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Reductive perturbation method for quasi one-dimensional nonlinear wave propagation. I
373
WAVE MOTION 12 (1990) 373-383 NORTH-HOLLAND
REDUCTIVE NONLINEAR
PERTURBATION METHOD WAVE PROPAGATION. I
FOR QUASI
ONE-DIMENSIONAL
Tosiya TANIUTI* Department Kensington,
Received
of Applied Mathematics, N.S. W. 2033, Australia
School of Mathematics,
University of New South Wales, P.O. Box 2,
2 February 1989
By means of an extension of the reductive perturbation method to quasi one-dimensional wave propagation it is shown that systems of nonlinear evolutional equations are reducible to the Kadomtsev and Petviashvili equation and the ZabolotskayaKhokhlov equation in the weakly/dispersive and dissipative cases, respectively, which are given in three-dimensional forms representing two-dimensional slow modulations.
1. Introduction In a previous paper of the present author [l], reductions of nonlinear evolutional systems with two space-variables were discussed by means of the reductive perturbation method [2,3]. The reductions were based on the assumption that the systems are weakly dispersive or weakly dissipative and reduce to a hyperbolic system of first order partial differential equations if higher order derivative terms are neglected. Consequently it is assumed that for one-dimensional propagation the systems are reducible to the Korteweg-de Vries (KdV) equation or the Burgers equation. It was then shown that for quasi-onedimensional propagation (with slow transverse modulation) the systems can be reduced to the KadomtsevPetviashvili (K.P.) equation [4] and the Zabolotskaya-Khokhlov (Z.K.) equation [5], which are twodimensional extensions of the KdV and Burgers equations, respectively. However, for the hyperbolic system a restricted form was assumed, and the system was not given in a sufficiently general form to be adaptable to various physical problems. In the present paper the restriction to the hyperbolic system is removed; also the space is extended to the three dimensions, and the systems so generalized are reduced to the three-dimensional (3D) K.P. equation and the 3D Z.K. equation. In the next section the hyperbolic system is reduced to the equation which governs the quasi one-dimensional propagation of a simple wave of small amplitude; of course, this is the hyperbolic limit of the K.P. and Z.K. equations. The reduction is based on the linear dispersion relation, namely the linear algebraic equation for a plane wave of infinitesimal amplitude is solved by means of an adiabatic perturbation method in terms of a smallness of the wavenumbers for the transverse modulation. The method is closely analogous to the stationary perturbation method in quantum mechanics; the frequency shift corresponds to the shift of the energy level. In this way the orderings of time scales and length scales are determined except for one arbitrary parameter, which is fixed by taking into account the higher order derivative terms responsible for the * Permanent address: The Japan Society of Plasma Sciences Nagoya 460, Japan.
0165-8641/90/903.50
@ 1990 - Elsevier Science Publishers
and Nuclear
Fusion Research,
B.V. (North-Holland)
4F Tosho
Bldg., Nishiki
2, 20-29,
374
T Taniutil Reductive perturbation method
dispersion or dissipation. Thus, in Sections 3 and 4 the weakly/dispersive and dissipative systems are reduced to the 3D K.P. and Z.K. equations, respectively. In Section 5 an exceptional case is discussed.
2. Reduction of the hyperbolic system We first consider the following system of equations
for a column vector
U with N components
ul,uZ,.--SUN,
Here each Aj is an N x N matrix function of U, and necessary analyticities with respect to U will be assumed. The system is assumed hyperbolic in a certain domain of the U space, 0, in the following sense. For any vector U in 0, consider the linear algebraic equation for SU,
-WI+; A'k. j=l
SU=O, >
where kj are real constants. Then all the roots of the secular equation det -WI+
i Ajkj = 0 j=l
are real for all 4; some of them may be degenerated, however the associated N eigenvectors are linearly independent so that the solution SU is given by a linear combination of them. In particular we observe the eigenvalue of A’, that is, eq. (2) for k2 = k3 = 0, which is written as (A’--A)R=O,
(4)
where R is the column eigenvector and the associated row vector will be denoted by L. Hereafter, eqs. (2) and (3) will be considered for a constant vector U,, and A’( U,,) will be denoted by A’,, then eq. (2) holds for a plane wave of infinitesimal amplitude and eq. (3) becomes the linear dispersion relation. On the other hand, in eq. (4), A’, A and R will be regarded as functions of U and it is assumed that there exists at least one non-degenerate eigenvalue satisfying the condition A’=V,A.
RZO,
(5)
where VU is the gradient with respect to U. genuinely non-linear, the simple wave of this steepening.) We first suppose that a wave is propagated direction, so that lk21- lk31< Ik11, or by means they may be expressed as lkll- s”,
~k2~-~k3~-~Y+”
(The wave mode under the condition (5) will be called mode propagating in the x,-direction can break itself by in the x,-direction while it varies slowly in the transverse of a parameter E ordering a smallness of the amplitude
(yaO,a>O),
where y (2 0) and a (> 0) are parameters to be determined later. Then eq. (2) is written as (H,+zH’)ry=n!I$
(6)
T. Taniuti / Reductive perturbation
in which H,, H’ and A are represented tH’=
Ho = A;,
5 A;(k,/k,), /=2
method
375
as (El=C),
A =o/k,,
and Y is proportional to 6U. It should be noted that the normalizations of 6U and W are arbitrary. The eigenvalue problem (6) may be solved by the adiabatic perturbation method which is familiar in quantum mechanics. Expanding Y and A in powers of i? as Y=Yo+EIY,+E12Y2+.
**,
(7)
A=A,+&i,+E^‘A,+~~~,
(8)
we have the sequence of equations:
(9)
H,Yo=&Yo, (Ho-A,)Y,+H’Yo=A,Y,,,
(10)
(Ho--A,)Y2+H’Y,-A,YY,=A2Y,,,
(11)
....
Equation (9) is identical with eq. (4) at U = U,,, namely A, = A,, and Y,, = &. For convenience, corresponding to Y,, the adjoint eigenvector Lo is denoted by ‘Doand the N eigenvectors {eon} (i = 1,. . . , IV) are normalized so that the orthonormal conditions hold, namely, (12)
(@On,Yclm)= %n.
By the assumption the N eigenvalues {A,,} of eq. (9) are real and the corresponding eigenvectors { YO,} are linearly independent. Each of {A,,} and {Yen} changes by the perturbation. We now consider the adiabatic change of an eigenstate of H,, which is non-degenerate and genuinely nonlinear. Let the state be specified as the pth state. Then the first order shift of Ao,, is readily obtained by eq. (lo), that is, (13)
Alp = (@,,H’Yo,) = (plH’\p). Also, representing y
=ct
1P
n
QIP as a linear combination
of {Y,,} and using eq. (12) we have
(nlH’/p)PO”+ cpyop,
(14)
A,--Aon
in which the pth term is not summed, while c, is an arbitrary constant. Introducing eqs. (13) and (14) into eq. (11) multiplied by @, enables us to obtain the second order shift of the pth eigenvalue, A
=
cr
2P ”
tilH’b)(nlH’l~)E &p
-A,.
+I
H’H’
Ho_no
Ip)-
(15)
Eqs. (13) and (15) yield the shift of the frequency due to the transverse modulation with k2 and 5, which takes the form
,i2~l&bW,-, Ii, (PI H:_A;op bMn/kl .
w = Aopkl+
376
T. Taniuti/
Reductive perturbation
method
This equation shows that the time evolution comprises the three different time scales of the orders of cv7, s-(Y+a), and c-_(Y+ZP);the first two terms on the right-hand side can be eliminated by introducing the transformation, rlr = s y+a(~, -(plA;(p)t),
5 = c y(x* - Aopt),
1= 2,3.
For the last term we require that it is balanced with the nonlinear frequency shift of the lowest order 6(hkJ - h;&k, - .sy+‘. Then (Yis fixed to 4. It is obvious that by this choice of (Ythe nonlinearity is in balance with the linear dispersion due to the transverse perturbation. On the other hand U must be expanded to be matched with ?P, while the order of 6U has been tacitly assumed as E, that is, ?P -O(E), hence eq. (7) leads us to assume u=
UO+EU,+E3’2U,,2+&2U2+*
* *.
(16)
We are now ready to make the reduction of eq. (1). Introducing by the equations
the coordinates
(17)
5 = cY(x, - A$), c y+“2(x, - (plA;(p)t),
171=
5, vf (I = 2,3) and r
I = 2,3,
(18)
T=&y+‘t,
(19)
and substituting eqs. (16)-(19) into eq. (l), we have the sequence of equations corresponding to eqs. (9)-( 10) (20)
(&-&,)u,,=O, cHO
-
AOp)
(HO-AO,)U2,,+
u3,2,C
+
i I=2
Ul,,+
(21)
(~iA;kdl U,,,, = 0,
[Ai-
6
. P,a)Ou,,,+,i2
(A:,-(~lA;bW3,2,,,
=O,
(22)
in which the subscripts after the commas denote the differentiations. In analogy with the adiabatic perturbation let each Uj be given by a linear combination of !PO,,_Then the reduction goes parallel to that of eqs. (9)-(11). From eq. (20) it follows u 1 = u(‘+?@+ v 1.
(23)
Here u(‘) is a scalar function of 5, r and nl to be determined later; VI is an arbitrary vector independent of 6, however its ?P,,-component can be absorbed to the first term so that (%DoP * V,) = 0. Since the normalization of Y,, has not been specified, it may be fixed by equating a non-vanishing component of ly,, say I,&,,, to unity, then 8) becomes u1(‘) . Introducing eq. (23) into eq. (21) multiplied by c&,,, (n=l,...,N)yields U 3/U
- -C’ R { i2 [ (4 H
-
,i2(PIA~I v,,,,) = 0,
(24)
0
(25)
in which u(~‘~) is a scalar function undetermined in this step. However, when eq. (24) is integrated with respect t0 ( t0 get U3/2, the second term in the bracket on the right-hand side gives rise to the secular
T. Taniuti / Reductive perturbation
method
317
term, because VI is independent of 5. Hence V, must vanish, and eq. (25) becomes trivial. Substituting eqs. (23) and (24) (with V, = 0) in eq. (22) and using eq. (12), we obtain
(26) in which cl”
,“FJop IP).
= _ (pI
(27)
0
The reduction in the two-space dimensions was done in the previous paper [l]. However it is assumed that (P~AZ~P)= 0, consequently 7)2was given by Ey+1’2x2. As will be seen in the following example, this is quite a stringent restriction and does not hold in many physical problems. Example: gas dynamics The ideal gas dynamic equations
are given by v,,+v*
p,,+v-vp+pv*v=o,
vv+(c~/p)vp=o.
Here p is the mass density v = ( vX,v,, vz) the flow velocity, c, is the sound speed which is a function of p. In matrix form they are written as U,,+ i A’U,i=O, i=l
where v,
P
Z/P
vx
1
0 0 0
P
0
v, 0
0 vy
0 0
0
0
v,
VY
AZ’
i 0 cf/p 0
0 0
’
0
0
0
0
v, 0
II PO
G-0
.
VYO vzo
Then the linear dispersion relation becomes (w-k.
vo)‘[(u-k.
v~)~-c~~]=O,
that is, there are three eigenvalues w=k~v,*c,,~k~,
o=k.
v,,
’
VX1
A3=
Let U, be specified as
uo=
0
ii
Cl/P VZ 0
v, 0 00
v, 00 0
vz 0P
1*
378
T. Taniutij Reductive perturbation method
For the pth mode we may consider w = k - v,,+ c,lkl. If k, = (5, k,) is
and the third one is degenerate. small, w is approximated as
w = 00 . k+c,,k,+tc,,((k~lk,)+(kIlk,)), from which we can anticipate that (P~:/P) = GO,
(PIA~P) = vyo,
(PI&/P) = GO,
C’” = ~CsoSrm.
This is shown by means of the R.P.M. as follows. The eigenvalues and eigenvectors obtained.
I [I
of AA are readily
1
qo* =
*Go/P0 0
1
’
A”, = Go* Go,
0
0
ry,,=
0 1 ’
&=A:=
vxo,
0
@o*= [*Go/P0 1 0 w*tpo/2cso), Consequently
II 0
qo.%= ;
,
1
d&3=(0 0 1 O),
G&=(0
0 0 1).
we have
(@o&&I qo*> =0,
(@o+lAfl po3>= PO/~, (@03lA;l*o+)= doho, MMI&I PO+)= 0, which are introduced
into eq. (27) to give
p2 = (+I‘4~13)(3lAi+) = c ,2 so , Go + Go - Go likewise we find c33 = &o/2, Since the eigenvectors modulation gives rise interpreted as the self the transverse flows +
c*3= c3*=0 Vo3 and Iyo4yield v, and v, varying with x, respectively, we find that the transverse to the transverse flow. And the second order frequency shift $,,k:/ k, may be energy of a sound given by virtual transverse motion in the process the sound + the phonon, while the direct sound-sound transition is prohibited.
3. Reduction of the weakly dispersive system We now extend eq. (1) to
(28)
T. Tanhti
/ Reductive perturbation
method
379
Here K”’ are N x N matrix functions of U. If all the roots o of the equation -oI + ; A’kj + i j=l
i,j,l=
K”‘k,k,k,
=0
(29)
1
are real for any vector k of sufficiently small length, the system (28) is called weakly dispersive. Since (Y has been given by f, we have the orderings k, - eY, k2 - ks - Ed+“*, hence, eq. (6) is modified to [Ho+ e ‘/*HI+ E2YH”+. . .] t$f = AIJ?
(30)
in which e2YHlr= K;“( k,)* and other terms resulting from the third order derivatives are of higher order. It is natural to require that the third order dispersion is in the same order as that due to the transverse dispersion. Then y is determined by 2y - 1, that is, y = f. However, our purpose is not to solve eq. (30) but to make the reduction of the system (28). Hence we assume the expansion (16) for U and introduce the transformations (17)-(19) with y = i, 5 = E”*(x’ -L&t),
TI= s(x, -(p]A&)t),
7 = E3/*t.
Corresponding to the adiabatic assumption, it is assumed that (i) each q((i=1,$,2,$ ,...) are given by a linear combination of the eigenvectors of AA, { !P,,“} = {It,,}, the coefficients of which are functions of 5, nr and T. As will be seen later, the condition (ii) (plK”‘]p) # 0 is also required. Introducing the expansion and the transformations into eq. (28) we find that the equations for U, and U3,2 are the same as eqs. (20) and (21), respectively, and eq. (22) is supplemented by the third order derivatives: (Ho-&,)~2,,+
K,r+
u,
* (v,a),~,.,+,~2(Ab-(plA~lp))U,,2,,,+~11’U,,~~~=0,
(31)
where U, and I&,* are given by eqs. (23) and (24) with V, = 0. Then multiplying eq. (31) by GOP we obtain the 3D K.P. equation
in which Y= (pl~“‘lp). For the dispersive terms comprising the third order derivatives we can consider a number of different forms [2], such as
which leads to the 3D K.P. equation with v = (~~K~“G~“M~“~p);
380
7: Tan&i/
also, one of the space-derivatives
Reductive perturbation
method
may be replaced by the time derivative, for example,
U
yields
v = -AOp(pIKAIIp).
The same applies to the dissipative terms which will be discussed in the next section.
4. Reduction of the weakly dissipative system As another extension of eq. (1) the second order derivatives may be added to give (33)
We call this system weakly dissipative, if the equation -WI+
Ajk,+i
i
i
K”kikj
-0
(34)
i,j=l
j=l
admits at least one complex root and the imaginary parts of all the complex roots are negative for any which vector k of a sufficiently small Ikl. In this case azyH” in eq. (31) takes the form EYH”=iKA1kl, implies y = 1. By means of this choice of 7, the reduction goes parallel to that of eq. (28), provided that in place of the assumption (ii) (nlK”ln)=
-v*
is assumed for any non-degenerate
(35)
eigenvector of A’. And we obtain the 3D Z.K. equation
This result also shows that if all the eigenvalues condition for the weak dissipation.
of A’ are distinct, the condition
(35) is a sufficient
5. Exceptional case
If for an eigenvector, say, Rp, the condition (35) does not hold, that is, if (plK%)
=0
(37)
is valid, eq. (33) cannot be reduced to the Z.K. equation. In this case, choosing y = f enables us to reduce the system to a K.P. equation with a non-Hermitian term [l]. This is readily seen by observing that the non-vanishing contribution of E~H” appears first in its second order, which is to be in balance with the second order of &“*H’. We thus find that eq. (21) is altered to
which is solved for U3,2,5 to give (38)
T. Taniuti / Reductive perturbation
method
381
whilst eq. (22) becomes
which leads to the equation
Here v and
K~
are given by
K/= --(PI
A$;’
+ K;‘A;
IP>+(PIK:‘+
K~P).
K-&p
Since the product Cf=, Ktqf cannot be made a definite sign for all the vectors q = (q2, qJ even if sufficiently small, eq. (39) is not dissipative and admits growing modes, unless the condition K2 =
191
is
(40)
KJ = 0
holds, which however is not valid, in general. Hence, it may be said that when the condition (35) does not hold, in general, eq. (33) is not weakly dissipative. We thus find that if all the eigenvalues of A’ are distinct and the present orderings are valid, the condition (35) is the necessary and sufficient one for the weak dissipation. However, there exists the exceptional case that for a non-degenerate state both the conditions (37) and (40) hold, in which case eq. (39) reduces to the K.P. equation. (It may be noted that without the transverse modulation eq. (39) becomes the KdV equation regardless of eq. (40).) Since this does not prove that Im o = 0 in all order, it will require further considerations as to whether the system (33) can be weakly dispersive. Of course, there are such cases; for example if the matrices Aj are symmetric whilst K”‘ are antisymmetric, then i Ajk,+i i K”lcJc,
j=l
i.j
becomes Hermitian; in this case it is readily confirmed that the conditions (37) and (40) are satisfied for all eigenvectors. Finally we note that in correspondence to the condition (37), if the condition (pI~“‘lp) = 0 holds, by means of the choice y = f, eq. (28) is reduced to the following equation
I=2
I,m=2
(41)
Here v’ and C, are constants, their explicit forms may be deduced from v and K/ by noticing the correspondence among K ii and K”‘; however eq. (41) is a dispersive equation, hence the condition on I;; such as eq. (40) is not required. This equation has not been studied in any detail.
382
T. Taniuti / Reductive perturbation
method
6. Summary and remarks
In this paper we have extended the reductive perturbation method for one-dimensional propagation to quasi one-dimensional propagation in three-dimensional space. A crucial point in this extension is the reduction of the hyperbolic system (1) to the scalar equation (26), that is, the hyperbolic limit of the Z.K. and K-P. equations describes a weak simple wave with slow transverse modulations. Of course, in general this equation does not have a solution for all time. The weakly/dispersive and dissipative systems are introduced by supplementing to the system (1) the third and second order derivatives, respectively; the reductions of those systems to the 3D K.P. and K.Z. equations go parallel to that of the system (1). Also, there exists an exceptional case that the system with second order derivatives becomes dispersive. A generic theory for the exceptional case may not be significant, but it is desirable to study physical examples. The most interesting one may be the magnetoacoustic waves in two-fluid magnetohydrodynamics. In this case the reductions to the K.P. equation were given by De Vito and Pantano [6] for cold plasma and by Shah and Bruno [7] for warm plasma. However, both of these works deal with the two dimensional case under the restriction (plA&)=O, and the results cannot directly be extended to the three-dimensional case, which we will discuss in a next paper as an example to the present paper. In all of the reductions, a basic assumption is that the solution vector U, expanded as (16), is given by a linear combination of the eigenvectors of A& the coefficient of each eigenvector is a function of the strained variables 5, n,, T. The first order term of the expansion (16), U, , is proportional to an eigenvector tyop (of a nondegenerate state) and its coefficient u(r) is determined by the Z.K. or K.P. equation. In the ordinary case, the next order term Q2 is given as follows: each coefficient of lyO, (n # p) was given in terms of u(‘) by solving the algebraic equation (21), while that of lyop, u(~‘~), is governed by a linear partial differential equation which follows from the higher order equation (for Us,*) as a solvability condition. Thus the partial differential equations for the coefficient of ??Opare obtained successively as compatibility conditions, whilst the coefficients of Y,,, (n # p) are expressed in terms of lower order terms by solving algebraic equations, which may be deduced from the adiabatic perturbation on the linear dispersion relation. In the one-dimensional case the (linear) higher order equations admit secular terms, which for the dispersive system can be removed by a renormalization technique [8,9]. It will be a future problem as to whether such a procedure will work also in quasi one-dimensional propagation. The dispersive or dissipative system considered in this paper seems to be sufficiently general for applications to sound-like modes, because it is based on the hyperbolic system. It should however be remarked that in the present paper propagation in unbounded space has been considered. In some cases, effects of boundary conditions at side walls (in the transverse direction) will become significant; then the present method of reduction is not applicable or must be modified. As is well known, the K.P. equation was introduced originally for the shallow water wave [4, lo]. In this case the fluid is inviscid and incompressible, consequently the basic system is not hyperbolic, and the boundary condition at the surface of the water plays a crucial role for the time evolution. Hence the present method does not apply to this system, but will be applicable when the system is reduced to an equation of the Boussinesq-type which takes a form similar to eq. (28). The asymptotic reduction of the incompressible hydrodynamic equations for weakly nonlinear long internal waves was established by Grimshaw [ll]. It deals with various cases involving the effect of the Coriolis force due to the earth’s rotation. The method of reduction is entirely different from the present one, because of the conditions in the vertical direction and also due to the presence of side-walls in the horizontal direction. It may be noted that nevertheless for waves of the same (horizontal) length scales, the K.P. equation (with a linear term accounting for the rotation) is obtained after integrating over the vertical direction. This reflects the intrinsic similarity law of the K.P. equation.
383
T. Taniuti f Reductive perturbation method
Acknowledgement The present paper was completed University
during my stay at the Department
of Applied
Mathematics
of the
of New South Wales. I wish to thank Professor R. Grimshaw for reading through the manuscripts
as well as for the helpful
discussions,
and the University
also indebted to Professor M.J. Ablowitz for his useful Clarkson University, in regard to the exceptional case.
of New South Wales for financial comments,
support. I am
made during my short stay at the
References [l] T. Taniuti, “Reductive perturbation method for quasi-one-dimensional wave propagation”, in: A. Jeffrey, ed., Nonlinear Wave Motion, Longman, Boston (1989). [2] T. Taniuti and C.C. Wei, “Reductive perturbation method in nonlinear wave propagation”, J. Phys. Sot. Jpn. 24,941-944 (1968). [3] T. Taniuti and K. Nishihara, Nonlinear Waves, Pitman, London (1983). [4] B.B. Kadomtsev and V.I. Petviashvili, “On the stability of solitary waves in weakly dispersive media”, Sov. Phys. Dokl. 15, 539-541 (1970). [5] E.A. Zabolotskaya and R.V. Khokholov, “Quasi-plane waves in the nonlinear acoustics of confined beams”, Sov. Phys. Acoust. 15, 35-40 (1969). Cimento [6] M. De Vito and P. Pantano, “Deduction of the Kadomtsev-Petviashvili equation for magnetosonic waves”, Len. NUOLH) 49 58-62 (1984). [7] H.A. Shah and R Bruno, “Oblique propagation of nonlinear magnetosonic waves”, J. Plasma Phys. 37, 143-148 (1987). [8] Y. Kodama and T. Taniuti, “Higher order approximation in the reductive perturbation method, I, the weakly dispersive system”, J. Phys. Sot. Jpn. 45, 298-341 (1978). [9] Y. Kodama and T. Taniuti, “Higher order approximation in the reductive perturbation method, I, the weakly dispersive system, errata”, J. Phys. Sot. Jpn. 45, 1765-1766 (1978). [lo] M.J. Ablowitz and H. Segur, Solitons and Inverse Scattering Transform, SIAM, Philadelphia (1981). [ 111 R. Grimshaw, “Evolution equations for weakly nonlinear, long internal waves in a rotating fluid”, Stud. Appl. Math. 73, 7-33 (1985).
WAVE MOTION 12 (1990) 373-383 NORTH-HOLLAND
REDUCTIVE NONLINEAR
PERTURBATION METHOD WAVE PROPAGATION. I
FOR QUASI
ONE-DIMENSIONAL
Tosiya TANIUTI* Department Kensington,
Received
of Applied Mathematics, N.S. W. 2033, Australia
School of Mathematics,
University of New South Wales, P.O. Box 2,
2 February 1989
By means of an extension of the reductive perturbation method to quasi one-dimensional wave propagation it is shown that systems of nonlinear evolutional equations are reducible to the Kadomtsev and Petviashvili equation and the ZabolotskayaKhokhlov equation in the weakly/dispersive and dissipative cases, respectively, which are given in three-dimensional forms representing two-dimensional slow modulations.
1. Introduction In a previous paper of the present author [l], reductions of nonlinear evolutional systems with two space-variables were discussed by means of the reductive perturbation method [2,3]. The reductions were based on the assumption that the systems are weakly dispersive or weakly dissipative and reduce to a hyperbolic system of first order partial differential equations if higher order derivative terms are neglected. Consequently it is assumed that for one-dimensional propagation the systems are reducible to the Korteweg-de Vries (KdV) equation or the Burgers equation. It was then shown that for quasi-onedimensional propagation (with slow transverse modulation) the systems can be reduced to the KadomtsevPetviashvili (K.P.) equation [4] and the Zabolotskaya-Khokhlov (Z.K.) equation [5], which are twodimensional extensions of the KdV and Burgers equations, respectively. However, for the hyperbolic system a restricted form was assumed, and the system was not given in a sufficiently general form to be adaptable to various physical problems. In the present paper the restriction to the hyperbolic system is removed; also the space is extended to the three dimensions, and the systems so generalized are reduced to the three-dimensional (3D) K.P. equation and the 3D Z.K. equation. In the next section the hyperbolic system is reduced to the equation which governs the quasi one-dimensional propagation of a simple wave of small amplitude; of course, this is the hyperbolic limit of the K.P. and Z.K. equations. The reduction is based on the linear dispersion relation, namely the linear algebraic equation for a plane wave of infinitesimal amplitude is solved by means of an adiabatic perturbation method in terms of a smallness of the wavenumbers for the transverse modulation. The method is closely analogous to the stationary perturbation method in quantum mechanics; the frequency shift corresponds to the shift of the energy level. In this way the orderings of time scales and length scales are determined except for one arbitrary parameter, which is fixed by taking into account the higher order derivative terms responsible for the * Permanent address: The Japan Society of Plasma Sciences Nagoya 460, Japan.
0165-8641/90/903.50
@ 1990 - Elsevier Science Publishers
and Nuclear
Fusion Research,
B.V. (North-Holland)
4F Tosho
Bldg., Nishiki
2, 20-29,
374
T Taniutil Reductive perturbation method
dispersion or dissipation. Thus, in Sections 3 and 4 the weakly/dispersive and dissipative systems are reduced to the 3D K.P. and Z.K. equations, respectively. In Section 5 an exceptional case is discussed.
2. Reduction of the hyperbolic system We first consider the following system of equations
for a column vector
U with N components
ul,uZ,.--SUN,
Here each Aj is an N x N matrix function of U, and necessary analyticities with respect to U will be assumed. The system is assumed hyperbolic in a certain domain of the U space, 0, in the following sense. For any vector U in 0, consider the linear algebraic equation for SU,
-WI+; A'k. j=l
SU=O, >
where kj are real constants. Then all the roots of the secular equation det -WI+
i Ajkj = 0 j=l
are real for all 4; some of them may be degenerated, however the associated N eigenvectors are linearly independent so that the solution SU is given by a linear combination of them. In particular we observe the eigenvalue of A’, that is, eq. (2) for k2 = k3 = 0, which is written as (A’--A)R=O,
(4)
where R is the column eigenvector and the associated row vector will be denoted by L. Hereafter, eqs. (2) and (3) will be considered for a constant vector U,, and A’( U,,) will be denoted by A’,, then eq. (2) holds for a plane wave of infinitesimal amplitude and eq. (3) becomes the linear dispersion relation. On the other hand, in eq. (4), A’, A and R will be regarded as functions of U and it is assumed that there exists at least one non-degenerate eigenvalue satisfying the condition A’=V,A.
RZO,
(5)
where VU is the gradient with respect to U. genuinely non-linear, the simple wave of this steepening.) We first suppose that a wave is propagated direction, so that lk21- lk31< Ik11, or by means they may be expressed as lkll- s”,
~k2~-~k3~-~Y+”
(The wave mode under the condition (5) will be called mode propagating in the x,-direction can break itself by in the x,-direction while it varies slowly in the transverse of a parameter E ordering a smallness of the amplitude
(yaO,a>O),
where y (2 0) and a (> 0) are parameters to be determined later. Then eq. (2) is written as (H,+zH’)ry=n!I$
(6)
T. Taniuti / Reductive perturbation
in which H,, H’ and A are represented tH’=
Ho = A;,
5 A;(k,/k,), /=2
method
375
as (El=C),
A =o/k,,
and Y is proportional to 6U. It should be noted that the normalizations of 6U and W are arbitrary. The eigenvalue problem (6) may be solved by the adiabatic perturbation method which is familiar in quantum mechanics. Expanding Y and A in powers of i? as Y=Yo+EIY,+E12Y2+.
**,
(7)
A=A,+&i,+E^‘A,+~~~,
(8)
we have the sequence of equations:
(9)
H,Yo=&Yo, (Ho-A,)Y,+H’Yo=A,Y,,,
(10)
(Ho--A,)Y2+H’Y,-A,YY,=A2Y,,,
(11)
....
Equation (9) is identical with eq. (4) at U = U,,, namely A, = A,, and Y,, = &. For convenience, corresponding to Y,, the adjoint eigenvector Lo is denoted by ‘Doand the N eigenvectors {eon} (i = 1,. . . , IV) are normalized so that the orthonormal conditions hold, namely, (12)
(@On,Yclm)= %n.
By the assumption the N eigenvalues {A,,} of eq. (9) are real and the corresponding eigenvectors { YO,} are linearly independent. Each of {A,,} and {Yen} changes by the perturbation. We now consider the adiabatic change of an eigenstate of H,, which is non-degenerate and genuinely nonlinear. Let the state be specified as the pth state. Then the first order shift of Ao,, is readily obtained by eq. (lo), that is, (13)
Alp = (@,,H’Yo,) = (plH’\p). Also, representing y
=ct
1P
n
QIP as a linear combination
of {Y,,} and using eq. (12) we have
(nlH’/p)PO”+ cpyop,
(14)
A,--Aon
in which the pth term is not summed, while c, is an arbitrary constant. Introducing eqs. (13) and (14) into eq. (11) multiplied by @, enables us to obtain the second order shift of the pth eigenvalue, A
=
cr
2P ”
tilH’b)(nlH’l~)E &p
-A,.
+I
H’H’
Ho_no
Ip)-
(15)
Eqs. (13) and (15) yield the shift of the frequency due to the transverse modulation with k2 and 5, which takes the form
,i2~l&bW,-, Ii, (PI H:_A;op bMn/kl .
w = Aopkl+
376
T. Taniuti/
Reductive perturbation
method
This equation shows that the time evolution comprises the three different time scales of the orders of cv7, s-(Y+a), and c-_(Y+ZP);the first two terms on the right-hand side can be eliminated by introducing the transformation, rlr = s y+a(~, -(plA;(p)t),
5 = c y(x* - Aopt),
1= 2,3.
For the last term we require that it is balanced with the nonlinear frequency shift of the lowest order 6(hkJ - h;&k, - .sy+‘. Then (Yis fixed to 4. It is obvious that by this choice of (Ythe nonlinearity is in balance with the linear dispersion due to the transverse perturbation. On the other hand U must be expanded to be matched with ?P, while the order of 6U has been tacitly assumed as E, that is, ?P -O(E), hence eq. (7) leads us to assume u=
UO+EU,+E3’2U,,2+&2U2+*
* *.
(16)
We are now ready to make the reduction of eq. (1). Introducing by the equations
the coordinates
(17)
5 = cY(x, - A$), c y+“2(x, - (plA;(p)t),
171=
5, vf (I = 2,3) and r
I = 2,3,
(18)
T=&y+‘t,
(19)
and substituting eqs. (16)-(19) into eq. (l), we have the sequence of equations corresponding to eqs. (9)-( 10) (20)
(&-&,)u,,=O, cHO
-
AOp)
(HO-AO,)U2,,+
u3,2,C
+
i I=2
Ul,,+
(21)
(~iA;kdl U,,,, = 0,
[Ai-
6
. P,a)Ou,,,+,i2
(A:,-(~lA;bW3,2,,,
=O,
(22)
in which the subscripts after the commas denote the differentiations. In analogy with the adiabatic perturbation let each Uj be given by a linear combination of !PO,,_Then the reduction goes parallel to that of eqs. (9)-(11). From eq. (20) it follows u 1 = u(‘+?@+ v 1.
(23)
Here u(‘) is a scalar function of 5, r and nl to be determined later; VI is an arbitrary vector independent of 6, however its ?P,,-component can be absorbed to the first term so that (%DoP * V,) = 0. Since the normalization of Y,, has not been specified, it may be fixed by equating a non-vanishing component of ly,, say I,&,,, to unity, then 8) becomes u1(‘) . Introducing eq. (23) into eq. (21) multiplied by c&,,, (n=l,...,N)yields U 3/U
- -C’ R { i2 [ (4 H
-
,i2(PIA~I v,,,,) = 0,
(24)
0
(25)
in which u(~‘~) is a scalar function undetermined in this step. However, when eq. (24) is integrated with respect t0 ( t0 get U3/2, the second term in the bracket on the right-hand side gives rise to the secular
T. Taniuti / Reductive perturbation
method
317
term, because VI is independent of 5. Hence V, must vanish, and eq. (25) becomes trivial. Substituting eqs. (23) and (24) (with V, = 0) in eq. (22) and using eq. (12), we obtain
(26) in which cl”
,“FJop IP).
= _ (pI
(27)
0
The reduction in the two-space dimensions was done in the previous paper [l]. However it is assumed that (P~AZ~P)= 0, consequently 7)2was given by Ey+1’2x2. As will be seen in the following example, this is quite a stringent restriction and does not hold in many physical problems. Example: gas dynamics The ideal gas dynamic equations
are given by v,,+v*
p,,+v-vp+pv*v=o,
vv+(c~/p)vp=o.
Here p is the mass density v = ( vX,v,, vz) the flow velocity, c, is the sound speed which is a function of p. In matrix form they are written as U,,+ i A’U,i=O, i=l
where v,
P
Z/P
vx
1
0 0 0
P
0
v, 0
0 vy
0 0
0
0
v,
VY
AZ’
i 0 cf/p 0
0 0
’
0
0
0
0
v, 0
II PO
G-0
.
VYO vzo
Then the linear dispersion relation becomes (w-k.
vo)‘[(u-k.
v~)~-c~~]=O,
that is, there are three eigenvalues w=k~v,*c,,~k~,
o=k.
v,,
’
VX1
A3=
Let U, be specified as
uo=
0
ii
Cl/P VZ 0
v, 0 00
v, 00 0
vz 0P
1*
378
T. Taniutij Reductive perturbation method
For the pth mode we may consider w = k - v,,+ c,lkl. If k, = (5, k,) is
and the third one is degenerate. small, w is approximated as
w = 00 . k+c,,k,+tc,,((k~lk,)+(kIlk,)), from which we can anticipate that (P~:/P) = GO,
(PIA~P) = vyo,
(PI&/P) = GO,
C’” = ~CsoSrm.
This is shown by means of the R.P.M. as follows. The eigenvalues and eigenvectors obtained.
I [I
of AA are readily
1
qo* =
*Go/P0 0
1
’
A”, = Go* Go,
0
0
ry,,=
0 1 ’
&=A:=
vxo,
0
@o*= [*Go/P0 1 0 w*tpo/2cso), Consequently
II 0
qo.%= ;
,
1
d&3=(0 0 1 O),
G&=(0
0 0 1).
we have
(@o&&I qo*> =0,
(@o+lAfl po3>= PO/~, (@03lA;l*o+)= doho, MMI&I PO+)= 0, which are introduced
into eq. (27) to give
p2 = (+I‘4~13)(3lAi+) = c ,2 so , Go + Go - Go likewise we find c33 = &o/2, Since the eigenvectors modulation gives rise interpreted as the self the transverse flows +
c*3= c3*=0 Vo3 and Iyo4yield v, and v, varying with x, respectively, we find that the transverse to the transverse flow. And the second order frequency shift $,,k:/ k, may be energy of a sound given by virtual transverse motion in the process the sound + the phonon, while the direct sound-sound transition is prohibited.
3. Reduction of the weakly dispersive system We now extend eq. (1) to
(28)
T. Tanhti
/ Reductive perturbation
method
379
Here K”’ are N x N matrix functions of U. If all the roots o of the equation -oI + ; A’kj + i j=l
i,j,l=
K”‘k,k,k,
=0
(29)
1
are real for any vector k of sufficiently small length, the system (28) is called weakly dispersive. Since (Y has been given by f, we have the orderings k, - eY, k2 - ks - Ed+“*, hence, eq. (6) is modified to [Ho+ e ‘/*HI+ E2YH”+. . .] t$f = AIJ?
(30)
in which e2YHlr= K;“( k,)* and other terms resulting from the third order derivatives are of higher order. It is natural to require that the third order dispersion is in the same order as that due to the transverse dispersion. Then y is determined by 2y - 1, that is, y = f. However, our purpose is not to solve eq. (30) but to make the reduction of the system (28). Hence we assume the expansion (16) for U and introduce the transformations (17)-(19) with y = i, 5 = E”*(x’ -L&t),
TI= s(x, -(p]A&)t),
7 = E3/*t.
Corresponding to the adiabatic assumption, it is assumed that (i) each q((i=1,$,2,$ ,...) are given by a linear combination of the eigenvectors of AA, { !P,,“} = {It,,}, the coefficients of which are functions of 5, nr and T. As will be seen later, the condition (ii) (plK”‘]p) # 0 is also required. Introducing the expansion and the transformations into eq. (28) we find that the equations for U, and U3,2 are the same as eqs. (20) and (21), respectively, and eq. (22) is supplemented by the third order derivatives: (Ho-&,)~2,,+
K,r+
u,
* (v,a),~,.,+,~2(Ab-(plA~lp))U,,2,,,+~11’U,,~~~=0,
(31)
where U, and I&,* are given by eqs. (23) and (24) with V, = 0. Then multiplying eq. (31) by GOP we obtain the 3D K.P. equation
in which Y= (pl~“‘lp). For the dispersive terms comprising the third order derivatives we can consider a number of different forms [2], such as
which leads to the 3D K.P. equation with v = (~~K~“G~“M~“~p);
380
7: Tan&i/
also, one of the space-derivatives
Reductive perturbation
method
may be replaced by the time derivative, for example,
U
yields
v = -AOp(pIKAIIp).
The same applies to the dissipative terms which will be discussed in the next section.
4. Reduction of the weakly dissipative system As another extension of eq. (1) the second order derivatives may be added to give (33)
We call this system weakly dissipative, if the equation -WI+
Ajk,+i
i
i
K”kikj
-0
(34)
i,j=l
j=l
admits at least one complex root and the imaginary parts of all the complex roots are negative for any which vector k of a sufficiently small Ikl. In this case azyH” in eq. (31) takes the form EYH”=iKA1kl, implies y = 1. By means of this choice of 7, the reduction goes parallel to that of eq. (28), provided that in place of the assumption (ii) (nlK”ln)=
-v*
is assumed for any non-degenerate
(35)
eigenvector of A’. And we obtain the 3D Z.K. equation
This result also shows that if all the eigenvalues condition for the weak dissipation.
of A’ are distinct, the condition
(35) is a sufficient
5. Exceptional case
If for an eigenvector, say, Rp, the condition (35) does not hold, that is, if (plK%)
=0
(37)
is valid, eq. (33) cannot be reduced to the Z.K. equation. In this case, choosing y = f enables us to reduce the system to a K.P. equation with a non-Hermitian term [l]. This is readily seen by observing that the non-vanishing contribution of E~H” appears first in its second order, which is to be in balance with the second order of &“*H’. We thus find that eq. (21) is altered to
which is solved for U3,2,5 to give (38)
T. Taniuti / Reductive perturbation
method
381
whilst eq. (22) becomes
which leads to the equation
Here v and
K~
are given by
K/= --(PI
A$;’
+ K;‘A;
IP>+(PIK:‘+
K~P).
K-&p
Since the product Cf=, Ktqf cannot be made a definite sign for all the vectors q = (q2, qJ even if sufficiently small, eq. (39) is not dissipative and admits growing modes, unless the condition K2 =
191
is
(40)
KJ = 0
holds, which however is not valid, in general. Hence, it may be said that when the condition (35) does not hold, in general, eq. (33) is not weakly dissipative. We thus find that if all the eigenvalues of A’ are distinct and the present orderings are valid, the condition (35) is the necessary and sufficient one for the weak dissipation. However, there exists the exceptional case that for a non-degenerate state both the conditions (37) and (40) hold, in which case eq. (39) reduces to the K.P. equation. (It may be noted that without the transverse modulation eq. (39) becomes the KdV equation regardless of eq. (40).) Since this does not prove that Im o = 0 in all order, it will require further considerations as to whether the system (33) can be weakly dispersive. Of course, there are such cases; for example if the matrices Aj are symmetric whilst K”‘ are antisymmetric, then i Ajk,+i i K”lcJc,
j=l
i.j
becomes Hermitian; in this case it is readily confirmed that the conditions (37) and (40) are satisfied for all eigenvectors. Finally we note that in correspondence to the condition (37), if the condition (pI~“‘lp) = 0 holds, by means of the choice y = f, eq. (28) is reduced to the following equation
I=2
I,m=2
(41)
Here v’ and C, are constants, their explicit forms may be deduced from v and K/ by noticing the correspondence among K ii and K”‘; however eq. (41) is a dispersive equation, hence the condition on I;; such as eq. (40) is not required. This equation has not been studied in any detail.
382
T. Taniuti / Reductive perturbation
method
6. Summary and remarks
In this paper we have extended the reductive perturbation method for one-dimensional propagation to quasi one-dimensional propagation in three-dimensional space. A crucial point in this extension is the reduction of the hyperbolic system (1) to the scalar equation (26), that is, the hyperbolic limit of the Z.K. and K-P. equations describes a weak simple wave with slow transverse modulations. Of course, in general this equation does not have a solution for all time. The weakly/dispersive and dissipative systems are introduced by supplementing to the system (1) the third and second order derivatives, respectively; the reductions of those systems to the 3D K.P. and K.Z. equations go parallel to that of the system (1). Also, there exists an exceptional case that the system with second order derivatives becomes dispersive. A generic theory for the exceptional case may not be significant, but it is desirable to study physical examples. The most interesting one may be the magnetoacoustic waves in two-fluid magnetohydrodynamics. In this case the reductions to the K.P. equation were given by De Vito and Pantano [6] for cold plasma and by Shah and Bruno [7] for warm plasma. However, both of these works deal with the two dimensional case under the restriction (plA&)=O, and the results cannot directly be extended to the three-dimensional case, which we will discuss in a next paper as an example to the present paper. In all of the reductions, a basic assumption is that the solution vector U, expanded as (16), is given by a linear combination of the eigenvectors of A& the coefficient of each eigenvector is a function of the strained variables 5, n,, T. The first order term of the expansion (16), U, , is proportional to an eigenvector tyop (of a nondegenerate state) and its coefficient u(r) is determined by the Z.K. or K.P. equation. In the ordinary case, the next order term Q2 is given as follows: each coefficient of lyO, (n # p) was given in terms of u(‘) by solving the algebraic equation (21), while that of lyop, u(~‘~), is governed by a linear partial differential equation which follows from the higher order equation (for Us,*) as a solvability condition. Thus the partial differential equations for the coefficient of ??Opare obtained successively as compatibility conditions, whilst the coefficients of Y,,, (n # p) are expressed in terms of lower order terms by solving algebraic equations, which may be deduced from the adiabatic perturbation on the linear dispersion relation. In the one-dimensional case the (linear) higher order equations admit secular terms, which for the dispersive system can be removed by a renormalization technique [8,9]. It will be a future problem as to whether such a procedure will work also in quasi one-dimensional propagation. The dispersive or dissipative system considered in this paper seems to be sufficiently general for applications to sound-like modes, because it is based on the hyperbolic system. It should however be remarked that in the present paper propagation in unbounded space has been considered. In some cases, effects of boundary conditions at side walls (in the transverse direction) will become significant; then the present method of reduction is not applicable or must be modified. As is well known, the K.P. equation was introduced originally for the shallow water wave [4, lo]. In this case the fluid is inviscid and incompressible, consequently the basic system is not hyperbolic, and the boundary condition at the surface of the water plays a crucial role for the time evolution. Hence the present method does not apply to this system, but will be applicable when the system is reduced to an equation of the Boussinesq-type which takes a form similar to eq. (28). The asymptotic reduction of the incompressible hydrodynamic equations for weakly nonlinear long internal waves was established by Grimshaw [ll]. It deals with various cases involving the effect of the Coriolis force due to the earth’s rotation. The method of reduction is entirely different from the present one, because of the conditions in the vertical direction and also due to the presence of side-walls in the horizontal direction. It may be noted that nevertheless for waves of the same (horizontal) length scales, the K.P. equation (with a linear term accounting for the rotation) is obtained after integrating over the vertical direction. This reflects the intrinsic similarity law of the K.P. equation.
383
T. Taniuti f Reductive perturbation method
Acknowledgement The present paper was completed University
during my stay at the Department
of Applied
Mathematics
of the
of New South Wales. I wish to thank Professor R. Grimshaw for reading through the manuscripts
as well as for the helpful
discussions,
and the University
also indebted to Professor M.J. Ablowitz for his useful Clarkson University, in regard to the exceptional case.
of New South Wales for financial comments,
support. I am
made during my short stay at the
References [l] T. Taniuti, “Reductive perturbation method for quasi-one-dimensional wave propagation”, in: A. Jeffrey, ed., Nonlinear Wave Motion, Longman, Boston (1989). [2] T. Taniuti and C.C. Wei, “Reductive perturbation method in nonlinear wave propagation”, J. Phys. Sot. Jpn. 24,941-944 (1968). [3] T. Taniuti and K. Nishihara, Nonlinear Waves, Pitman, London (1983). [4] B.B. Kadomtsev and V.I. Petviashvili, “On the stability of solitary waves in weakly dispersive media”, Sov. Phys. Dokl. 15, 539-541 (1970). [5] E.A. Zabolotskaya and R.V. Khokholov, “Quasi-plane waves in the nonlinear acoustics of confined beams”, Sov. Phys. Acoust. 15, 35-40 (1969). Cimento [6] M. De Vito and P. Pantano, “Deduction of the Kadomtsev-Petviashvili equation for magnetosonic waves”, Len. NUOLH) 49 58-62 (1984). [7] H.A. Shah and R Bruno, “Oblique propagation of nonlinear magnetosonic waves”, J. Plasma Phys. 37, 143-148 (1987). [8] Y. Kodama and T. Taniuti, “Higher order approximation in the reductive perturbation method, I, the weakly dispersive system”, J. Phys. Sot. Jpn. 45, 298-341 (1978). [9] Y. Kodama and T. Taniuti, “Higher order approximation in the reductive perturbation method, I, the weakly dispersive system, errata”, J. Phys. Sot. Jpn. 45, 1765-1766 (1978). [lo] M.J. Ablowitz and H. Segur, Solitons and Inverse Scattering Transform, SIAM, Philadelphia (1981). [ 111 R. Grimshaw, “Evolution equations for weakly nonlinear, long internal waves in a rotating fluid”, Stud. Appl. Math. 73, 7-33 (1985).