- Email: [email protected]
On approximate methods for constructing the evolution equations
Inr. J. Men-Ltneor .Mechana. Prmtcd m Grrat Bntam.
Vol. ?I. No. I. pp. 15-25. 1986
oO?O-7462.86 53.00 + 00 Pcrgamon Press Ltd.
ON APPROXIMATE METHODS FOR CONSTRUCTING THE EVOLUTION EQUATIONS E. PELINOVSKY~,V. FRIDMAN: and J. ENGELBRECHT~ of Applied Physics, U.S.S.R.Acad Sci, 46 Ulyanov St., 603600 Gorki. U.S.S.R.:
tfnstitute $Radiophysical Research Institute, 25 Lyadov St., 603600 Gorki. U.S.S.R. and §Institute of Cybernetics, Academy of Science of the Estonian S.S.R.. 21 Akadeemia St.. 200104 Tallinn, Estonia.
U.S.S.R. (Receired 3 July 1984;
received for publication
7 March
1985)
Abstract-Three methods are used to construct the non-linear evolution equations governing the propagation of single waves: the iterative, the asymptotic and the spectral. The paper gives a comparative analysis of these methods including the problems of the convergence and the correctness. The model second-order non-linear equation is chosen as a basic one and the corresponding evolution equations are derived by means ofall these methods. It is shown that although the evolution equations of the first order are the same within all of three presented methods, the next approximations differ and the best convergence to the exact solution is obtained by using the iterative method.
1. INTRODUCTION Wave propagation theory is historically most closely related to the development of the approximate methods for solving partial differential and/or integrodifferential equations (systems of equations). However, even in the linear approach the exact solutions describing the dependence of field variables on initial conditions are rare. The number of exactly solvable non-linear problems is certainly much smaller [ 11. This is the main reason why the approximate methods are so intensely developed for both linear and non-linear problems. Generally speaking, the approximate methods used in wave propagation analysis may be divided into three main groups: (i) approximate analysis of the exact solution [2,3]; (ii) perturbative analysis of the solution with small (slow) derivation from a fixed and known one [4-61; (iii) simplification of equation (mathematical models) describing the process [2,7-91. The group (i) takes the exact solutions written in the integral form as the basis and the approximate solutions are obtained by means of some classical approximation method, for example, by means of the method of stationary phase. This method, first proposed by Kelvin for solving the wave pattern formation after a moving ship is novv considered to be a classical one. Unfortunately it is practically impossible to get the explicit formulae for non-linear problems; that is a great drawback of this physically well grounded method. The methods of group (ii) are better in this sense because the non-linear problems are also solvable. The fixed solutions are usually taken in the form of stationary plane waves and the formerly constant coefficients are considered to be slowly perturbed; that permits one to use the series representation containing the small parameter E. The perturbation methods are widely used [4,5] but the main disadvantage of such methods is in the secularity of the additional terms and usually the perturbed solution may be used only in a certain space-time interval. The methods of group (iii) do not simplify the solutions but the equations governing the wave process paying at this stage no attention to the solutions themselves. The set of small parameters connected either with the initial conditions or with the coefficients of initial equations is used and the perturbation method together with the method of deformed coordinates is applied. The best results are obtained when the wave process is separated into single waves described by separate equations. These equations are usually called evolution equations governing the distortion of a single wave along the properly chosen characteristic (ray). The best example of such an evolution equation is the well-known Korteweg-de Vries equation. There are several methods to simplify the governing equations and this paper presents an attempt to classify them giving also a comparative analysis of the known methods. Three methods-the iterative, the asymptotic and the spectral-are briefly
E. PELINOVSKYet al.
16
described in Section 2. Section 3 deals with the comparison of these methods. The model second-order non-linear equation I_?*11
d211
a*u*
ax*
ax*
---_=E_
at*
(1.1)
is chosen as the basic one and the corresponding evolution equations are derived by means of all these methods. Here u is the field variable and E the small parameter. The accuracy of the solutions is analysed and the results are compared with each other. It is shown that although the evolution equations of the first order are the same within all of the three methods presented, the next approximations differ and the best convergence to the exact solution is obtained by using the iterative method. This is not the first attempt to demonstrate the efficiency of separate methods in solving a certain non-linear equation. In [6] several perturbation methods were compared by the solution of the non-linear second-order equation with dissipation. The extensive analysis of the solution of the non-linear Boussinesq equation is given in the recent monograph by Jeffrey and Kawahara [7]. The main difference of this paper with regard to [6,7 a.o.] is in the concentration on evolution equations only, i.e. single wave processes are described. It is difficult to overestimate the importance of the evolution equations in contemporary mathematical physics, therefore the convergence and the correctness of the evolution equations must be clearly stated. The choice of the model equation (1.1) is based on the existence of the exact solution that simplifies considerably the analysis. The paper forms actually a summary of a longer research paper that will be published in the future. 2. THE
APPROXIMATE
METHODS
In this Section the approximate methods for constructing the evolution equations are briefly discussed. The methods under consideration are: (i) the iterative, (ii) the asymptotic and (iii) the spectral method. The one-dimensional processes are analysed only in order to obtain clear physical background. 2.1. The iterative method In our opinion, this approach gives the best physical understanding of the process [lo, 111. Let the wave propagation be described by the system
zg+A(r,%)g =
&{U,X,t,~,T,E)
(2.la)
1.
(2.lb)
e
5 =
Et,
x =
EX,
EC
Here U is the n-vector of field variables, I is the unit matrix, A is the n x n matrix, I? is a nonlinear integro-differential operator. The eigenvalues of matrix A are all assumed to be real and different. It means that the system (2.la) with E = 0 is a hyperbolic one. The initial and boundary conditions for (2.la) are given in dependence ofp and will not be discussed here. In case E = 0 the system (2.la) has the solution n
U = C Vi(X - E.it)ri
(2.2)
i=l
where vi is the scalar function, r is the right eigenvector of matrix A for eigenvalue i (the velocity) given by Ar = E.r,
detlA - ).I1 = 0.
(2.3)
We represent the initial system (2.la) in the normal form [12], changing the variable U to V by making use of the formula
u=
YV
(2.4)
17
On approximate methods for constructing the evolution equations
where Y is the n x n matrix formed by the linearly independent eigenvectors ri of matrix A. As far as A is dependent on “slow” coordinates T and ;c,the eigenvectors ri and the eigenvalues 1bidepend also on these coordinates. Substituting (2.4) into (2.la) we get
-av at
(!?+Z)v]
++Y-p-
= - Ef[v,x,t,X,q&].
(2.5)
Here 8 is the diagonal ‘matrix with the elements ii (T,x) and Y-i is the inverse matrix of Y, consisting of the left eigenvectors Ii of A. The usual normalisation is here assumed: lirj = ⅈ 6, is the Kronecker symbol. According to (2.4) and (2.5), the solution may be given as a superposition of waves, each one propagating (at E = 0) along its characteristics and interacting with others only due to the perturbation f # 0. In many cases only m < n waves ri are generated by the initial and boundary conditions. When the initial condition is localized then the wave process is soon decomposed into the separate independent waves, so that in the far field each of them may be analysed separately. In this case the vector V contains m “basic” waves and other n-m waves may be calculated by the usual perturbation scheme. For every si, i = 0, 1, 2,... the equation (2.5) yields
ap
EO. .x
+
&T)g) =0
p)s = 0,
ap
s= m +
T)z, =&f;:{up
&’ :- ar + %i(X,
adl) ad*' -$ + ux,T)“-=&f,{u(ll’)... ax
)...
(2.6a)
i = l,...,m l,...,n
(2.6b)
IQ,0 ,... O,%,S,O)
(2.7a)
o’,“,o
(2.7b)
)...
. .. .. .. . aup acp En: dt + li(X,T)z = &f{C(“),... L$‘,L$;i)
~+%s(x,T)~=&~{u’~’ ,... (n) (nu,,u,+1
O;x,qO}
,... Lf-“;%,T,&} 1) )...
(2.8a)
uj:-“;&T,E}.
In every step the system is decomposed into two groups: (i) m coupled non-linear evolution equations (2.8a) for “basic” waves and (ii) n-m linear independent evolution equations (2.8b) with known right-hand sides. Now it is clear that this approach leads to the iterative procedure for m interacting waves. The main subcase is m = 1. Let $) = C, i.i = i., fi = j. Then (2.8a) yields
2 +i.(r,g =Ef{c, c(“-‘) ,x, T, E}. The first approximation needs E = 0, therefore C, = 0. The structure of the equation (2.9) depends certainly on the functionalF in the initial system (2.la). Usually it is given in the form of a sum of derivatives and integrals from the field variables. Using (2.5) for calculating the right-hand side in (2.9) the functionfmay be represented also in the form of a sum of certain derivatives and integrals [lo, 111. The general form may be written as h-‘dx+sh’“‘+s
ah(l) + Ea2hc2) -zx ax2
+ E _aw + . ..= o sx3
E.
18
PELINOVSKY et al.
where /I(‘)are certain functions oft’. The equation (2.10) is the evolution equation of the first order for a single wave. 2.2. The asymptotic method We shall start from the initial system (2.la). The solution is sought in the form of the asymptotic series [8,9] u =u,(g,?)
+ sU1(5,r) + &%J2(5,5) + .**
(2.11)
where the independent variables are either 5 = x - At,
T =
EC
(2.12)
or ( = x - At,
t =
EX.
(2.13)
For the case of a homogeneous and stationary medium as shown in [9], the transformation (2.12) must be used for the problem with the initial conditions and the transformation (2.13) for the problem with the boundary conditions. In this way the evolution equation is always solved with the initial conditions. Substituting (2.11) into (2.la) and using (2.12) we get the sequence: & =
0:
au0
P- %- =O &=
(2.14a)
1: (2.14b)
E = 2: (2.14~)
Here the variational derivative c@/sUo and @/as are determined approximations are, in principle, equivalent to (2.14c). The equation (2.14a) yields
uo =
Lg,t)r
at E = 0. The higher
(2.15)
where u is a scalar function and r, as above, the right eigenvector of A determined i.. As far as det(A - 1.1) = 0, then the solution of (2.14b) exists only for the certain right-hand side and the orthogonality conditions give
+-zfj+o where I is the corresponding
(2.16)
left eigenvector. Substituting (2.15) into (2.16), we obtain
-&
=
j(c) = l@:(w)
(2.17)
On approximate methods for constructing the evolution equations
19
i.e. the evolution equation of the first order. It is clear that (2.17) and (2.9) coincide if (2.9) is rewritten using (2.12). It means that the evolution equations obtained using the iterative and asymptotic methods coincide in the first approximation. It must be pointed out, however, that the asymptotic series (2.11) has a serious advantage over the usual perturbation series with dependence only on 5. in the last case U, cannot be bounded for the far field and the series is not convergent. In case of asymptotic series (2.1 l), U0 is determined so that Ui is bounded (cf. (2.15) and (2.16)). The next approximation needs Ui in the explicit form. From (2.14b) and (2.16) we have (2.18) where Uie is the forced solution of (2.14b) and uI is a scalar function, determined from the orthogonality condition for the right hand side of (2.14c). As above, this condition means that Uz is bounded. Substituting (2.18) into (2.14~) and multiplying with 1 we get the evolution equation of the second order
au,
-
a7
=
I~(U,, +clr) + 1:.
(2.19)
Next approximations are obtained in the same way. It is important to underline that the evolution equations of higher approximations than the first found by iterative and asymptotic methods differ. However, if the multiple scales are used then the influence of the forcedsolutionsUIO,UzO,... may be replaced by the terms dr/‘&i, where ri = &t, i = 1,2,. . . . 2.3. The spectral method This method is convenient for waves in dispersive media and gives an excellent possibility to explain the decomposition into single waves (one-wave approach) 19,131, Let the system (2.la) be rewritten in a generalized form
where i is a linear differential operator (a matrix depending on r3U/&), ti is a non-linear operator. The wave process is decomposed into the eigenvalues of the linear (E = 0) problem U = c [dk Urt$‘exp[i(wrt
(2.21)
- kx)]
where r$’ is determined by i(--ik)r;:
= -io$‘r;:
(2.22a)
det li( - ik) + io? II = 0.
(2.22b)
Here Ur is the spectral amplitude, and m is the mode number. We assume CJJ = Q’(t). Substituting (2.21) into (2.20) and performing the inverse Fourier transform in the coordinate we get the infinite system of equations for the spectral amphtudes diJr -=E dt
c dk, dkz c r; ;r,’Q: K; mf,maf
x exp[i(oz$ + I@; - oZ’ft&k,
+ kz - k)]
(2.23)
where the matrix-coefficients V;:r; r; are determined by the given operator M and depend only upon the wave numbers k and the mode numbers m. As far as the Fourier integrals may contain also the given fields of perturbation, the equations (2.23) are able to describe not only the interaction between single waves but also the interaction with perturbation fields. No general methods for solving the equations (2.23) are known, therefore we seek possible
E. PELINOVSKYef al.
20
simplifications. Evidently the strongest interaction of spectral amplitudes occurs when the synchronic conditions (for quadratic non-linearity) kl + kz - k = 0, ’
A = CU”’ kl + we; - 0;: = 0
(2.24)
are fulfilled, or at least A 1 E.If the initial system is of high order (the number of modes is big) then practically always it is possible to satisfy (2.24). Therefore the decomposition into single waves is possible if v”kk,kzmlmz
0
=
t
ml.2
f
(2.25)
m
and in this case the wave with wave number m is propagating without any dependence on other modes. The spectral method gives the excellent possibility to examine the accuracy of the onewave approach that is comparatively difficult by making use of the iterative method and the asymptotic methods. The crucial points are the synchronic conditions following from the dispersion relationship without the detailed analysis of the non-linear interaction. If the one-wave approach is possible, then the equation (2.23) yields to (index m is neglected!)
duk
-
dt
=
dki dkz %k, k2uk, ukz exp(iAt)b(k, + k2 - k).
(2.26)
Now the Fourier transform must be performed in order to obtain the evolution equation that generally speaking is of an integrodifferential form. Further details will be given in Section 3. 3. THE
COMPARISON
OF
THE
METHODS
3.1. The statement of the problem and its exact solution We shall start from the model non-linear wave equation
a2u ---=
a2u
at2
ax2
a2u2
(3.1)
yp.
It is equivalent to the system (2.la) with
(3.2)
The equation (3.1) or the system (2.la) with (3.2) permits the Riemann solution. Assuming ti = c(u), the governing system may be represented as (3.3a) 4
g;
^
+ (1 + 2ELI)g
= 0.
(3.3b)
The nontrivial solution of the system (3.3) exists provided the system determinant is equal to zero. This condition yields = 1+
c’ =
&f&-l
[(l +
(3.4a)
2EU
243’2
-
I].
On approximate methods for constructing the evolution equations
21
Now it is easy to find the exact solution from (3.3). For the wave propagating in the direction x > 0 we obtain
u = F[x - V(u)t]
(3Sa)
V(u) = (1 + 2&U)“2
(3Sb)
where F is an arbitrary function determined by the initial conditions. 3.2. The iterative method We shall represent the governing system (2.la) with (3.2) in the normal form. Introducing 1 u=,(w-w-),
,+w-w_,
(3.6)
-;-&(w+W-)2
(3.7a)
the governing system takes the form g+g=
aw_ aw_ E a - - - = -&(w at ax
+
w-)2.
The iterative procedure (see Section 2.1) gives in the zeroth approximation with E = 0 and
(3.7b)
the system (3.7)
w = w(x - t),
w- =o
(3.8)
u = u(x - t),
v = u.
(3.9)
that is equivalent to
This coincides with the exact solution in case E = 0. The first approximation
is governed by
(3.10a)
aw_ ---=_-*
at
a~_
E
ax
4
aw2 ax
(3.10b)
Because the initial system is hyperbolic, the evolution equation (3.10a) has the typical form of Riemann wave equation with the velocity
v, =
1 + ;w.
(3.11)
Noting that w = 2u + O(E), the final result is
v, = 1 + EU.
(3.12)
This result is easily obtained from the exact solution (3.5b) after series representation with two terms kept in it. In order to construct the evolution equation of the second
E. PELINOVSKYet
12
al.
approximation we need w_ with the correctness of E. After integration determined by
of (3.10b) w- is
3 w_ = --_)v2 8
(3.13)
and substituting this result into (3.7a) the sought evolution equation takes the form (3.14) The velocity V2 is now easily determined as v2 =
1 +
;w- ;e2w2
(3.15)
and (3.6) yields 1 11 ‘y
TW
3 -
(3.16)
&
or 1 U’21 2U + -&l12. 2
(3.17)
In terms of u the velocity V2 may be calculated by
v, =
(3.18)
1 + EM- +,2.
Once more, this result is easily obtained from the exact solution (3.5b) keeping three terms in series. It is easily understood that the correctness of the solution obtained by the iterative The evolution equation of the nth method increases with higher approximations. approximation is the representation of the initial equation for the Riemann wave with correctness of s”. The exact hyperbolic equations are correct in a bounded time up to t,--the shock wave formation time. The evolution equations are correct for t c t,, with maxlt, - t,J c O(E”).
(3.19)
3.3. The asymptotic method Introducing the independent variables (=x-t,
7 =
Et
(3.20)
the governing system is rewritten in the form
au ati
-g-g= au
au
-@+Zf=
-“t
all
au
au2
a7
ag
----_-_.
(3.21a)
(3.21b)
The unknown functions are represented in series u, v =
(h t.0)+ E(U1,VI) + E2(l12, VJ + . ‘.
(3.22)
On approximate methods for constructing the evolution equations
In the zeroth approximation
23
the system (3.21) yields (3.23a)
_
&fJ+ au0-
at
at
D0 =
uo(5,7).
0
(3.23b)
with the trivial result
The first approximation
(3.24)
is governed by
-_aut au1 --ali0
ac:+T&=
(3.25a)
a7
(3.25b) and the second by -- au2
av,
-- au1
au2
(3.26a)
a7
at +a(=
au2
-z+af=-~-2$po,s1).
(3.26b)
The system (3.23) corresponds to the system (2.14a), the system (3.25) to the system (2.14b) etc. The orthogonality condition (2.16) gives the evolution equation of the first approximation
g?+.,!22=0 a<
(3.27)
that coincides with (3.10a) with correctness of E. The formula (2.18) gives
Ul =
U+(&T),
u+(T,z) +
u1 =
$5
(3.28)
and the equation (2.19) takes now the form
au
s
+
a
-&uou+)
au0
I
-u2 - = 0. 2 ar:
-
(3.29)
Its solution is u+
1-u
1
2
2O
(3.30)
and, consequently 1
u1 =
-II& 2
ur = uf.
(3.31)
E. PELINOVSKY et al.
24
Now it is easily concluded that the asymptotic method gives the small additions to the main solution uo, and its solution is continuous for t < t,, t, = tl. Therefore the condition max It, - t,l < O(E) determines the time interval when the asymptotic (3.19)].
(3.32)
evolution equations are correct
[cf.
3.4. Spectral method The solution of the governing equation (3.1) is sought in the form of the Fourier integral (2.21) for colliding waves u=
s
{&(t)exp[(ik(x
- t)] + &(t)exp[(ik(x
+ t)] + c.c.}dk
(3.33)
where cc. is the complex conjugate. Substituting (3.33) into (3.1) we get the equations governing the spectral amplitudes Ak(t) and &(t). For example, for A,(t) it reads
d&s
dt=
- ;-dk,
dk,({b(k,
+ kz - k)(k, + kz)’ [Ak,Akl
+ &,B,,
exp i(k, + k2 + k)t + Ak,Bkl exp i(k2 - k, + k)t
+ AkJ&
exp i(kl - k2 + k)t]j + {6(k, - k2 - k)(k, - k2)2
[Ak,A,*,expi(k, - k, + k)t + B,,Bk*,exp i(kl - k2 + k)t + Ak,Bt2 exp i(k - kl - k,)t]j
+ CC.).
(3.34)
Here A*, B* are complex conjugates of the spectral amplitudes. According to the properties of b-functions, only the first terms in figure brackets have the resonance character. Therefore if in the first approximation it is possible to neglect all terms containing expiAt, A = kl & k2 T k, the equation (3.34) yields dAk ’ = - f dt
f
dki dk2{Ak,Ak,b(k,
+ k2 - k) + Ak,A;16(kl
- k2 - k) + c.c.}.
(3.35)
This equation does not include the amplitude of another wave &Jr) and is solvable with respect to Ak. Coming back to the wave field through the transformation c=
s
dkA,(r) exp ik(x - t)
(3.36)
we get in the one-wave approximation (3.37)
It is easily concluded that after the transformation
(3.20) this equation is transferred to (3.38)
that coincides exactly with the evolution equation (3.27) obtained using the asymptotic method. However, in this sense the nonexistence of interaction between the colliding waves is clearly shown that was not obvious using the other methods described above. The correctness of the evolution equation (3.38) is the same as described in Section 3.3 and its solution needs no comments.
On approximate methods for constructing the evolution equations
25
REFERENCES 1. W. F. Ames, Non-linear Partial Differential Equations in Engineering. Academic Press, New York (1965). 2. G. Whitham, Linear and Non-linear Waves. Wiley-Interscience, New York (1974). 3. M. B. Vinogradova, 0. V. Rudenko and A. P. Sukhorukov, Theory of Waces. Nauka, Moscow (1979) (in Russian). 4. M. G. Van Dyke, Perturbation Methods in Fluid Mechanics. Academic Press, New York (1964). 5. A. F. Nayfeh, Perturbation Methods. Wiley-Interscience, New York (1973). 6. A. H. Nayfeh, A comparison of perturbation methods for non-linear hyperbolic waves. In Singular Perturbations and Asymptotics (Edited by R. E. Meyer and S. V. Parter), p. 223. Academic Press, New York (1980). 7. A. Jeffrey and T. Kawahara, Asymprotic Methods in Non-linear Wane Theory. Pitman, London (1982). 8. T. Taniuti and K. Nishihara, Non-linear Waces. Pitman, London (1983). 9. J. Engelbrecht, Non-linear Wave Processes of Deformation in Solids. Pitman, London (1983). 10. L. A. Ostrovsky and E. N. Pelinovsky, Approximate equations for waves in media with small non-linearity and dispersion. Prikl. Mat. Mekh. 38, 121 (1974). 11. K. A. Gorschkov, L. A. Ostrovsky and E. N. Pelinovsky, Some problems of asymptotic theory of non-linear waves. Proc. IEEE 62, 1511 (1974). 12. R. Courant and D. Hilbert, Methods of Marhemotical Physics, Vol. II. Wiley-Interscience, New York (1963). 13. Yu. Z. Miropolsky, Dynamics of Internal Gracitationnl Waces in the Ocean. Gidrometeoizdat, MOSCOW (1981) (in. Russian).
Vol. ?I. No. I. pp. 15-25. 1986
oO?O-7462.86 53.00 + 00 Pcrgamon Press Ltd.
ON APPROXIMATE METHODS FOR CONSTRUCTING THE EVOLUTION EQUATIONS E. PELINOVSKY~,V. FRIDMAN: and J. ENGELBRECHT~ of Applied Physics, U.S.S.R.Acad Sci, 46 Ulyanov St., 603600 Gorki. U.S.S.R.:
tfnstitute $Radiophysical Research Institute, 25 Lyadov St., 603600 Gorki. U.S.S.R. and §Institute of Cybernetics, Academy of Science of the Estonian S.S.R.. 21 Akadeemia St.. 200104 Tallinn, Estonia.
U.S.S.R. (Receired 3 July 1984;
received for publication
7 March
1985)
Abstract-Three methods are used to construct the non-linear evolution equations governing the propagation of single waves: the iterative, the asymptotic and the spectral. The paper gives a comparative analysis of these methods including the problems of the convergence and the correctness. The model second-order non-linear equation is chosen as a basic one and the corresponding evolution equations are derived by means ofall these methods. It is shown that although the evolution equations of the first order are the same within all of three presented methods, the next approximations differ and the best convergence to the exact solution is obtained by using the iterative method.
1. INTRODUCTION Wave propagation theory is historically most closely related to the development of the approximate methods for solving partial differential and/or integrodifferential equations (systems of equations). However, even in the linear approach the exact solutions describing the dependence of field variables on initial conditions are rare. The number of exactly solvable non-linear problems is certainly much smaller [ 11. This is the main reason why the approximate methods are so intensely developed for both linear and non-linear problems. Generally speaking, the approximate methods used in wave propagation analysis may be divided into three main groups: (i) approximate analysis of the exact solution [2,3]; (ii) perturbative analysis of the solution with small (slow) derivation from a fixed and known one [4-61; (iii) simplification of equation (mathematical models) describing the process [2,7-91. The group (i) takes the exact solutions written in the integral form as the basis and the approximate solutions are obtained by means of some classical approximation method, for example, by means of the method of stationary phase. This method, first proposed by Kelvin for solving the wave pattern formation after a moving ship is novv considered to be a classical one. Unfortunately it is practically impossible to get the explicit formulae for non-linear problems; that is a great drawback of this physically well grounded method. The methods of group (ii) are better in this sense because the non-linear problems are also solvable. The fixed solutions are usually taken in the form of stationary plane waves and the formerly constant coefficients are considered to be slowly perturbed; that permits one to use the series representation containing the small parameter E. The perturbation methods are widely used [4,5] but the main disadvantage of such methods is in the secularity of the additional terms and usually the perturbed solution may be used only in a certain space-time interval. The methods of group (iii) do not simplify the solutions but the equations governing the wave process paying at this stage no attention to the solutions themselves. The set of small parameters connected either with the initial conditions or with the coefficients of initial equations is used and the perturbation method together with the method of deformed coordinates is applied. The best results are obtained when the wave process is separated into single waves described by separate equations. These equations are usually called evolution equations governing the distortion of a single wave along the properly chosen characteristic (ray). The best example of such an evolution equation is the well-known Korteweg-de Vries equation. There are several methods to simplify the governing equations and this paper presents an attempt to classify them giving also a comparative analysis of the known methods. Three methods-the iterative, the asymptotic and the spectral-are briefly
E. PELINOVSKYet al.
16
described in Section 2. Section 3 deals with the comparison of these methods. The model second-order non-linear equation I_?*11
d211
a*u*
ax*
ax*
---_=E_
at*
(1.1)
is chosen as the basic one and the corresponding evolution equations are derived by means of all these methods. Here u is the field variable and E the small parameter. The accuracy of the solutions is analysed and the results are compared with each other. It is shown that although the evolution equations of the first order are the same within all of the three methods presented, the next approximations differ and the best convergence to the exact solution is obtained by using the iterative method. This is not the first attempt to demonstrate the efficiency of separate methods in solving a certain non-linear equation. In [6] several perturbation methods were compared by the solution of the non-linear second-order equation with dissipation. The extensive analysis of the solution of the non-linear Boussinesq equation is given in the recent monograph by Jeffrey and Kawahara [7]. The main difference of this paper with regard to [6,7 a.o.] is in the concentration on evolution equations only, i.e. single wave processes are described. It is difficult to overestimate the importance of the evolution equations in contemporary mathematical physics, therefore the convergence and the correctness of the evolution equations must be clearly stated. The choice of the model equation (1.1) is based on the existence of the exact solution that simplifies considerably the analysis. The paper forms actually a summary of a longer research paper that will be published in the future. 2. THE
APPROXIMATE
METHODS
In this Section the approximate methods for constructing the evolution equations are briefly discussed. The methods under consideration are: (i) the iterative, (ii) the asymptotic and (iii) the spectral method. The one-dimensional processes are analysed only in order to obtain clear physical background. 2.1. The iterative method In our opinion, this approach gives the best physical understanding of the process [lo, 111. Let the wave propagation be described by the system
zg+A(r,%)g =
&{U,X,t,~,T,E)
(2.la)
1.
(2.lb)
e
5 =
Et,
x =
EX,
EC
Here U is the n-vector of field variables, I is the unit matrix, A is the n x n matrix, I? is a nonlinear integro-differential operator. The eigenvalues of matrix A are all assumed to be real and different. It means that the system (2.la) with E = 0 is a hyperbolic one. The initial and boundary conditions for (2.la) are given in dependence ofp and will not be discussed here. In case E = 0 the system (2.la) has the solution n
U = C Vi(X - E.it)ri
(2.2)
i=l
where vi is the scalar function, r is the right eigenvector of matrix A for eigenvalue i (the velocity) given by Ar = E.r,
detlA - ).I1 = 0.
(2.3)
We represent the initial system (2.la) in the normal form [12], changing the variable U to V by making use of the formula
u=
YV
(2.4)
17
On approximate methods for constructing the evolution equations
where Y is the n x n matrix formed by the linearly independent eigenvectors ri of matrix A. As far as A is dependent on “slow” coordinates T and ;c,the eigenvectors ri and the eigenvalues 1bidepend also on these coordinates. Substituting (2.4) into (2.la) we get
-av at
(!?+Z)v]
++Y-p-
= - Ef[v,x,t,X,q&].
(2.5)
Here 8 is the diagonal ‘matrix with the elements ii (T,x) and Y-i is the inverse matrix of Y, consisting of the left eigenvectors Ii of A. The usual normalisation is here assumed: lirj = ⅈ 6, is the Kronecker symbol. According to (2.4) and (2.5), the solution may be given as a superposition of waves, each one propagating (at E = 0) along its characteristics and interacting with others only due to the perturbation f # 0. In many cases only m < n waves ri are generated by the initial and boundary conditions. When the initial condition is localized then the wave process is soon decomposed into the separate independent waves, so that in the far field each of them may be analysed separately. In this case the vector V contains m “basic” waves and other n-m waves may be calculated by the usual perturbation scheme. For every si, i = 0, 1, 2,... the equation (2.5) yields
ap
EO. .x
+
&T)g) =0
p)s = 0,
ap
s= m +
T)z, =&f;:{up
&’ :- ar + %i(X,
adl) ad*' -$ + ux,T)“-=&f,{u(ll’)... ax
)...
(2.6a)
i = l,...,m l,...,n
(2.6b)
IQ,0 ,... O,%,S,O)
(2.7a)
o’,“,o
(2.7b)
)...
. .. .. .. . aup acp En: dt + li(X,T)z = &f{C(“),... L$‘,L$;i)
~+%s(x,T)~=&~{u’~’ ,... (n) (nu,,u,+1
O;x,qO}
,... Lf-“;%,T,&} 1) )...
(2.8a)
uj:-“;&T,E}.
In every step the system is decomposed into two groups: (i) m coupled non-linear evolution equations (2.8a) for “basic” waves and (ii) n-m linear independent evolution equations (2.8b) with known right-hand sides. Now it is clear that this approach leads to the iterative procedure for m interacting waves. The main subcase is m = 1. Let $) = C, i.i = i., fi = j. Then (2.8a) yields
2 +i.(r,g =Ef{c, c(“-‘) ,x, T, E}. The first approximation needs E = 0, therefore C, = 0. The structure of the equation (2.9) depends certainly on the functionalF in the initial system (2.la). Usually it is given in the form of a sum of derivatives and integrals from the field variables. Using (2.5) for calculating the right-hand side in (2.9) the functionfmay be represented also in the form of a sum of certain derivatives and integrals [lo, 111. The general form may be written as h-‘dx+sh’“‘+s
ah(l) + Ea2hc2) -zx ax2
+ E _aw + . ..= o sx3
E.
18
PELINOVSKY et al.
where /I(‘)are certain functions oft’. The equation (2.10) is the evolution equation of the first order for a single wave. 2.2. The asymptotic method We shall start from the initial system (2.la). The solution is sought in the form of the asymptotic series [8,9] u =u,(g,?)
+ sU1(5,r) + &%J2(5,5) + .**
(2.11)
where the independent variables are either 5 = x - At,
T =
EC
(2.12)
or ( = x - At,
t =
EX.
(2.13)
For the case of a homogeneous and stationary medium as shown in [9], the transformation (2.12) must be used for the problem with the initial conditions and the transformation (2.13) for the problem with the boundary conditions. In this way the evolution equation is always solved with the initial conditions. Substituting (2.11) into (2.la) and using (2.12) we get the sequence: & =
0:
au0
P- %- =O &=
(2.14a)
1: (2.14b)
E = 2: (2.14~)
Here the variational derivative c@/sUo and @/as are determined approximations are, in principle, equivalent to (2.14c). The equation (2.14a) yields
uo =
Lg,t)r
at E = 0. The higher
(2.15)
where u is a scalar function and r, as above, the right eigenvector of A determined i.. As far as det(A - 1.1) = 0, then the solution of (2.14b) exists only for the certain right-hand side and the orthogonality conditions give
+-zfj+o where I is the corresponding
(2.16)
left eigenvector. Substituting (2.15) into (2.16), we obtain
-&
=
j(c) = l@:(w)
(2.17)
On approximate methods for constructing the evolution equations
19
i.e. the evolution equation of the first order. It is clear that (2.17) and (2.9) coincide if (2.9) is rewritten using (2.12). It means that the evolution equations obtained using the iterative and asymptotic methods coincide in the first approximation. It must be pointed out, however, that the asymptotic series (2.11) has a serious advantage over the usual perturbation series with dependence only on 5. in the last case U, cannot be bounded for the far field and the series is not convergent. In case of asymptotic series (2.1 l), U0 is determined so that Ui is bounded (cf. (2.15) and (2.16)). The next approximation needs Ui in the explicit form. From (2.14b) and (2.16) we have (2.18) where Uie is the forced solution of (2.14b) and uI is a scalar function, determined from the orthogonality condition for the right hand side of (2.14c). As above, this condition means that Uz is bounded. Substituting (2.18) into (2.14~) and multiplying with 1 we get the evolution equation of the second order
au,
-
a7
=
I~(U,, +clr) + 1:.
(2.19)
Next approximations are obtained in the same way. It is important to underline that the evolution equations of higher approximations than the first found by iterative and asymptotic methods differ. However, if the multiple scales are used then the influence of the forcedsolutionsUIO,UzO,... may be replaced by the terms dr/‘&i, where ri = &t, i = 1,2,. . . . 2.3. The spectral method This method is convenient for waves in dispersive media and gives an excellent possibility to explain the decomposition into single waves (one-wave approach) 19,131, Let the system (2.la) be rewritten in a generalized form
where i is a linear differential operator (a matrix depending on r3U/&), ti is a non-linear operator. The wave process is decomposed into the eigenvalues of the linear (E = 0) problem U = c [dk Urt$‘exp[i(wrt
(2.21)
- kx)]
where r$’ is determined by i(--ik)r;:
= -io$‘r;:
(2.22a)
det li( - ik) + io? II = 0.
(2.22b)
Here Ur is the spectral amplitude, and m is the mode number. We assume CJJ = Q’(t). Substituting (2.21) into (2.20) and performing the inverse Fourier transform in the coordinate we get the infinite system of equations for the spectral amphtudes diJr -=E dt
c dk, dkz c r; ;r,’Q: K; mf,maf
x exp[i(oz$ + I@; - oZ’ft&k,
+ kz - k)]
(2.23)
where the matrix-coefficients V;:r; r; are determined by the given operator M and depend only upon the wave numbers k and the mode numbers m. As far as the Fourier integrals may contain also the given fields of perturbation, the equations (2.23) are able to describe not only the interaction between single waves but also the interaction with perturbation fields. No general methods for solving the equations (2.23) are known, therefore we seek possible
E. PELINOVSKYef al.
20
simplifications. Evidently the strongest interaction of spectral amplitudes occurs when the synchronic conditions (for quadratic non-linearity) kl + kz - k = 0, ’
A = CU”’ kl + we; - 0;: = 0
(2.24)
are fulfilled, or at least A 1 E.If the initial system is of high order (the number of modes is big) then practically always it is possible to satisfy (2.24). Therefore the decomposition into single waves is possible if v”kk,kzmlmz
0
=
t
ml.2
f
(2.25)
m
and in this case the wave with wave number m is propagating without any dependence on other modes. The spectral method gives the excellent possibility to examine the accuracy of the onewave approach that is comparatively difficult by making use of the iterative method and the asymptotic methods. The crucial points are the synchronic conditions following from the dispersion relationship without the detailed analysis of the non-linear interaction. If the one-wave approach is possible, then the equation (2.23) yields to (index m is neglected!)
duk
-
dt
=
dki dkz %k, k2uk, ukz exp(iAt)b(k, + k2 - k).
(2.26)
Now the Fourier transform must be performed in order to obtain the evolution equation that generally speaking is of an integrodifferential form. Further details will be given in Section 3. 3. THE
COMPARISON
OF
THE
METHODS
3.1. The statement of the problem and its exact solution We shall start from the model non-linear wave equation
a2u ---=
a2u
at2
ax2
a2u2
(3.1)
yp.
It is equivalent to the system (2.la) with
(3.2)
The equation (3.1) or the system (2.la) with (3.2) permits the Riemann solution. Assuming ti = c(u), the governing system may be represented as (3.3a) 4
g;
^
+ (1 + 2ELI)g
= 0.
(3.3b)
The nontrivial solution of the system (3.3) exists provided the system determinant is equal to zero. This condition yields = 1+
c’ =
&f&-l
[(l +
(3.4a)
2EU
243’2
-
I].
On approximate methods for constructing the evolution equations
21
Now it is easy to find the exact solution from (3.3). For the wave propagating in the direction x > 0 we obtain
u = F[x - V(u)t]
(3Sa)
V(u) = (1 + 2&U)“2
(3Sb)
where F is an arbitrary function determined by the initial conditions. 3.2. The iterative method We shall represent the governing system (2.la) with (3.2) in the normal form. Introducing 1 u=,(w-w-),
,+w-w_,
(3.6)
-;-&(w+W-)2
(3.7a)
the governing system takes the form g+g=
aw_ aw_ E a - - - = -&(w at ax
+
w-)2.
The iterative procedure (see Section 2.1) gives in the zeroth approximation with E = 0 and
(3.7b)
the system (3.7)
w = w(x - t),
w- =o
(3.8)
u = u(x - t),
v = u.
(3.9)
that is equivalent to
This coincides with the exact solution in case E = 0. The first approximation
is governed by
(3.10a)
aw_ ---=_-*
at
a~_
E
ax
4
aw2 ax
(3.10b)
Because the initial system is hyperbolic, the evolution equation (3.10a) has the typical form of Riemann wave equation with the velocity
v, =
1 + ;w.
(3.11)
Noting that w = 2u + O(E), the final result is
v, = 1 + EU.
(3.12)
This result is easily obtained from the exact solution (3.5b) after series representation with two terms kept in it. In order to construct the evolution equation of the second
E. PELINOVSKYet
12
al.
approximation we need w_ with the correctness of E. After integration determined by
of (3.10b) w- is
3 w_ = --_)v2 8
(3.13)
and substituting this result into (3.7a) the sought evolution equation takes the form (3.14) The velocity V2 is now easily determined as v2 =
1 +
;w- ;e2w2
(3.15)
and (3.6) yields 1 11 ‘y
TW
3 -
(3.16)
&
or 1 U’21 2U + -&l12. 2
(3.17)
In terms of u the velocity V2 may be calculated by
v, =
(3.18)
1 + EM- +,2.
Once more, this result is easily obtained from the exact solution (3.5b) keeping three terms in series. It is easily understood that the correctness of the solution obtained by the iterative The evolution equation of the nth method increases with higher approximations. approximation is the representation of the initial equation for the Riemann wave with correctness of s”. The exact hyperbolic equations are correct in a bounded time up to t,--the shock wave formation time. The evolution equations are correct for t c t,, with maxlt, - t,J c O(E”).
(3.19)
3.3. The asymptotic method Introducing the independent variables (=x-t,
7 =
Et
(3.20)
the governing system is rewritten in the form
au ati
-g-g= au
au
-@+Zf=
-“t
all
au
au2
a7
ag
----_-_.
(3.21a)
(3.21b)
The unknown functions are represented in series u, v =
(h t.0)+ E(U1,VI) + E2(l12, VJ + . ‘.
(3.22)
On approximate methods for constructing the evolution equations
In the zeroth approximation
23
the system (3.21) yields (3.23a)
_
&fJ+ au0-
at
at
D0 =
uo(5,7).
0
(3.23b)
with the trivial result
The first approximation
(3.24)
is governed by
-_aut au1 --ali0
ac:+T&=
(3.25a)
a7
(3.25b) and the second by -- au2
av,
-- au1
au2
(3.26a)
a7
at +a(=
au2
-z+af=-~-2$po,s1).
(3.26b)
The system (3.23) corresponds to the system (2.14a), the system (3.25) to the system (2.14b) etc. The orthogonality condition (2.16) gives the evolution equation of the first approximation
g?+.,!22=0 a<
(3.27)
that coincides with (3.10a) with correctness of E. The formula (2.18) gives
Ul =
U+(&T),
u+(T,z) +
u1 =
$5
(3.28)
and the equation (2.19) takes now the form
au
s
+
a
-&uou+)
au0
I
-u2 - = 0. 2 ar:
-
(3.29)
Its solution is u+
1-u
1
2
2O
(3.30)
and, consequently 1
u1 =
-II& 2
ur = uf.
(3.31)
E. PELINOVSKY et al.
24
Now it is easily concluded that the asymptotic method gives the small additions to the main solution uo, and its solution is continuous for t < t,, t, = tl. Therefore the condition max It, - t,l < O(E) determines the time interval when the asymptotic (3.19)].
(3.32)
evolution equations are correct
[cf.
3.4. Spectral method The solution of the governing equation (3.1) is sought in the form of the Fourier integral (2.21) for colliding waves u=
s
{&(t)exp[(ik(x
- t)] + &(t)exp[(ik(x
+ t)] + c.c.}dk
(3.33)
where cc. is the complex conjugate. Substituting (3.33) into (3.1) we get the equations governing the spectral amplitudes Ak(t) and &(t). For example, for A,(t) it reads
d&s
dt=
- ;-dk,
dk,({b(k,
+ kz - k)(k, + kz)’ [Ak,Akl
+ &,B,,
exp i(k, + k2 + k)t + Ak,Bkl exp i(k2 - k, + k)t
+ AkJ&
exp i(kl - k2 + k)t]j + {6(k, - k2 - k)(k, - k2)2
[Ak,A,*,expi(k, - k, + k)t + B,,Bk*,exp i(kl - k2 + k)t + Ak,Bt2 exp i(k - kl - k,)t]j
+ CC.).
(3.34)
Here A*, B* are complex conjugates of the spectral amplitudes. According to the properties of b-functions, only the first terms in figure brackets have the resonance character. Therefore if in the first approximation it is possible to neglect all terms containing expiAt, A = kl & k2 T k, the equation (3.34) yields dAk ’ = - f dt
f
dki dk2{Ak,Ak,b(k,
+ k2 - k) + Ak,A;16(kl
- k2 - k) + c.c.}.
(3.35)
This equation does not include the amplitude of another wave &Jr) and is solvable with respect to Ak. Coming back to the wave field through the transformation c=
s
dkA,(r) exp ik(x - t)
(3.36)
we get in the one-wave approximation (3.37)
It is easily concluded that after the transformation
(3.20) this equation is transferred to (3.38)
that coincides exactly with the evolution equation (3.27) obtained using the asymptotic method. However, in this sense the nonexistence of interaction between the colliding waves is clearly shown that was not obvious using the other methods described above. The correctness of the evolution equation (3.38) is the same as described in Section 3.3 and its solution needs no comments.
On approximate methods for constructing the evolution equations
25
REFERENCES 1. W. F. Ames, Non-linear Partial Differential Equations in Engineering. Academic Press, New York (1965). 2. G. Whitham, Linear and Non-linear Waves. Wiley-Interscience, New York (1974). 3. M. B. Vinogradova, 0. V. Rudenko and A. P. Sukhorukov, Theory of Waces. Nauka, Moscow (1979) (in Russian). 4. M. G. Van Dyke, Perturbation Methods in Fluid Mechanics. Academic Press, New York (1964). 5. A. F. Nayfeh, Perturbation Methods. Wiley-Interscience, New York (1973). 6. A. H. Nayfeh, A comparison of perturbation methods for non-linear hyperbolic waves. In Singular Perturbations and Asymptotics (Edited by R. E. Meyer and S. V. Parter), p. 223. Academic Press, New York (1980). 7. A. Jeffrey and T. Kawahara, Asymprotic Methods in Non-linear Wane Theory. Pitman, London (1982). 8. T. Taniuti and K. Nishihara, Non-linear Waces. Pitman, London (1983). 9. J. Engelbrecht, Non-linear Wave Processes of Deformation in Solids. Pitman, London (1983). 10. L. A. Ostrovsky and E. N. Pelinovsky, Approximate equations for waves in media with small non-linearity and dispersion. Prikl. Mat. Mekh. 38, 121 (1974). 11. K. A. Gorschkov, L. A. Ostrovsky and E. N. Pelinovsky, Some problems of asymptotic theory of non-linear waves. Proc. IEEE 62, 1511 (1974). 12. R. Courant and D. Hilbert, Methods of Marhemotical Physics, Vol. II. Wiley-Interscience, New York (1963). 13. Yu. Z. Miropolsky, Dynamics of Internal Gracitationnl Waces in the Ocean. Gidrometeoizdat, MOSCOW (1981) (in. Russian).