., where / ! " , /x
/ ! " . /x
(85)
Bearing in mind the Riemann system (62) there result r"r() and s"s() whereby the representations of the general solution (63) and (64) in terms of the characteristic parameters and can be obtained. As far as the Cauchy problem is concerned, it is convenient to normalize and as follows:
(82) at "0,
we get 1 log #GI (S), p" 2G
(83)
1 G( , S)" , G
(84)
where G is a constant and GI is an arbitrary func tion while F and K specialize to F"0, K(S)"Glog G $G GI (S). 4. Wave interaction An inherent feature of the present approach is that solving a given initial and/or boundary value problem to the reduced model (39) or to the hodo-
"x
and "x ,
(86)
whereupon the initial data for r and s at "0 can be expressed in the form: r"r(x ), s"s(x ) for!R(x (#R. (87) Once the initial value problem for r and s has been given the relations (63) and (64) determine the functions ¹ (), ¹ (), () and (). In view of describing in the ( , x )-plane the interaction of two opposite travelling simple waves and according to the analysis carried out in [27], we assume that at "0, the right travelling pulse occupies the region x )x )!x and the left travelling pulse the J D region x )x )x (see Fig. 1). Both pulses D P propagate into a region of constant state where r"r and s"s . We also require that r(x ) and
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290
¹ (x )"!x #(x )(Rx ) !H (R(x ), s ))
(x )"(x )
(x )"!
V
d , (R()!H ())() (92)
V
d , (R()!H ())()
V
d . (R()!H ())()
(93)
(94)
For x )x , J (95)
¹ "0, Fig. 1. Qualitative description of wave interaction in the ( , x ) plane for the general class of p"p( , S) laws de"ned by the conditions (52), (53) and (59).
¹ "¹ (x )"!x J J J # (r !(H ) )
s(x ) are continuous. Thus, at "0
VJ
d , (R()!H ())() (96)
(88)
x #¹ J , (x )" r !(H )
(97)
(89)
x #¹ J . (x )" ((H ) !r )
(98)
r"
s"
R(x ), x )x )!x , J D r otherwise, S(x ), x )x )x , D P s otherwise,
where R(x ) and S(x ) are di!erentiable functions. Owing to (88) and (89) from the solutions (63) and (64) we obtain: For x )x , D
For x )x )x , D P (99)
¹ "0, ¹ (x )"S(x )
¹ "¹ "0, x (x )" , r !(H ) x (x )" . ((H ) !r )
! (90)
For x )x )!x , J D ¹ "0,
(91)
(x )"
V
d , (H ()!r )()
V
S() d , (H ()!r )()
V
d , (H ()!r )()
(x )"!
V
d . (H ()!r )
(100)
(101)
(102)
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For x *x , P (103)
¹ "0, ¹ "¹ (x )"s P P
VP
d , (H ()!r )() VP
S() d , ! (H ()!r )()
x #((H /s)#(H !r )/s) ¹ P , (x )" (H !r ) (105) x #(H /s) ¹ P . (x )" (r !H )
(106)
There are "ve distinct regions in the ( , x )-plane where r and s are nonzero (see Fig. 1). The left travelling pulse traverses regions I , II and III . In * * region I we have * x !r ",
(107)
so that the pulse is a simple wave. It interacts with the left travelling pulse in region II where the pulse is no longer a simple wave. Finally, in region III * the left travelling pulse is a simple wave characterized by the following relation: x !r "!¹ . J
(108)
The comparison of (107) and (108) shows that the simple wave which emerges from the interaction region is identical with that produced by the initial conditions: at "0,
r"
s"
R(x ), x )x )!x , J D r otherwise,
A similar argument for the right travelling pulse in regions I and III leads to the results: 0 0 in region I 0 x !H () ",
(104)
(109)
S(x #¹ ), x #¹ )x )x #¹ , J D J P J s otherwise. (110)
Thus, the only e!ect of the interaction on the left travelling emerging wave is to change the origin of x in the original pulse.
291
(111)
in region III , 0 H x !H () "!() ()¹ . P s
(112)
Therefore, if the material response function p"p( , S) satis"es the conditions (52), (53)) and (59), unlike the case of the left travelling pulse in general, the right travelling simple wave emerges from region II with an altered pro"le, unless p( , S) is required to obey further restrictions. In [26}28] there were thoroughly investigated classes of 2;2 quasilinear hyperbolic systems allowing for wave interaction where the emerging pulse keeps its own original pro"le. The completely exceptional systems (CEX) [34,35] are therein included and that case here corresponds to the model law (70) with "!1 (case (i4) of Section 3) for which I ) d H "0. However, apart from the von-KaH rmaH n}Tsien law which makes CEX the subsystem (39), within the present reduction framework it is a simple matter to ascertain that provided p( , S) adopts the form (70) with O!1 (cases (i1)}(i3) or the form (83) we have I ) d H O0 and H "H (r) whence, because of the relation (112), the pulse of characteristic velocity is una!ected by the interaction process as well (see Fig. 2). On the latter cases of model constitutive laws a remark is in order. Quasilinear hyperbolic 2;2 systems of physical interest exhibiting only one exceptional characteristic eigenvalue do not usually occur. Nevertheless the system (39) is not expected here to have a physical meaning but it represents only an intermediate although crucial step of our approach for determining exact wave-like solutions to the original system of two-dimensional adiabatic #uid dynamics. In order to illustrate how the Cauchy problem (88) and (89) generates a wave interaction process in the (x , x , t)-space, "rst we observe that the characteristic curves C? : ( , x )"const. and
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Fig. 2. Case of the model laws given by the relations (70) or (83): the interaction does not a!ect the pro"les of both emerging pulses.
C@ : ( , x )"const. are included in the family of characteristic curves of the system (34) and the latter ones, because of the underlying invariance transform into characteristic surfaces of the system (28)}(31), (see [2]). Consequently r"r() and s"s() result to be invariant on corresponding characteristic surfaces in the (x , x , t)-space. Hence the solutions U"U(r(), s()) determined to (28)}(31) within the present theoretical framework are expressed in terms of two-wave variables. These solutions also involve arbitrary functions so that they can be used for describing multidimensional interaction of two simple waves provided, suitable initial/boundary data are associated with the system (28)}(31). In passing we notice that the simple wave-like exact solutions which are determined to hyperbolic systems of PDEs in more space coordinates by other methods of approach (e.g. see [36]), depend upon only one variable so that, in general, they are not of direct use for studying multidimensional wave interaction. It is straightforward to see that the Cauchy problem (88) and (89) corresponds in the (x , x , t)-space to prescribe the data on the plane x !t" 0, where the two original pulses are assumed to be located in the strips x )x )!x and P D
Fig. 3. The interaction in the (x , x , t)-space: the pulse travel ling at characteristic velocity " emerges with an altered pro"le.
x )x )x , respectively. Outside of these strips D J a constant state is assumed to hold. Bearing in mind that U"U(r(), s()), by direct extension to the present multidimensional case of the analysis which was previously worked out in the !x plane it is easy to ascertain that in the strips de"ned by the characteristic planes x !(H ) (x !t)"x , J x !(H ) (x !t)"!x D and by the characteristic planes
(113)
x !r (x !t)"x , D x !r (x !t)"x , (114) P respectively, there result to be two separate regions of plane (simple) waves initiated at x "t (see Fig. 3). These pulses interact in a region where they
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293
if the two waves interact before the occurrence of any singularity or shock formation.
5. Conclusions and 5nal remarks
Fig. 4. The pro"le of the pulse travelling at characteristic velocity " is not in#uenced by the interaction.
are no longer plane waves. Later the pulses emerge from the interaction region as plane waves. Obviously, if the material response function p"p ( , S) is of form (70) or (83) then the pro"les of both emerging pulses are not a!ected by the interaction (Fig. 4). In the latter case the interaction process in point is similar to the one occurring in non-linear dispersive media when two solitons collide but emerge with unchanged pro"le from the interaction. Of course, the non-linear hyperbolic waves are subjected to the pro"le distorsion. However, in the present context the plane wave propagating with the exceptional characteristic speed (37) never evolves into a non-linear shock after a "nite time has elapsed whereas for the other wave a pro"le breakdown has to be expected. Hence, the analysis of pulse interaction under interest makes sense only
We have presented a systematic approach to determine a class of solutions of the system (1) and which, to a certain extent, exhibit wave features as the Riemann invariants associated with 2;2 hyperbolic systems do. The approach proceeds in two steps. First the invariance of the autonomous system (1) under the action of the group (3) and (4) permits to reduce it to a hyperbolic system of form (6). Next, searching for the solutions (8) to the system (6) leads to considering the overdetermined set of Eqs. (9) and (10), where (9) is assumed to represent a 2;2 system determining u and u while (10) has to be identically satis"ed. Hence, through the relations (8) a class of exact solutions to (1) is obtained. Of course, the "rst step based upon invariance reduction is not to be considered if the system (1) involves only two independent variables. Since U"U(u , u ), the structure of the solu tions to system (1) considered herein resembles that of the partially invariant (double wave) solutions studied in [2,37,38], with the functions u and u playing the role of wave parameters. However, on this subject we observe "rst that, because of the invariance reduction (3) and (4), the solutions (8) are in fact invariant solutions of the system (1). Furthermore, unlike the procedure worked out to determine double-wave solutions here we do not aim at analyzing all the di!erential conditions arising from the compatibility of the overdetermined system (9) and (10) but rather we treat (10) as a set of supplementary equations [35,39}41] which must be identically satis"ed by any solutions u and u of the system (9) as well as by the functions u "u (u , u ). That permits us to construct soluFY FY tions to systems of form (1) by means of the wellestablished methods of integrating 2;2 hyperbolic systems and which are of more direct use for solving problems of interest in non-linear wave propagation. A striking feature of the present approach is that the hyperbolicity of the system (6) (or equivalently,
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of the system (1)), induces the hyperbolicity of the subsystem (9) whose characteristic velocities inherit all the properties of the corresponding characteristic eigenvalues of the system (6). By assuming strict hyperbolicity, the integration of (9) is accomplished by means of the hodograph transformation. Within such a theoretical framework the two-dimensional motion of an adiabatic #uid was investigated. Di!erent structural requirements can be made to the system (9) in order to integrate in a closed form the associated hodograph equations and to provide exact solutions involving arbitrary functions which can "t with prescribed initial and/or boundary value problems. However, our attention was mainly focused on the existence of an exceptional characteristic eigenvalue resulting from a double eigenvalue (exceptional) of the spectrum of the characteristic velocities of the system of balance equations (28)}(31). In such a case several classes of solutions are provided by the hodograph method. In particular we considered solutions which represent in ( , x )-plane two opposite travelling waves which interact strongly on colliding. In general, only the wave travelling with the exceptional velocity emerges with una!ected pro"le from the interaction region. However, if the material response function p"p ( , S) adopts the form (70) or (83) then the pro"les of both waves are not altered by the interaction process. These solutions through the relations (5) and (8) generate a class of exact solutions of the original system (28)}(31) and describe in the (x , x , t)-space two non-linear waves which were initiated as plane waves and exhibit in interacting a behaviour similar to that occurring in non-linear dispersive media when two solitons collide. Finally, as the solutions of the 2;2 system (9) can be superimposed via the hodograph transformation, we remark that the solutions to systems of form (1) which are obtained by means of the present approach in fact obey a non-linear superposition principle. Acknowledgements This work was supported in part by Consiglio Nazionale delle Ricerche through grants No.
96.03851.CT01 and No. 97.00888.CT01 and by Ministero della Universita` e della Ricerca Scienti"ca e Tecnologica (MURST-fondi 40% and 60%).
References [1] C. Rogers, W.F. Shadwick, BaK clund Transformations and Their Applications, Academic Press, New York, 1982. [2] L.V. Ovsiannikov, Group Analysis of Di!erential Equations, Academic Press, New York, 1982. [3] N.N. Janenko, B.L. Rozdestvenskii, System of quasilinear equations and their applications to gas dynamics, Amer. Math. Soc. Providence, RI 55 (1983). [4] L. Dresner, Similarity Solutions of Non-Linear Partial Di!erential Equations, Res. Notes in Math., Vol. 88, Pitman, London, 1983. [5] P.J. Olver, Applications of Lie Groups of Di!erential Equations, Springer, New York, 1986. [6] G.W. Bluman, S. Kumei, Symmetries and Di!erential Equations, Springer, New York, 1989. [7] C. Rogers, W.F. Ames, Nonlinear Boundary Value Problems in Science and Engineering, Academic Press, New York, 1989. [8] W.F. Ames, C. Rogers (Eds.), Nonlinear Equations in the Applied Sciences, Academic Press, Boston, 1992. [9] W.I. Fushchych, W.M. Shtelen, N.I. Serov, Symmetry Analysis and Exact Solutions of Nonlinear Equations of Mathematical Physics, Kluwer Academic Publishers, Dordrecht, 1993. [10] N.H. Ibragimov (Ed.), CRC Handbook of Lie Group Analysis of Di!erential Equations, Vols. 1 & 2, CRC Press, Boca Raton, FL, 1994. [11] P.J. Olver, Symmetry and explicit solutions of partial di!erential equations, Appl. Numer. Math. 10 (1992) 307}324. [12] S.V. Meleshko, Di!erential constraints and one-parameter Lie-BaK cklund groups, Sov. Math. Dokl. 28 (1983) 37}41. [13] D. Fusco, N. Manganaro, A class of linearizable models and generation of material response functions to nonlinear hyperbolic heat conduction, J. Math. Phys. 32 (1991) 3043}3046. [14] D. Fusco, N. Manganaro, Generation of exact solutions to a class of quasilinear hyperbolic models via reduction techniques, Math. Comput. Modelling 18 (1993) 101}106. [15] D. Fusco, N. Manganaro, A method for determining exact solutions to a class of non-linear models based on introduction of di!erential constraints, J. Math. Phys. 35 (1994) 3659}3669. [16] A. Donato, F. Oliveri, Linearization procedure of nonlinear "rst order systems of PDE's by means of canonical variables related to Lie groups of point transformations, J. Math. Anal. Appl. 188 (1994) 552}568. [17] A. Donato, F. Oliveri, When nonautonomous equations are equivalent to autonomous ones, Appl. Anal. 58 (1995) 313}323.
C. Curro% , D. Fusco / International Journal of Non-Linear Mechanics 37 (2002) 281}295 [18] D. Fusco, Reduction Methods for 2 ; 2 Hyperbolic Systems of First Order PDEs, Quaderno C.N.R.-G.N.F.M. Vol. 48, 1995. [19] A. Donato, F. Oliveri, How to build up variable transformations allowing one to map non-linear hyperbolic equations into autonomous or linear ones, Transport Theory Statist. Phys. 25 (1996) 303}322. [20] D. Fusco, N. Manganaro, A method for "nding exact solutions to hyperbolic systems of "rst order PDEs, IMA J. Appl. Math. 57 (1996) 223}242. [21] D. Fusco, N. Manganaro, M. Migliardo, Pulse-like exact solutions for a model describing nerve pulse transmission, Stud. Appl. Math., to appear. [22] S. C[ anicH , B.L. Key"tz, Quasi-one-dimensional Riemann problems and their role in self-similar two-dimensional problems, Arch. Rational Mech. Anal. 144 (1998) 233}258. [23] G. Boillat, La Propagation des Ondes, Gauthier-Villars, Paris, 1965. [24] A. Je!rey, Quasilinear Hyperbolic Systems and Waves, Pitman Publishing, London, 1976. [25] R. Courant, K.O. Friedrichs, Supersonic Flows and Shock Waves, Interscience Publ, New York, 1962. [26] B.R. Seymour, E. Varley, Exact solutions describing soliton-like interactions in a non dispersive medium, SIAM J. Appl. Math. 42 (1982) 804}821. [27] C. Curro`, D. Fusco, On a class of quasilinear hyperbolic reducible systems allowing for special wave interactions, J. Appl. Math. Phys. ZAMP 38 (1987) 580}594. [28] C. Curro`, D. Fusco, N. Manganaro, Wave features of a class of reducible nonhomogeneous and nonautonomous models, Appl. Anal. 57 (1995) 47}62. [29] J.K. Hunter, A. Majda, R. Rosales, Resonantly interacting weakly non-linear hyperbolic waves. II: Several space variables, J. Stud. Appl. Math. 75 (1986) 187}226.
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[30] J.K. Hunter, J.B. Keller, Weakly nonlinear high frequency waves, Comm. Pure Appl. Math. XXXVI (1983) 547}569. [31] D.F. Parker, G.A. Maugin (Eds.), Recent Developments in Surface Acoustic Waves, Springer, Berlin, 1988. [32] G. Boillat, Chocs caracteH ristiques, C. R. Acad. Sci., Paris 274 A (1972) 1018}1021. [33] A. Greco, On the strict exceptionality for a subsonic #ow, Atti II Congresso AIMETA, Napoli 4 (1974) 127}134. [34] G. Boillat, T. Ruggeri, Characteristic shocks: completely and strictly exceptional systems, Boll. Un. Mat. Ital. (5) 15 A (1978) 197}204. [35] G. Boillat, in: T. Ruggeri (Ed.), Non Linear Fields and Waves in CIME Course Recent Mathematical Methods in Nonlinear Wave Propagation, Lecture Notes in Mathematics, Vol. 1640, Springer, Berlin, 1995, pp. 103}152. [36] A.M. Grundland, in: C. Rogers, T.B. Moodie (Eds.), Riemann Invariants in Wave Phenomena: Modern Theory and Applications, Elsevier Science Publishers, North-Holland, Amsterdam, 1984, pp. 123}152. [37] S.V. Meleshko, Homogeneous autonomous systems with three independent variables, J. Appl. Math. Mech. 58 (1994) 857}863. [38] A.M. Grundland, J. Tafel, Nonclassical symmetry reduction and Riemann wave solutions, J. Math. Anal. Appl. 198 (1996) 879}892. [39] G. Boillat, Sur l'existence et la recherche d'eH quations de conservation suppleH mentaires pour les syste`mes hyperboliques, C. R. Acad. Sci., Paris 278 A (1974) 909}912. [40] T. Ruggeri, A. Strumia, Main "eld and convex covariant density for quasi-linear hyperbolic systems, Ann. Inst, H. PoincareH A XXXIV (1981) 65}84. [41] I. MuK ller, T. Ruggeri, Extended Thermodynamics, Vol. 37, Springer, New York, 1993.
A reduction method to quasilinear hyperbolic systems of multicomponent "eld PDEs with application to wave interaction Carmela Curro`*, Domenico Fusco Dipartimento di Matematica, Universita di Messina, Salita Sperone 31, 98166 Messina, Italy Received 1 July 2000; accepted 12 October 2000 Dedicated to the memory of Professor A. Donato, our unforgettable friend and colleague
Abstract A reduction method is worked out for determining a class of exact solutions with inherent wave features to quasilinear hyperbolic homogeneous systems of N'2 "rst-order autonomous PDEs. A crucial point of the present approach is that in the process the original set of "eld equations induces the hyperbolicity of an auxiliary 2;2 subsystem and connection between the respective characteristic velocities can be established. The integration of this auxiliary subsystem via the hodograph method and through the use of the Riemann invariants provides the searched solutions to the full governing system. These solutions also represent invariant solutions associated with groups of translation of space/time coordinates and involving arbitrary functions that can be used for studying non-linear wave interaction. Within such a theoretical framework the two-dimensional motion of an adiabatic #uid is considered. For appropriate model pressure}entropy}density laws, we determine a solution to the governing system of equations which describes in the 2#1 space two non-linear waves which were initiated as plane waves, interact strongly on colliding but emerge with una!ected pro"le from the interaction region. These model material laws include the classical pressure}entropy}density law which is usually adopted for a polytropic #uid. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Hyperbolic systems; Double wave solutions; Reduction methods; Non linear wave interactions; Model constitutive laws
1. Introduction Over the last decades, the development of reduction techniques was proved to be a useful tool for determining exact solutions to non-linear system of
* Corresponding author. Tel.: #39-090-676-5067; fax: #39090-3502. E-mail addresses: [email protected] (C. Curro`), [email protected] (D. Fusco).
PDEs. In general, these procedures are based either upon group methods (classical or non-classical) or upon `ad hoca and direct approaches which permit to look for solutions of special form to a given governing model. A large body of literature is available on this subject so that any list of references is very far from being exhaustive. However, the guidelines along which the di!erent investigations were carried on can be found in [1}19]. As far as the application of reduction techniques to solving problems of physical interest is
0020-7462/02/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 0 ) 0 0 1 1 2 - 8
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concerned, it is worth noticing that often the reduction process also provides a mathematical vehicle for characterizing special classes of material response functions (model constitutive laws) which permit to obtain in a closed form the solution of classes of non-linear initial and/or boundary value problems connected with the existence of relevant wave pro"les. On that concern recent contributions within the framework of non-linear wave propagation can be found in [20}22]. In this paper we consider a quasi-linear hyperbolic system of "rst-order PDEs of the form AA
U "0, x A
(1)
where x ("0,2, m) denote the independent A variables; and U"U(x )3R, is a column vector A representing the "eld while AA are N-order square matrices. Moreover, a subscript means for partial derivative with respect to the indicated variable. We assume throughout the system (1) to be hyperbolic [23,24] with respect to the time coordinate x "t, that is for each unit space vector n"(n ), i"1,2, m, the equation G K det(!A#A )"0, A " AGn L L G G
(2)
has N real roots and the matrix A has a comG L plete set of left and right eigenvectors. It is well known that if N"2 and m"1, under the assumption of strict hyperbolicity the classical hodograph transformation interchanging the role of dependent and independent variables reduces the system (1) to a linear form for which the solutions can be searched by means of well-established techniques [25]. In many cases of physical interest further use of reduction approaches permits to characterize classes of 2;2 models allowing for special wave interactions [26}28]. Of course, when N'2 and/or m'1 the hodograph transformation is no longer valid and classes of exact solutions to (1) can be obtained by means of suitable group reductions. However, for investigating the evolution of non-linear wave processes and mainly wave interaction, solutions to the model in point are
usually found by means of asymptotic or approximate methods [29}31]. Here, our main aim is to outline a framework for obtaining exact solutions to (1) which also have distinguishing wave features as those exhibited by the solutions of the 2;2 systems given in terms of the classical Riemann invariants. Owing to the invariance under the action of the translation group, in Section 2 "rst we reduce the system (1) to an autonomous, homogeneous and hyperbolic system of PDEs involving two independent variables. Furthermore to the latter system a class of exact solutions is found through the integration of an auxiliary 2;2 hyperbolic model via a hodograph transformation whence solutions in a closed form to the original system of "eld equations (1) are obtained. In the process we show the connections between the characteristic velocities of the system (1) and the di!erent hyperbolic models which arise from the reduction. The material response functions which are involved in a given set of governing equations are required to adopt special form (model constitutive laws) in order to make consistent the approach in point. This aspect is illustrated in Section 3 where two-dimensional motion of an adiabatic #uid is considered. Although in the present theoretical setting several cases can be studied, later our attention is focused on the existence of solutions which describe special wave interaction in the #uid #ow.These solutions are consistent with model constitutive laws which include, as a particular case, the classical pressure}density relation characterizing a polytropic #uid [25].
2. Reduction procedure The autonomous system (1) is invariant under the action of the one-parameter () in"nitesimal translation group x "x , A A
(3)
x "x # , O, N N N
(4)
where are constants and x is a chosen space or N A time coordinate. The invariant solutions of (1)
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where, in line with (8), we put
corresponding to (3) and (4) are of the form U"U(x , ), " x N N A N$A
(5)
and they are de"ned by the system AA
U U #AI "0, x A
u , I " u u
(6)
BI BM
, u
, AI I U"
CI CM
(11)
with BI "[B ], CI "[C ], JK JK (7)
Because of the hyperbolicity of the system (1) the matrix A is non-singular whence if x "x , the A system (6) also results to be hyperbolic. Alternatively, if the time coordinate x is included in the similarity variable then the system (6) is hyperbolic with respect to the direction provided that det(AI )O0 which is tantamount to require that none of the eigenvalues of ( AN) with respect N$ N to A coincides with . In the following we will assume the latter case to hold. Owing to the invariance, the characteristic surfaces of the system (1) corresponding to the class of solutions (5) transform into characteristic curves of the system (6), (see [2]). Of course, the characteristic velocities of the latter system are obtained by those of (1) evaluated at the solutions of (6). In view of obtaining solutions to (1) in a closed form by means of hodograph-like techniques we search for the following class of solutions to (6): u "u (x , ), u "u (x , ), A A u "u (u , u ), h"3,2, N. FY FY
V"
AA I U"
where AI " ( AN). N N$A
283
(8)
Insertion of (8) into (6) yields the overdetermined system V V BI #CI "0, x A
(9)
V V BM #CM "0, x A
(10)
BM "[B ], CM "[C ], IYK IYK
(12)
u FY , B "AA #AA JK JFY u JK K u FY , C "AI #AI JK JK JFY u K
(13)
u FY , B "AA #AA IYK IYK IYFY u K u FY , C "AI #AI IYK IYK IYFY u K l, m"1, 2; k, h"3,2, N.
(14)
Next, we assume that the two equations (9) are independent while we require the remaining N!2 equations (10) to be identically satis"ed by any solution V"V(x , ) of the system (9). Hence the A following compatibility conditions must be ful"lled: ¸BI "BM , ¸CI "CM ,
(15)
¸"[¸ ] h"3,2, N, l"1, 2, FYJ
(16)
where the matrix ¸ is a Lagrange multiplier. The overdetermined set of 4(N!2) conditions (15) characterizes the N!2 functions u "u (u , u ), FY FY along with the 2(N!2) components of the matrix ¸ in order that any solution V"V(x , ) of (9) A provides through (8) a solution of the original system (1).
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We prove the following. Proposition. The hyperbolicity of the system (6) induces the hyperbolicity of the system (9) with respect to the -direction. Proof. Owing to the assumed hyperbolicity, at any solution of the system (6) the characteristic equation det(AA !AI )"0
(17)
yields N real eigenvalues '(U) (I"1,2, N) of AA to which there correspond N left and N right eigenvectors spanning the Euclidean space E,. Thus, bearing in mind the consistency conditions (15), the use of a well-known property of linear algebra shows that at the class of solutions (8), Eq. (17) specializes to det(AA !AI )"det(BI !CI ) det(Q!P),
(18)
Q"[Q ], Q "AA !¸ AA , FYIY FYIY FYIY FYJ JIY
(19)
P"[P ], P "AI !¸ AI , FYIY FYIY FYIY FYJ JIY
(20)
k, h"3,2, N, whereupon two of the characteristic speeds, say and , of the spectrum of 's are the eigenvalues of BI with respect to the matrix CI . Of course, and are selected among the 's by the relations (8) and by the pair of equations (9) which have been assumed to be independent. Let lH (" , ), be a left eigenvector of AA : lH(AA !AI )"0.
(21)
If we split lH into two parts lH,[lK H, lM H], lK H3R, lM H3R,\ a right multiplying of (21) by I U yields (lK H#lM H¸) (BI !CI )"0,
(22)
whereupon l H"lK H#l H¸ is the left eigenvector of the matrix BI corresponding to the eigenvalue . Since the left eigenvectors lH and lH of AA are linearly independent, the system (9) results to be hyperbolic.
Remark. According to the above proposition we notice that if dI H(" , ), is a right eigenvector of BI (BI !CI )dI H"0,
(23)
then dH"(I U)dI H
(24)
is the right eigenvector of AA associated with the eigenvalue : (AA !AI )dH"0,
(25)
whence ) dH"I ) dI H,
"
u
u
2
u ,
(26)
so that the genuine non-linearity or the exceptionality of an eigenvalue is preserved by the reduction process which was developed hitherto. Since we aim at constructing exact solutions to (1) which exhibit a wave behaviour, we require (9) to be strictly hyperbolic that is tantamount to assume the matrix BI to admit two real and distinct eigenvalues and , whereupon via the hodo graph transformation it is possible to determine the solutions of (9) by means of the Riemann invariants [24,25]:
r" l H dU, s" l H dU.
(27)
Therefore, the insertion of V(r, s) into the relations (8) gives rise, through (3)}(5), to a class of solutions to (1) which describe in the x -space wave processes A in terms of the wave fronts associated with the characteristic velocities and , respectively. On this concern in Section 3, our attention will be focused on investigating special wave interaction processes for the system of balance equations of adiabatic two-dimensional #uid dynamics.
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Setting "x !t, is a constant, in the present case, (6) specializes to
3. Adiabatic two-dimensional 6uid The adiabatic two-dimensional motion of a #uid is described by the following set of PDEs: u v #u #v # # "0, t x x x x
(28)
u u u 1 p 1 p S #u #v # # "0, t x x x S x (29) v v v 1 p 1 p S #u #v # # "0, x x x S x t (30) S S S #u #v "0, x x t
(31)
where " (x , x , t) is the mass density, S"S(x , x , t) is the entropy density, while u"u(x , x , t) and v"v(x , x , t) are the compo nents of the #uid velocity along the axes x and x , respectively; moreover, p"p( , S) denotes the pressure and the condition p/ '0 is required to hold in order the governing model in point to be hyperbolic [23,24]. The system of equations (28)}(31) can be cast in the form (1) with
U"
u v
, A"I,
285
(32)
U U (A!I) #A "0. x
(34)
Hereafter it is assumed that none of the eigenvalues of the matrix A coincides with , whereupon (34) can also be written in the form U U #A "0, x
(35)
where A"(A!I)\A. In passing we notice that the solutions U( , x ), we are considering herein include the travelling wave solutions of the governing system of a onedimensional #uid motion in the x direction. The eigenvalues of the matrix A are given by
v(u!)$(p/ (v#(u!)!p/ ) " , ! (u!)!p/ (36)
v " u!
(double).
(37)
Since the system (34) as the system of equations (28)}(31) can be written in conservative form, the eigenvalue turns out to be exceptional [32]. In view of the approach which was outlined in Section 2 we search for solutions of the following form to the system (35):
S
u
/N A" M /M 0
A"
0
0
0
0 /N M /1 , v 0
0
0
0
u
v
0
0
0
v
0
0
/N M /M 0
0
v
0
0
/N M /1 v
u
u"U( , S), v"V( , S).
.
(38)
By assuming " ( , x ) and S"S( , x ) to be any solution of the 2;2 system V V #D "0, x
(39)
where (33)
V"
S
, D"
H 0
H
V U
\J
(40)
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286
where
and
V p (U!)V b " , b "! ,
U V #a , H "! a #a U V H "! a #a #a , S S
(41)
V , c "! (U!)
(U!) a "! ,
(42)
V p p a "! , "(U!)! . (U!) S
(43)
In the present case the compatibility conditions (15) give rise to the overdetermined set of equations
U U V U #a #(a !b ) V !b !b "0,
a
(44)
a
u"U( , S)"#(G cos M,
(50)
v"V( , S)"(G sin M,
(51)
where 1 p d #F(S),
(52)
M( , S)" (45)
(49)
We notice that the Lagrange multiplier matrix ¸ which is involved in the relations (15) here specializes to BM because in the system (40) BI is the unit matrix. A direct inspection shows that the integration of the set of conditions (44)}(47) yields
G( , S)"!2
U V U a ! !b S U!
1 p c "! . (U!) S
V U V #a V #(a !c ) !c "0,
(48)
1 p c "! , (U!)
V (U!)V , a " , a "!
a
1 p V p b " , b " , S
K(S)G
1 2G
!
G
G 2G # d .
(53)
F(S) and K(S) are arbitrary functions, the condition (/ ) ( G)'0 is required to hold and the following equation must be ful"lled:
U V U # a !b !b #a "0, S
(46)
V U V V V #a !c #a "0, S S
(47)
M M tan M G M tan M M p # # S 2G S G S 1 G p 1 p ! ! "0. 2 G S G S
(54)
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Taking into account (50) and (51), it is easily seen that the matrix D admits the eigenvalues (M/ )#sin M cos M "H " , cos M#( /2G)(G/ )
whose integration yields the general solution x !H (r, s) "w (r, s),
(63)
x !r "w (r, s),
(64)
where
U "tan M " V!
(55)
to which there correspond the left eigenvectors
H , H !tan M
l H " 0
1 , tan M!H
H w (r, s)"(r!H ) !¹ !¹ s
(56)
w (r, s)"(H !r) !¹ H !(H !r)¹ !¹ s s
and the right eigenvectors
d H "
1 0
,
H d H " , tan M!H
, S
(58)
whereupon M"M(p)
(59)
and Eq. (54) results to be identically satis"ed. Owing to the condition (58) we have &l H so that the Riemann variables (27) in the present case are given by r"tan M, s"S.
(60)
Thus through the standard hodograph transformation [25]: " (r, s), x "x (r, s),
(61)
the system (39) can be reduced to the linear system x x "r , "H (r, s) , r r s s
(H ) ds, ! r¹ ds# ¹ s s
(57)
so that (39) is strictly hyperbolic. The eigenvalues and result from the characteristic velocities (36) and (37), respectively. Since (37) is exceptional, in line with the remark of Section 2, here the consistency of the full reduction process requires that ) d H "0, "
(H ) ds, (65) !H ¹ ds# ¹ s s
l H " 1
287
d¹ (r)" , dr
d¹ (s)" , ds
(66)
(67)
¹ "¹ (r), ¹ "¹ (s),
(68)
while
1 (r, s)" exp r!H (r, s)
dr . r!H (r, s)
(69)
The functions (x , ) and S(x , ) which are de "ned by (63) and (64) along with the relations (50) and (51) provide an exact solution of the system (28)}(31). It is worth noticing that the arbitrariness of the functions ¹ (s), ¹ (r) makes the class of solutions in point #exible to "t with initial and/or boundary conditions. That is relevant to our later analysis of wave interaction. The form of the response function p"p( , S) which allows the reduction approach developed hitherto to hold is selected by the consistency of the overdetermined set of conditions (52), (53) and (59). Several cases of model pressure}density}entropy law are considered below: (i) Assume
(62)
p( , S)"p (S) #p (S).
(70)
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288
(i1) (1. Substitution of (70) into (52) and (53), taking into account that M"M(p), gives rise to
2 G( , S)" p (S) \#F(S), 1!
p "p "const.,
M(p)"
1 F(S)"0, K(S)"k G 2
1# ln p (S), 1!
2 G( , S)" p (S) \, 1! 1 M(p)"G 2
(71)
1# 1# #F (p!p )\ $ arctan 1! !1
k Garctan
k "const.,
(76)
(1!)#(1!)F (p!p )\ , (!1)
(77)
where p (S) is arbitrary and the following condition must be ful"lled:
1# 2 ln 1! 1!
!1 # ln p!p #k
(72)
or, alternatively, to p "p "const., F(S)"F [p (S)] '0, K(S)"k "const., F "const., 2 G( , S)" p (S) \#F(S), 1!
(73)
p!p (
(!1)F #1
\ .
Of course, relation (70) with '1 includes as a very special case the p! law where is given by "(2n#1)/(2n!1), n3N. These values of the adiabatic exponent were considered in [25] for obtaining explicit integration of the hodograph equations associated with the governing system of an isentropic one-dimensional gas dynamics. (i3) "1. Here we have p"p "const.,
M(p)"
k Garctan
1# 1 #F (p!p )\ $ 1! 2
1# ln 1!
((1!)#F (1!)(p!p )\ #(1! (74) ((1!)#F (1!)(p!p )\ !(1!
with p (S) arbitrary. (i2) '1. Here the consistency of (52), (53) and (59) yields p "p "const., F(S)"F [p (S)] , F "const.,
F(S)"!p (S)[2 ln p #F ], K(S)"k "const., F "const., G( , S)"!2p (S)ln #F(S),
(78)
M(p)"k G(1#F #2 ln(p!p ) $arctan(1#F #2 ln(p!p ),
(79)
where p (S) is arbitrary and the following condition must be ful"lled:
K(S)"k "const., (75)
p!p ((exp(!1!F ).
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(i4) "!1 (Von-KaH rmaH n}Tsien law) In the present case we have p"p "const., F F(S)"! , K(S)"k "const., p (S) F "const., p (S) G( , S)"! #F(S),
(80)
F . M(p)"k $arctan p!p
(81)
As well known [33}35], the relation (70) with "!1 makes the system of equations (28)}(31) completely exceptional and so the subsystem (9) does. As far as the double sign in the M(p) expression is concerned, a direct inspection shows that only the positive case must be considered, otherwise the eigenvalue does not result to be exceptional. (ii) By assuming M(p)"$G p,
289
graph system (62) provides an exact solution of a corresponding initial and/or boundary value problem associated with the original set of the governing equations (28)}(31). Within such a context, here we study simple wave interaction in the ( , x )-plane by means of the general solutions (63) and (64) and later we consider the corresponding wave processes which are described by the full set of equations governing the #uid motion. Along the lines of the approach worked out in [26] and later in a more systematic way in [27], it is appropriate for studying pulse interaction to proceed in terms of the characteristic curves CH , CH de"ned, respectively, by ( , x )"const. and ( , x )"const., where / ! " , /x
/ ! " . /x
(85)
Bearing in mind the Riemann system (62) there result r"r() and s"s() whereby the representations of the general solution (63) and (64) in terms of the characteristic parameters and can be obtained. As far as the Cauchy problem is concerned, it is convenient to normalize and as follows:
(82) at "0,
we get 1 log #GI (S), p" 2G
(83)
1 G( , S)" , G
(84)
where G is a constant and GI is an arbitrary func tion while F and K specialize to F"0, K(S)"Glog G $G GI (S). 4. Wave interaction An inherent feature of the present approach is that solving a given initial and/or boundary value problem to the reduced model (39) or to the hodo-
"x
and "x ,
(86)
whereupon the initial data for r and s at "0 can be expressed in the form: r"r(x ), s"s(x ) for!R(x (#R. (87) Once the initial value problem for r and s has been given the relations (63) and (64) determine the functions ¹ (), ¹ (), () and (). In view of describing in the ( , x )-plane the interaction of two opposite travelling simple waves and according to the analysis carried out in [27], we assume that at "0, the right travelling pulse occupies the region x )x )!x and the left travelling pulse the J D region x )x )x (see Fig. 1). Both pulses D P propagate into a region of constant state where r"r and s"s . We also require that r(x ) and
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290
¹ (x )"!x #(x )(Rx ) !H (R(x ), s ))
(x )"(x )
(x )"!
V
d , (R()!H ())() (92)
V
d , (R()!H ())()
V
d . (R()!H ())()
(93)
(94)
For x )x , J (95)
¹ "0, Fig. 1. Qualitative description of wave interaction in the ( , x ) plane for the general class of p"p( , S) laws de"ned by the conditions (52), (53) and (59).
¹ "¹ (x )"!x J J J # (r !(H ) )
s(x ) are continuous. Thus, at "0
VJ
d , (R()!H ())() (96)
(88)
x #¹ J , (x )" r !(H )
(97)
(89)
x #¹ J . (x )" ((H ) !r )
(98)
r"
s"
R(x ), x )x )!x , J D r otherwise, S(x ), x )x )x , D P s otherwise,
where R(x ) and S(x ) are di!erentiable functions. Owing to (88) and (89) from the solutions (63) and (64) we obtain: For x )x , D
For x )x )x , D P (99)
¹ "0, ¹ (x )"S(x )
¹ "¹ "0, x (x )" , r !(H ) x (x )" . ((H ) !r )
! (90)
For x )x )!x , J D ¹ "0,
(91)
(x )"
V
d , (H ()!r )()
V
S() d , (H ()!r )()
V
d , (H ()!r )()
(x )"!
V
d . (H ()!r )
(100)
(101)
(102)
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For x *x , P (103)
¹ "0, ¹ "¹ (x )"s P P
VP
d , (H ()!r )() VP
S() d , ! (H ()!r )()
x #((H /s)#(H !r )/s) ¹ P , (x )" (H !r ) (105) x #(H /s) ¹ P . (x )" (r !H )
(106)
There are "ve distinct regions in the ( , x )-plane where r and s are nonzero (see Fig. 1). The left travelling pulse traverses regions I , II and III . In * * region I we have * x !r ",
(107)
so that the pulse is a simple wave. It interacts with the left travelling pulse in region II where the pulse is no longer a simple wave. Finally, in region III * the left travelling pulse is a simple wave characterized by the following relation: x !r "!¹ . J
(108)
The comparison of (107) and (108) shows that the simple wave which emerges from the interaction region is identical with that produced by the initial conditions: at "0,
r"
s"
R(x ), x )x )!x , J D r otherwise,
A similar argument for the right travelling pulse in regions I and III leads to the results: 0 0 in region I 0 x !H () ",
(104)
(109)
S(x #¹ ), x #¹ )x )x #¹ , J D J P J s otherwise. (110)
Thus, the only e!ect of the interaction on the left travelling emerging wave is to change the origin of x in the original pulse.
291
(111)
in region III , 0 H x !H () "!() ()¹ . P s
(112)
Therefore, if the material response function p"p( , S) satis"es the conditions (52), (53)) and (59), unlike the case of the left travelling pulse in general, the right travelling simple wave emerges from region II with an altered pro"le, unless p( , S) is required to obey further restrictions. In [26}28] there were thoroughly investigated classes of 2;2 quasilinear hyperbolic systems allowing for wave interaction where the emerging pulse keeps its own original pro"le. The completely exceptional systems (CEX) [34,35] are therein included and that case here corresponds to the model law (70) with "!1 (case (i4) of Section 3) for which I ) d H "0. However, apart from the von-KaH rmaH n}Tsien law which makes CEX the subsystem (39), within the present reduction framework it is a simple matter to ascertain that provided p( , S) adopts the form (70) with O!1 (cases (i1)}(i3) or the form (83) we have I ) d H O0 and H "H (r) whence, because of the relation (112), the pulse of characteristic velocity is una!ected by the interaction process as well (see Fig. 2). On the latter cases of model constitutive laws a remark is in order. Quasilinear hyperbolic 2;2 systems of physical interest exhibiting only one exceptional characteristic eigenvalue do not usually occur. Nevertheless the system (39) is not expected here to have a physical meaning but it represents only an intermediate although crucial step of our approach for determining exact wave-like solutions to the original system of two-dimensional adiabatic #uid dynamics. In order to illustrate how the Cauchy problem (88) and (89) generates a wave interaction process in the (x , x , t)-space, "rst we observe that the characteristic curves C? : ( , x )"const. and
292
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Fig. 2. Case of the model laws given by the relations (70) or (83): the interaction does not a!ect the pro"les of both emerging pulses.
C@ : ( , x )"const. are included in the family of characteristic curves of the system (34) and the latter ones, because of the underlying invariance transform into characteristic surfaces of the system (28)}(31), (see [2]). Consequently r"r() and s"s() result to be invariant on corresponding characteristic surfaces in the (x , x , t)-space. Hence the solutions U"U(r(), s()) determined to (28)}(31) within the present theoretical framework are expressed in terms of two-wave variables. These solutions also involve arbitrary functions so that they can be used for describing multidimensional interaction of two simple waves provided, suitable initial/boundary data are associated with the system (28)}(31). In passing we notice that the simple wave-like exact solutions which are determined to hyperbolic systems of PDEs in more space coordinates by other methods of approach (e.g. see [36]), depend upon only one variable so that, in general, they are not of direct use for studying multidimensional wave interaction. It is straightforward to see that the Cauchy problem (88) and (89) corresponds in the (x , x , t)-space to prescribe the data on the plane x !t" 0, where the two original pulses are assumed to be located in the strips x )x )!x and P D
Fig. 3. The interaction in the (x , x , t)-space: the pulse travel ling at characteristic velocity " emerges with an altered pro"le.
x )x )x , respectively. Outside of these strips D J a constant state is assumed to hold. Bearing in mind that U"U(r(), s()), by direct extension to the present multidimensional case of the analysis which was previously worked out in the !x plane it is easy to ascertain that in the strips de"ned by the characteristic planes x !(H ) (x !t)"x , J x !(H ) (x !t)"!x D and by the characteristic planes
(113)
x !r (x !t)"x , D x !r (x !t)"x , (114) P respectively, there result to be two separate regions of plane (simple) waves initiated at x "t (see Fig. 3). These pulses interact in a region where they
C. Curro% , D. Fusco / International Journal of Non-Linear Mechanics 37 (2002) 281}295
293
if the two waves interact before the occurrence of any singularity or shock formation.
5. Conclusions and 5nal remarks
Fig. 4. The pro"le of the pulse travelling at characteristic velocity " is not in#uenced by the interaction.
are no longer plane waves. Later the pulses emerge from the interaction region as plane waves. Obviously, if the material response function p"p ( , S) is of form (70) or (83) then the pro"les of both emerging pulses are not a!ected by the interaction (Fig. 4). In the latter case the interaction process in point is similar to the one occurring in non-linear dispersive media when two solitons collide but emerge with unchanged pro"le from the interaction. Of course, the non-linear hyperbolic waves are subjected to the pro"le distorsion. However, in the present context the plane wave propagating with the exceptional characteristic speed (37) never evolves into a non-linear shock after a "nite time has elapsed whereas for the other wave a pro"le breakdown has to be expected. Hence, the analysis of pulse interaction under interest makes sense only
We have presented a systematic approach to determine a class of solutions of the system (1) and which, to a certain extent, exhibit wave features as the Riemann invariants associated with 2;2 hyperbolic systems do. The approach proceeds in two steps. First the invariance of the autonomous system (1) under the action of the group (3) and (4) permits to reduce it to a hyperbolic system of form (6). Next, searching for the solutions (8) to the system (6) leads to considering the overdetermined set of Eqs. (9) and (10), where (9) is assumed to represent a 2;2 system determining u and u while (10) has to be identically satis"ed. Hence, through the relations (8) a class of exact solutions to (1) is obtained. Of course, the "rst step based upon invariance reduction is not to be considered if the system (1) involves only two independent variables. Since U"U(u , u ), the structure of the solu tions to system (1) considered herein resembles that of the partially invariant (double wave) solutions studied in [2,37,38], with the functions u and u playing the role of wave parameters. However, on this subject we observe "rst that, because of the invariance reduction (3) and (4), the solutions (8) are in fact invariant solutions of the system (1). Furthermore, unlike the procedure worked out to determine double-wave solutions here we do not aim at analyzing all the di!erential conditions arising from the compatibility of the overdetermined system (9) and (10) but rather we treat (10) as a set of supplementary equations [35,39}41] which must be identically satis"ed by any solutions u and u of the system (9) as well as by the functions u "u (u , u ). That permits us to construct soluFY FY tions to systems of form (1) by means of the wellestablished methods of integrating 2;2 hyperbolic systems and which are of more direct use for solving problems of interest in non-linear wave propagation. A striking feature of the present approach is that the hyperbolicity of the system (6) (or equivalently,
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C. Curro% , D. Fusco / International Journal of Non-Linear Mechanics 37 (2002) 281}295
of the system (1)), induces the hyperbolicity of the subsystem (9) whose characteristic velocities inherit all the properties of the corresponding characteristic eigenvalues of the system (6). By assuming strict hyperbolicity, the integration of (9) is accomplished by means of the hodograph transformation. Within such a theoretical framework the two-dimensional motion of an adiabatic #uid was investigated. Di!erent structural requirements can be made to the system (9) in order to integrate in a closed form the associated hodograph equations and to provide exact solutions involving arbitrary functions which can "t with prescribed initial and/or boundary value problems. However, our attention was mainly focused on the existence of an exceptional characteristic eigenvalue resulting from a double eigenvalue (exceptional) of the spectrum of the characteristic velocities of the system of balance equations (28)}(31). In such a case several classes of solutions are provided by the hodograph method. In particular we considered solutions which represent in ( , x )-plane two opposite travelling waves which interact strongly on colliding. In general, only the wave travelling with the exceptional velocity emerges with una!ected pro"le from the interaction region. However, if the material response function p"p ( , S) adopts the form (70) or (83) then the pro"les of both waves are not altered by the interaction process. These solutions through the relations (5) and (8) generate a class of exact solutions of the original system (28)}(31) and describe in the (x , x , t)-space two non-linear waves which were initiated as plane waves and exhibit in interacting a behaviour similar to that occurring in non-linear dispersive media when two solitons collide. Finally, as the solutions of the 2;2 system (9) can be superimposed via the hodograph transformation, we remark that the solutions to systems of form (1) which are obtained by means of the present approach in fact obey a non-linear superposition principle. Acknowledgements This work was supported in part by Consiglio Nazionale delle Ricerche through grants No.
96.03851.CT01 and No. 97.00888.CT01 and by Ministero della Universita` e della Ricerca Scienti"ca e Tecnologica (MURST-fondi 40% and 60%).
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