Multimode solutions of first-order quasilinear systems obtained from Riemann invariants. Part I

0.58 MB·English

by  A. M. Grundland

Multimode solutions of an ideal plastic flow.
A.M. Grundland∗and V. Lamothe†
2012-01-23
2
Abstract
1
0 ThispapercontainsananalysisofmultimodesolutionsexpressibleintermsofRiemanninvariants
2
obtained from complex characteristic elements. A variant of the conditional symmetry method for

n constructing this type of solution is proposed. Examples of applications of the proposed approach
a toanonstationaryirrotationalidealplasticflowin(2+1)dimensionsarediscussedindetail. Several
J newclassesofsolutions, someofthembounded, havebeenobtainedinclosedform. Foraparticular

3 case of a stationary ideal plastic flow, extrusion dies have been drawn.
2
MSC: 35B06 · 35F50 · 35F20 · 75Cxx

] keywords: conditional symmetries; Riemann invariants; multimode solution; ideal plasticity.
h
p
- 1 Introduction
h
t
a
Riemann waves represent a very important class of solution of nonlinear first-order systems of PDEs.
m
They are ubiquitous in the differential equations of mathematical physics, since they appear in all multi-
[
dimensional hyperbolic systems and constitute their elementary solutions. Their characteristic feature is

1 that, in most cases, they are expressible only in implicit form. For a homogeneous hyperbolic quasilinear
v system of first-order PDEs,
9
9 ∂uα
7 Aiµ(u) =0, i=1,...,p, α=1,...,q, µ=1,...,m, (1)
α ∂xi
4
.
1 (where A1,...,Ap) are q×m matrix functions of an unknown function u. Here we adopt the convention
0 that repeated indices are summed unless one of them is in a bracket.
2
1
: ARiemannwavesolutionisdefinedbytheequationu=f(r(x,u)),wheref :R→Rq andthefunction
v r(x,u)=λ (u)xi is called the Riemann invariant of the vector λ satisfying the equation ker(cid:0)λ Ai(u)(cid:1)(cid:54)=
i i i
X 0. These solutions have rank at most equal to one. They are building blocks for constructing more
r general types of solutions describing nonlinear superpositions of many waves (k-waves), which are very
a
interesting from the physical point of view. Until now, the only way to approach this task was through
the generalized method of characteristics (GMC) [3, 7, 21, 22, 29, 30] and more recently through the
∗Centre de Recherche Mathématiques, Université du Montréal, C.P. 6128, Succc. Centre-ville, Montréal, (QC) H3C
3J7,Canada,andDépartementdemathématiquesetinformatiques,UniversitéduQuébec,Trois-Rivières,(QC)G9A5H7,
Canada,email:[email protected]
†Département de Mathématiques et Statisque, Université de Montréal, C.P. 6128, Succc. Centre-ville, Montréal, (QC)
H3C3J7,Canada,email:[email protected]
1
conditionalsymmetrymethod(CSM)[1,6,13,15,16,28]. TheGMCreliesontreatingRiemanninvariants
as new dependent variables (which remain constant along the appropriate characteristic curves of the
initialsystem(1)andconstituteasetofinvariantsoftheAbelianalgebraofsomevectorfieldX =ξi(u)∂i
a a x
with λaξi = 0 for 1 ≤ a ≤ k < p. This leads to the reduction of the dimension of the problem. The
i a
most important theoretical results obtained with the use of the GMC or CSM [16] include the finding
of necessary and sufficient conditions for the existence of Riemann k-waves in multidimensional systems.
It was shown [29] that these solutions depend on k arbitrary functions of one variable. Some criteria
were also found [29] for determining the elastic or nonelastic character of the superposition between
Riemann waves described by hyperbolic systems, which is particularly useful in physical applications.
In applications to fluid dynamics and nonlinear field theory, many new and interesting results were
obtained [2, 3, 10, 11, 12, 22, 26, 30, 32, 33]. Both the GMC and CSM methods, like all other techniques
for solving PDEs, have their limitations. This fact has motivated the authors to search for the means
of constructing larger classes of multiple wave solutions expressible in terms of Riemann invariants by
allowing the introduction of complex integral elements in the place of real simple integral elements (with
which the solutions of hyperbolic systems are built [29]). This idea originated from the work of S.
Sobolev [31] in which he solved the wave equation by using the associated complex wave vectors. We are
particularly interested in the construction of nonlinear superpositions of elementary rank−k solutions
(modes) and the proposed analysis indicates that the language of conditional symmetries is an effective
tool for this purpose. This approach is applied to the nonstationary irrotational flow of a ideal plastic
material in its elliptic region.
The organization of this paper is as follows. Section 2 contains a detailed account of the generalized
methodofcharacteristicsforfirst-orderquasilinearsystemsofPDEsinmanydimensionsbasedoncomplex
characteristicelements. Insection3weformulatetheproblemofmultimodesolutionsexpressibleinterms
ofRiemanninvariantsbymeansofagrouptheoreticalapproach. Thisallowsustoformulatethenecessary
and sufficient conditions for constructing these types of solutions. Section 4 presents an application of
the developed approach to the system of equations describing a nonstationary irrotational flow of ideal
plastic materials in (2+1) dimensions. Several new classes of solutions are obtained in closed form. We
obtain several bounded solutions. For the stationary case, limiting ourselves to the region where the
gradient catastrophe does not occur, we have drawn extrusion dies and the flow inside the tools. Section
5 summarizes the obtained results and contains some suggestions regarding some further developments.
2 Complex characteristic elements.
Themethodologicalapproachassumedinthissectionisbasedonthegeneralizedmethodofcharacteristics
whichhasbeenextensivelydeveloped(e.g. in [3,9,14,17,29]andreferencestherein)formultidimensional
homogeneous and nonhomogeneous hyperbolic systems of first-order PDEs. A specific feature of that
approach is an algebraic and geometric point of view. An algebraization of systems of PDEs was made
possible by representing the general integral elements as linear combinations of some special elements
associated with those vector fields which generate characteristic curves in the spaces of independent
variables X and dependent variables U, respectively. The introduction of these elements (called simple
integral elements) proved to be very useful for constructing certain classes of rank-k solutions in closed
form. These integral elements proved to correspond to Riemann wave solutions in the case of nonelliptic
systems and serve to construct multiple waves (k-waves) as a superposition of several single Riemann
waves.
2
The generalized method of characteristics for solving quasilinear hyperbolic first-order systems of can
be extended to the case of complex characteristic elements [29]. These elements were introduced not
only for elliptic systems but also for hyperbolic systems by allowing the wave vectors to be the complex
solutions of the dispersion relation associated with the initial system of equations.
Thestartingpointistomakeanalgebraization,accordingto[3,14,29],ofafirst-ordersystemofPDEs
(1) in p independent and q dependent variables written in its matrix form
Ai(u)u =0, i=1,...,p,
i
(2)
x=(x1,...,xp)∈X ⊆Rp, u=(u1,...,uq)∈U ⊆Rq,
whereA1,...,Ap arem×q matrixfunctionsofuandwedenotethepartialderivativesbyuα =∂uα/∂xi.
i
The matrix Lα satisfying the conditions
i
uα ∈(cid:8)Lα :AisLα =0, s=1,...,q(cid:9)
i i α i
at some open given point u ∈ U is called an integral element of the system (2). This matrix L =
0
(cid:0)∂uα/∂xi(cid:1) is a matrix of the tangent mapping du:X →T U given by the formula
u
X (cid:51)(δxi)→(δuα)∈T U, where δuα =uαδxi.
u i
The tangent mapping du(x) determines an element of linear space L(X,T U), which can be identified
u
with the tensor product T U ⊗X∗, (where X∗ is the dual space of X, i.e. the space of linear forms). It
u
is well known [3, 14, 29] that each element of this tensor product can be represented as a finite sum of
simple tensors of the form
L=γ⊗λ,
where λ ∈ X∗ is a covector and γ = γα ∂ ∈ T U is a tangent vector at the point u ∈ U. Hence, the
∂uα u
integral element Lα is called a simple element if rank(Lα) = 1. To determine a simple integral element
i i
Lα we have to find a vector field γ ∈T U and a covector λ∈X∗ satisfying the so-called wave relation
i u
(cid:0)λ Ai(u)(cid:1)γα =0. (3)
i
The necessary and sufficient condition for the existence of a nonzero solution γ for the equation (3) is
rank(cid:0)λ Ai(u)(cid:1)<min(m,q). (4)
i
This relation is known as the dispersion relation. If the covector λ = λ (u)dxi satisfies the dispersion
i
relation(4)thenthereexistsapolarizationvectorγ ∈T U satisfyingthewaverelation(3). Thealgebraic
u
approachwhichhasbeenusedin[3,14,29]forhyperbolicsystemsofequations(2)allowstheconstruction
of certain classes of k-wave solutions admitting k arbitrary functions of one variable. The replacement
of the simple real element Lα by the matrix of derivatives uα in the system of equations (2) allows us to
i i
construct more general classes of solutions by replacing the real elements with complex ones. A specific
form of solution u(x) of (2) is postulated for which the tangent mapping du(x) is a sum of a complex
element and its complex conjugate
duα(x)=ξ(x)γα(u)λ (u)dxi+ξ¯(x)γ¯α(u)λ¯ (u)dxi, (5)
i i
3
where γ =(γ1,...,γq)∈Cq and λ=(λ ,...,λ )∈Cp satisfy
1 p
λ Ais(u)γα =0, λ¯ Ais(u)γ¯α =0.
i α i α
Here the quantity ξ(x) (cid:54)= 0 is treated as a complex function of the real variables x. In what follows we
assume that the vectors γ and γ¯ are linearly independent. The proposed form of solution (5) is more
general than the one proposed in [18, 29] for which the derivatives uα are represented by a real simple
i
element leading to a simple Riemann wave solution. To distinguish this situation with the one proposed
in (5), we call the solution associated with a complex element and its complex conjugate a simple mode
solution. Similarly, as in the case of k-waves [29], we have to find the necessary and sufficient conditions
for the existence of solutions of type (5). These conditions are called involutivity conditions.
Firstwederiveanumberofnecessaryconditionsonthevectorfields(γ,λ)andtheircomplexconjugate
(γ¯,λ¯) as a requirement for the existence of rank-2 (simple mode) solutions of the original system (2).
Namely, closing (5) by exterior differentiation, we obtain the following 2-forms
γα(dξ∧λ+ξdλ)+ξdγ∧λ+γ¯α(dξ¯∧λ¯+ξ¯dλ¯)+ξ¯dγ¯∧λ¯ =0 (6)
which have to satisfy (5). Using (5) we get
dλ=ξ¯λ¯∧λ +ξλ∧λ¯ , dγ =ξ¯γ ⊗λ¯+γ ⊗dλ, (7)
,γ¯ ,γ ,γ¯ ,λ
where we have used the following notation
∂ ∂
λ =γα λ, γ =γ¯α γ.
,γ ∂uα ,γ¯ ∂uα
Substituting (7) into the prolonged system (6) we obtain
γ⊗(cid:2)dξ∧λ+ξξ¯λ¯∧λ (cid:3)+γ¯⊗(cid:2)dξ¯∧λ¯+ξξ¯λ∧λ¯ (cid:3)+ξξ¯[γ,γ¯]⊗λ∧λ¯ =0, (8)
,γ¯ ,γ
whenever the differential (5) holds. The commutator of vector fields γ and γ¯, is denoted by
[γ,γ¯]=(γ,γ¯) +γ λ −γ¯ λ ,
u ,λi i,γ¯ ,λi i,γ
while by (γ,γ¯) we denote a part of the commutator which contains the differentiation with respect to
u
the variables uα, i.e.
(γ,γ¯) =γ¯α ∂ γ(u,λ)| −γα ∂ γ¯(u,λ¯)(cid:12)(cid:12) .
u ∂uα λ=const. ∂uα λ¯=const.
Let Φ be an annihilator of the vectors γ and γ¯, i.e.
<ω(cid:121)γ >=0, <ω(cid:121)γ¯ >=0, ω ∈Φ=An{γ,γ¯}.
Here,bytheparenthesis<ω(cid:121)γ >,wedenotethecontractionofthe1-formω ∈T∗U withthevectorfield
u
γ ∈ T U and similarly for the vector field γ¯ ∈ T U. Multiplying the equations (8) by the 1-form ω ∈ Φ
u u
we get
ξξ¯<ω(cid:121)[γ,γ¯]>λ∧λ¯ =0. (9)
4
Welookforthecompatibilityconditionforwhichthesystem(9)doesnotprovideanyalgebraicconstraints
on the coefficients ξ and ξ¯. This postulate means that the profile of the simple modes associated with
γ ⊗λ and γ¯⊗λ¯ can be chosen in an arbitrary way for the initial (or boundary) conditions. It follows
from (9) that the commutator of the vector fields γ and γ¯ is a linear combination of γ and γ¯, where the
coefficients are not necessarily constant, i.e.
[γ,γ¯]∈span{γ,γ¯}.
So, the Frobenius theorem is satisfied. That is, at every point u of the space of dependent variables U,
0
there exists a tangent surface S spanned by the vector fields γ and γ¯ passing through the point u ∈U.
0
Moreover the above condition implies that there exists a complex-valued function α such that
[γ,γ¯]=αγ−α¯γ¯, (10)
since
[γ,γ¯]=[γ¯,γ]=−[γ,γ¯]
holds.
Next, from the vector fields γ and γ¯ defined on U-space, we can construct the coframe Ψ, that is the
set of 1-forms σ and σ defined on U satisfying the conditions
1 2
<σ (cid:121)γ >=1, <σ (cid:121)γ¯ >=0, σ ,σ ∈Ψ,
1 1 1 2
(11)
<σ (cid:121)γ >=0, <σ (cid:121)γ¯ >=1.
2 2
Substituting (7) and (10) into the prolonged system (8) and multiplying by σ and σ respectively, we
1 2
get
(i) dξ∧λ+ξ(cid:0)ξ¯λ¯∧λ +ξλ∧λ¯ (cid:1)+αξξ¯λ∧λ¯ =0,
,γ¯ ,γ
(12)
(ii) dξ¯∧λ¯+ξ¯(cid:0)ξλ∧λ¯ +ξ¯λ¯∧λ (cid:1)+α¯ξξ¯λ¯∧λ=0.
,γ ,γ¯
If dξ = dξ¯ then equation (12.i) is a complex conjugate of (12.ii). So one can consider one of them,
say (12.i). We look for the condition of integrability such that the system (12.i) does not impose any
restriction on the form of the coefficients ξ and ξ¯. By exterior multiplication of (12.i) by λ, we obtain
λ ∧λ¯∧λ=0 (13)
,γ¯
and its respective complex conjugate equation
λ¯ ∧λ∧λ¯ =0. (14)
,γ
Notethatifconditions(10),(13)and(14)aresatisfied,thenfromtheCartanlemmathesystem(12)and
consequently the set of 1-forms (8) has nontrivial solutions for dξ and dξ¯. So under these circumstances
the following result holds [29].
Proposition 1 The necessary condition for the Pfaffian system (5) to possess a simple mode solution is
that the following two constraints on the complex-valued vector fields γ and λ be satisfied
(i) [γ,γ¯]=αγ−α¯γ¯,
for any complex-valued function α∈C
(ii) λ ,γ¯ ∈span(cid:8)λ,λ¯(cid:9).
,γ¯ ,γ
5
Note that the conditions (10), (13) and (14) are strong requirements on the functions γ =(cid:0)γ1,...,γq(cid:1)
and λ = (λ ,...,λ ) and their complex conjugates which satisfy the wave relation (3). Consequently,
1 p
the postulated form of a one-mode solution u(x) of the elliptic system (2), required by the generalized
method of characteristics (GMC) [21, 29], is such that all first-order derivatives of u(x) with respect
to xi are decomposable in the form (5). This restriction is a strong limitation on the admissible class
of solutions of (2). So far, there have been few, if any, known examples of such solutions. Therefore,
it is worth developing this idea by weakening the integrability conditions (10), (13) and (14) in order
to construct multi-mode solutions. In the next section we propose an alternative way of constructing
multi-mode solutions (expressed in terms of Riemann invariants) which are obtained from a version of
theconditionalsymmetrymethod(CSM)[16]byadaptingittotheellipticsystems. Thisconstitutesthe
objective of this paper.
3 Conditional symmetry method and multimode solutions in terms
of Riemann Invariants.
Inthissection,weexaminecertainaspectsoftheconditionalsymmetrymethodinthecontextofRiemann
invariants for which the wave vectors are complex solutions of the dispersion relation (4) associated with
the original system (2). Let us consider a nondegenerate first-order quasilinear system of PDEs (2) in its
matrix form
A1(u)u +...+Ap(u)u =0, (15)
1 p
where A1,...,Ap are m×q real-valued matrix functions of u. Let us suppose that there exist k linearly
independent non-conjugate complex-valued wave vectors
λA(u)=(cid:0)λA(u),...,λA(u)(cid:1)∈Cp, λA (cid:54)=λ¯B ∀ B ∈{1,...,k}, A=1,...,k <p (16)
1 p
which satisfy the dispersion relation (4). One should note that in (16) we do not require indices A(cid:54)=B,
which means that real wave vectors are excluded from our consideration (we do not consider here the
mixed case for wave vectors involving real and complex wave vectors). Under the above hypotheses, k
wave vectors (16) and their complex conjugates
λ¯A(u)=(cid:0)λ¯A(u),...,λ¯A(u)(cid:1)∈Cp, A=1,...,k,
1 p
satisfy the dispersion relation (4). In what follows it is useful to introduce the notation c.c. which means
the complex conjugate of the previous term or equation. This notation is convenient for computational
purposes allowing the presentation of some expressions in abbreviated form.
Let us suppose that there exists a unique solution u(x) of the equations (1) of the form
u=f(r1(x,u),...,rk(x,u),r¯1(x,u),...,r¯k(x,u))+c.c., (17)
where the complex-valued functions rA,r¯A : Rp×Rq → C are called the Riemann invariants associated
respectively to wave vectors λA, λ¯A and are defined by
rA(x,u)=λA(u)xi, r¯A(x,u)=λ¯A(u)xi, A=1,...,k. (18)
i i
6
Note that the functions u(x) are defined implicitly in terms of uα,xi, rA and r¯A. For any function
f : Ck → Cq and its complex conjugate, the equation (17) determines a unique real function u(x) on a
neighborhood of the origin x=0. The Jacobi matrix of derivatives of u(x) is given by
(cid:18) (cid:20)(cid:18)∂f ∂f¯(cid:19) ∂r (cid:21)(cid:19)−1(cid:18)(cid:18)∂f ∂f¯(cid:19) (cid:19)
∂u=(uα)= I − + +c.c. + λ+c.c. , (19)
i q ∂r ∂r ∂u ∂r ∂r
or equivalently as
∂u=(cid:18)∂f + ∂f¯(cid:19)(cid:0)M1λ+M2λ¯(cid:1)+c.c. (20)
∂r ∂r
where the k×k matrices M1 and M2 are defined by
(cid:20) ∂r (cid:18)∂f ∂f¯(cid:19)
M1 = I − +
k ∂u ∂r ∂r
∂r (cid:18)∂f ∂f¯(cid:19)(cid:18) ∂r¯(cid:18)∂f ∂f¯(cid:19)(cid:19)−1 ∂r¯(cid:18)∂f ∂f¯(cid:19)(cid:21)−1
− + I − + + , (21)
∂u ∂r¯ ∂r¯ k ∂u ∂r¯ ∂r¯ ∂u ∂r ∂r
∂r (cid:18)∂f ∂f¯(cid:19)(cid:18) ∂r¯(cid:18)∂f ∂f¯(cid:19)(cid:19)−1
M2 =−M1 + I − + ,
∂u ∂r¯ ∂r¯ k ∂u ∂r¯ ∂r¯
∂f (cid:18)∂fα(cid:19) ∂f (cid:18)∂fα(cid:19)
= ∈Cq×k, = ∈Cq×k, λ=(λA)∈Ck×p, (22)
∂r ∂rA ∂r¯ ∂r¯A i
∂r (cid:18)∂rA(cid:19) (cid:18)∂λA (cid:19)
= = i xi ∈Ck×q, r =(r1,...,rk), (23)
∂u ∂uα ∂uα
andtheirrespectiveconjugateequations. ThematricesI andI aretheq×q andk×k identitymatrices
q k
respectively. WeusetheimplicitfunctiontheoremtoobtainthefollowingconditionsensuringthatrA,r¯A
and uα are expressible as graphs over some open subset D ⊂Rp,
(cid:18) (cid:20)(cid:18)∂f ∂f¯(cid:19) ∂r (cid:21)(cid:19)
det I − + +c.c. (cid:54)=0 (24)
q ∂r ∂r ∂u
or
(cid:18) ∂r (cid:18)∂f ∂f¯(cid:19)(cid:19) (cid:34) ∂r¯(cid:18)∂f ∂f¯(cid:19)
det I − + (cid:54)=0, and det I − +
k ∂u ∂r ∂r k ∂u ∂r¯ ∂r¯
(25)
∂r¯(cid:18)∂f ∂f¯(cid:19)(cid:18) ∂r (cid:18)∂f ∂f¯(cid:19)(cid:19)−1 ∂r (cid:18)∂f ∂f¯(cid:19)(cid:35)
− + I − + + (cid:54)=0
∂u ∂r ∂r k ∂u ∂r ∂r ∂u ∂r¯ ∂r¯
So the inverse matrix in (19) or in (20) is well-defined in the vicinity of x=0, since
∂r ∂r¯
=0, =0 at x=0. (26)
∂u ∂u
Note that on the hypersurface defined by equation (17) and one of the expressions (24) or (25) equal to
zero, the gradient of the function u(x) becomes infinite for some value of xi. So the multimode solution,
givenbyexpression(17)losesitssenseonthishypersurface. Consequently, sometypesofdiscontinuities,
i.e. shock waves can occur. In what follows, we search for solutions defined on a neighborhood of x=0
7
under the assumption that the conditions (24) or (25) are different from zero. This means that if the
initial data is sufficiently small, then there exists a time interval [t ,T], T > t (where we denote the
0 0
independent variables by t = x0,x˜ = x1,...,xn where p = n+1) in which the gradient catastrophe for
the solution u(t,x˜) of the system (2) does not take place [30].
Note that the Jacobian matrix of u(x) has at most rank equal to 2k. It follows that the proposed
solution (17) is also at most of rank-2k. Its image is a 2k-dimensional submanifold S in the first jet
2k
space J1 =J1(X×U).
Let us introduce now a set of p−2k linearly independent vectors ξ :Rq →Cp defined by
a
ξ (u)=(cid:0)ξ1(u),...,ξp(u)(cid:1), a=1,...,p−2k, (27)
a a a
satisfying the orthogonality conditions
λAξi =0, λ¯Aξi =0, A=1,...,k, a=1,...,p−2k, (28)
i a i a
for a set of 2k linearly independent wave vectors (cid:8)λ1,...,λk,λ¯1,...,λ¯k(cid:9). It should be noted that the
set (cid:8)λ1,...,λk,λ¯1,...,λ¯k,ξ1,...,ξp−2k(cid:9) forms a basis for the space of independent variables X. Note
also, that the vectors ξ are not uniquely defined since they obey the homogeneous conditions (28). As a
a
consequence of equation (19) or (20), the graph Γ={x,u(x)} is invariant under the family of first-order
differential operators
∂
X =ξi(u) , a=1,...,p−2k, (29)
a a ∂xi
defined on X ×U space. Since the vector fields X do not include vectors tangent to the direction of u,
a
they form an Abelian distribution on X×U space, i.e.
[X ,X ]=0, a,b=1,...,p−2k. (30)
a b
Theset(cid:8)r1,...,rk,r¯1,...,r¯k,u1,...uq(cid:9)constitutesacompletesetofinvariantsoftheAbelianalgebra
Lgeneratedbythevectorfields(29). Sogeometrically,thecharacterizationoftheproposedsolution(17)
of equations (15) can be interpreted in the following way. If u(x) is a q-component function defined on a
neighborhood of the origin x=0 such that the graph of the solution Γ={(x,u(x))} is invariant under a
setofp−2kvectorfieldsX withtheorthogonalityproperty(28),thenforsomefunctionf theexpression
a
u(x)isasolutionofequation(17). Hencethegroup-invariantsolutionsofthesystem(15)consistofthose
functions u=f(x) which satisfy the overdetermined system composed of the initial system (15) together
with the invariance conditions
ξiuα =0, i=1,...,p, a=1,...,p−2k, (31)
a i
ensuring that the characteristics of the vector fields X are equal to zero.
a
It should be noted that, in general, the conditions (31) are weaker than the differential constraints
(5) required by the generalized method of characteristics, since the latter method is subjected to the
algebraic conditions (3). In fact, equations (31) imply that there exist complex-valued matrix functions
8
Φα(x,u)andΦα(x,u)definedonthefirstjetspaceJ =J(X×U)suchthatallfirstderivativesofuwith
A A
respect to xi are decomposable in the following way
uα =Φα(x,u)λA+Φα(x,u)λ¯A (32)
i A i A i
where
(cid:18) (cid:20)(cid:18)∂f ∂f¯(cid:19) ∂r (cid:21)(cid:19)−1(cid:18)∂f ∂f¯(cid:19)
Φα = I − + +c.c. +
A q ∂r ∂r ∂u ∂r ∂r
(33)
α (cid:18) (cid:20)(cid:18)∂f ∂f¯(cid:19) ∂r (cid:21)(cid:19)−1(cid:18)∂f ∂f¯(cid:19)
Φ = I − + +c.c. +
A q ∂r ∂r ∂u ∂r¯ ∂r¯
or
Φα =(cid:18)∂f + ∂f¯(cid:19)M1+(cid:18)∂f + ∂f¯(cid:19)M2
A ∂r ∂r ∂r¯ ∂r¯
(34)
Φα =(cid:18)∂f + ∂f¯(cid:19)M1+(cid:18)∂f + ∂f¯(cid:19)M2
A ∂r¯ ∂r¯ ∂r ∂r
The matrices ΦαλA and Φαλ¯A appearing in equation (32) do not necessarily satisfy the wave relation
A i A i
(3). As a result of this fact, the restrictions on the initial data at t = 0 are eased, so we are able to
consider more diverse types of modes in the superpositions than in the case of the generalized method of
characteristics described in section 2.
Let us now proceed to solve the overdetermined system composed of equations (15) and differential
constraints (31)
Aiµ(u)uα =0, ξi(u)uα =0. (35)
α i a i
Substituting (19) or (20) into (15) yields the trace condition
(cid:32) (cid:18) (cid:20)(cid:18)∂f ∂f¯(cid:19) ∂r (cid:21)(cid:19)−1(cid:18)(cid:18)∂f ∂f¯(cid:19) (cid:19)(cid:33)
tr Aµ I − + +c.c. + λ+c.c. =0 (36)
q ∂r ∂r ∂u ∂r ∂r
or
tr(cid:18)Aµ(cid:20)(cid:18)∂f + ∂f¯(cid:19)(cid:0)M1λ+M2λ¯(cid:1)+c.c(cid:21)(cid:19)=0 (37)
∂r ∂r
onthewavevectorsλandλ¯ andonthefunctionsf andf¯,whereA1,...,Aq arep×q matrixfunctionsof
u (i.e. Aµ =(Aµi(u))∈Rp×q,µ=1,...,m). For the given initial system of equations (15), the matrices
α
Aµ areknownfunctionsofuandthetraceconditions(36)or(37)areconditionsonthefunctionsf,f¯,λ,
λ¯ (or on ξ due to the orthogonality conditions (28)). From the computational point of view, it is useful
to split xi into xiA and xia and to choose a basis for the wave vector λA and λ¯A such that
λA =dxiA +λAdxia, λ¯A =dxiA +λ¯Adxia, A=1,...,k, (38)
ia ia
where (i ,i ) is a permutation of (1,...,p). So, the expressions (23) become
A a
∂rA ∂λA ∂r¯A ∂λ¯A
= iaxia, = iaxia. (39)
∂uα ∂uα ∂uα ∂uα
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Substituting (39) into (36) (or (37)) yields
tr(cid:18)Aµ(cid:0)I −Q xia(cid:1)−1(cid:18)∂f + ∂f¯(cid:19)Λ(cid:19)=0, µ=1...,m, (40)
q a ∂R ∂R
or
tr(cid:18)Aµ(cid:18)∂f + ∂f¯(cid:19)(cid:0)I −K xia(cid:1)−1Λ(cid:19)=0, µ=1...,m, (41)
∂R ∂R 2k a
where R=(r1,...,rk,r¯1,...,r¯k)T and
(cid:18)∂f ∂f¯(cid:19)∂λ (cid:18)∂f ∂f¯(cid:19)∂λ¯ (cid:18)∂f ∂f¯(cid:19)
Q = + ia + + ia = + η ∈Cq×q,
a ∂r ∂r ∂u ∂r¯ ∂r¯ ∂u ∂R ∂R a
 ∂λia (cid:16)∂f + ∂f¯(cid:17) ∂λia (cid:16)∂f + ∂f¯(cid:17)  (cid:18)∂f ∂f¯(cid:19) (42)
Ka = ∂∂λ¯uia (cid:16)∂∂fr + ∂∂fr¯(cid:17) ∂∂λ¯uia (cid:16)∂∂fr¯ + ∂∂fr¯¯(cid:17) =ηa ∂R + ∂R ∈C2k×2k,
∂u ∂r ∂r ∂u ∂r¯ ∂r¯
and for simplicity of notation, we note
∂λ 
(cid:32)λ(cid:33) ∂uia ∂f (cid:18)∂f ∂f(cid:19)
Λ= ∈C2k×p, η = ∈C2k×q, = , ∈Cq×2k, (43)
λ¯ a ∂λ¯  ∂R ∂r ∂r¯
ia
∂u
fori fixedandi =1,...,p−1. In(42)the2k×2k matrixK isdefinedintermsofthek×k subblocks
A a a
of the form ∂λ /∂u(cid:0)∂f/∂r+∂f¯/∂r(cid:1), where η is a matrix form of the block ∂λ /∂u over the block
∂λ¯ /∂u. Thenoiatation(∂f/∂r,∂f/∂r¯)representsathematrixformedoftheleftblocka∂f/∂r andtheright
a
block ∂f/∂r¯. Note that the functions rA, r¯A and xia are functionally independent in a neighborhood of
x = 0 and the matrix functions Aµ, ∂f/∂r, ∂f/∂r¯, ∂f¯/∂r, ∂f¯/∂r¯, Q and K depend on r and r¯only.
a a
So, equations (40) (or (41)) have to be satisfied for any value of coordinates xia. As a consequence, we
have some constraints on these matrix functions. From the Cayley-Hamilton theorem, we know that for
any n×n invertible matrix M, the expression (M−1detM) is a polynomial in M of order (n−1). Thus,
using the tracelessness of the expression Aµ(cid:0)Iq−Qaxia(cid:1)−1(∂f/∂R)Λ, we can replace equations (40) by
the following condition
(cid:18) (cid:18)∂f ∂f¯(cid:19) (cid:19)
tr AµQ + Λ =0, where Q=adj(I −Q xia)∈Cq×q, (44)
∂R ∂R q a
where adjM denotes the adjoint of the matrix M. As a consequence the matrix Q is a polynomial of
order (q −1) in xia. Taking (44) and all its partial derivatives with respect to xia (with r, r¯ fixed at
x=0), we obtain the following conditions for the matrix functions f(r,r¯) and λ(f(r,r¯))
(cid:18) (cid:18)∂f ∂f¯(cid:19) (cid:19)
tr Aµ + Λ =0, µ=1,...,m, (45)
∂R ∂R
(cid:18) (cid:18)∂f ∂f¯(cid:19) (cid:19)
tr AµQ ...Q + Λ =0, (46)
(a1 as) ∂R ∂R
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