Physica 18D (1986) 455-463
North-Holland, Amsterdam
MULTI-SCALE EXPANSIONS IN THE THEORY OF SYSTEMS INTEGRABLE
BY THE INVERSE SCATI'ERING TRANSFORM
V.E. ZAKHAROV and E.A. KUZNETSOV
Landau Institute of Theoretical Physics, USSR Academy of Sciences, Moscow, USSR
and Institute of Automation and Electrometry, USSR Academy of Sciences, Novosibirsk, USSR
It is shown that using multi-scale expansions conventionally employed in the theory of nonlinear waves one can transform
systems integrable by the IST method into other systems of this type.
1. Introduction
The majority of nonlinear differential equations
integrable by the inverse scattering transform (IST)
and having physical applications represents differ-
ent models of interacting waves. In the wave the-
ory for nonlinear media there is a familiar totality
of methods, which are applied in such cases when
there exists two or several considerably differing
scales. This totality is known under the general
name of the multi-scale expansion method. This
method allows to obtain for such situations new
equations which differ from the initial ones and
are more adequate to the given problem (see, e.g.
[1, 21).
So, if it is already known that the solution
represents a set of one or more quasimonochro-
matic wave packets of a small amplitude, it is
reasonable to turn to a set of equations for com-
plex envelopes of these packets. A characteristic
packet size and wavelength play a role of different
scales for this problem. If, on the contrary, we are
interested in extremely long-wave weak nonlinear
oscillations, the problem can be simplified by ex-
panding the wave dispersion law in the neighbour-
hood of the wave number k equal to zero, then by
passing to the moving coordinate system we can
eliminate the terms linear with respect to a wave
number-the KdV equation is usually derived in
this way.
It is important to note that these versions of the
multiscale expansion method are structurally
rough- they are equally applied both to the inte-
grable and nonintegrable systems. If the initial
system is nonintegrable, the result can be both
integrable and nbnintegrable. But if we treat the
integrable system properly, we again must get
from it an integrable system. That is why we
became interested in application of multi-scale
expansions to the integrable systems. We hoped
thus to obtain new integrable formerly unknown
systems.
In reality we found few such systems. One of
them is shown in the end of the present paper.
However, application of multi-scale expansions to
the integrable systems in most cases leads to one
of the classical equations which at one time made
famous the inverse scattering transform-the
nonlinear Schrtidinger equation (NSE), the
Korteweg-deVries equation (KdV), the Davey-
Stewartson equation (DS), the system of N-waves.
This emphasizes once more the universality of the
enumerated systems. This fact explains publication
of this paper in the present collection of works
surveying almost twenty years of history of IST
(beginning from the first papers by M. Kruskal
and coworkers). We should like to show that in the
first stage of development of this method many
opportunities have been missed. Thus, from the
fact of the applicability of IST to the KdV equa-
0167-2789/86/$03.50 © Elsevier Science Publishers B.V.
(North-Holland Physics Publishing Division)
456
V.E. Zakharov and E.A. Kuznetsov / Systems integrable by the inverse scattering transform
tions it is not difficult to predict the application of
this method to solution of the NSE. This can be
made easier basing on a matrix version of the KdV
equation. Using the latter it is easy to determine
the applicability of IST to the N-wave problem
and to a vector version of the NSE. The integrabil-
ity of NSE is also derived from the integrability of
many other one-dimensional equations (Bous-
sinesq equation, sine-Gordon equation and so on).
The integrability of the Davey-Stewartson equa-
tion and a two-dimensional N-wave problem is
derived similarly from the applicability of IST to
the Kadomtsev-Petviashvili equation (KP).
We should also note, that the method we use is
related to the known "averaging method" by
Whitham [1] developed also in the works of
Flaschka, Forest and McLaughlin [3]. This method
allows to describe solutions of integrable systems
which are locally close to finite-zone solutions.
However, our method is more elementary and we
can advance further on; in particular we can real-
ize a correspondence between integrable systems
not only on the level of the equations but also on
the level of their integrating procedures
(L-A
pairs, Marchenko equations and so on).
2. Derivation of the nonlinear Schr6dinger equation
from the scalar Korteweg-deVries equation
Consider the scalar KdV equation
u t + 6uu x
+ uxx x = 0. (2.1)
The solution of it can be found in the form
U= ~ Un einO, U*=U_n, O=k3t+kx.
rl= --00
(2.2)
Here
k > 0 is an arbitrary number. Equating to
zero coefficients at every
e in0
separately, we get an
infinite set of equations for u.,
O 0 ink)
( _O_i + i n k 3 ) u , + ( _ff_dx + :3
Un
t°
+ -O--d + ink E
UqUn-q =
O.
(2.3)
q= -oo
It is sufficient to consider the case n > 0. Let us
introduce a parameter e to these equations, assum-
ing
u,=e""o,(x',t').
(2.4)
Here
x'=e(x+3k2t), t'=-6e2kt, a o=2,
(2.5)
a,=a_,=n
(n > 1).
Substitute (2.4), (2.5) into (2.3). Expression (2.3)
becomes an infinite series with respect to a. Let
a -o 0 and vanish the terms at minimal (for every
n) power of parameter e. The system (2.3) is
significantly reduced as e--* 0 converting into ex-
plicit expressions for corresponding V~ (except for
the case n = 1). In particular, we have
2 1
V0=- ~-~IVll 2,
V2=-~V12.
(2.6)
The equation for V 1 remains differential. It has the
form
8V 1 i 82V1
St' ¢ 2 8x ,---T +
i(VoV 1 + GVI*) = 0. (2.7)
Let us introduce the function ~ =
V~/k.
Now,
taking into account (2.6), (2.5) we have
1 O2~k
1~12~k = 0. (2.8)
iq~t' -~ 2 8x ,2
Thus, we got the nonlinear Schr&:linger equation
from the KdV equation. This fact is quite natural
because expression (1.2) at e---, 0 represents a
quasi-monochromatic weakly nonlinear wave
packet whose complex envelope should be de-
scribed by the nonlinear SchriSdinger equation.
It is less clear that correspondence between KdV
and NSE can be established on the level of the
integrable linear systems.
The KdV equation (2.1) is a compatibility con-
dition for the following overdetermined system:
L¢= Ox-- 5+u 0+72¢=0,
(2.9)
Mq~= (-O-t & + 4~x3 +
= 0. (2.10)
V.E. Zakharov and E.A. Kuznetsov / Systems integrable by the inverse scattering transform
457
As before, let us represent u in the form (2.2), and
for • let us assume
• = ~'~ ~k,, e i"°/:. (2.11)
Here the summing up is taken over the even n.
Introduce the parameter e into (2.11)
~. = E (l"l -1)/20.(X", t').
(2.12)
The choice of • in the form of (2.11), (2.12) is
associated with the fact that for periodic potentials
in the limit of e ~ 0 points of the spectrum for the
operator L, hE = (n + 1)2k2, define the position
of the forbidden zones. For potentials in the form
of a quasimonochromatic wave (2.2) the reflection
coefficient in these points has sharp peaks corre-
sponding to the strong resonance subbarrier reflec-
tion.
Let us consider eq. (2.9) near the first resonance
)~ = k/2.
Let
k
h = ~- + e/~. (2.13)
Substituting (2.1), (2.11) into (2.9) and (2.10) and
equating to zero coefficients at all e ine/2 in turn,
we get two infinite sets of equations for 0, (for the
reason of brevity we do not insert them here).
Each system represents an infinite series in powers
of the parameter e. Consider the system arising
from eq. (2.9) and put e ---, 0 in it.
All the equations except corresponding n = +__ 1
are converted into explicit expressions for 0,. In
particular, we have
V1
O- 1
• 3 = VI01
0_ 3 = (2.14)
2k 2 , 2k 2
At n= +lwefind
i
001
--ff~xt --[-~O_1 =/.LO1,
.
00_ 1
--l~q-l~*Ol~---/.LO_ 1.
(2.15)
In the system (2.15) we find out the L-operator for
the nonlinear Schrt~linger equation (2.8). Simi-
larly, substituting (2.2), (2.11), (2.12) into (2.10),
consider n = + 1 and let e ~ 0. The terms of zero
and first order in e are cancelled. Calculating the
terms of second order we should take into account
terms in the form of V1"O 3 and V10_ 3. Using
formulae (2.4) after simple calculations we finally
find
[
i(1
0
-1---
2 ~k*
-i(Vo+ I~12)
O_1
= 0. (2.16)
Systems (2.15) and (2.16) are compatible when the
quantity V0= -2bkl 2 is in complete correspon-
dence with (2.6) and ~k obeys NSE.
Let us observe a correspondence between the
KdV equation and the NS equation on the level of
equations of the inverse scattering problem. Con-
sider the Marchenko equation
fx °
r(x,z)+F(x+z)+ K(x,s)F(s+z)ds=O.
(2.17)
The potential
u(x)=
2dK(x,
x)/dx
in the form
of a quasimonochromatic wave packet with a mean
wave number k corresponds to the following choice
of F(~):
F(~) = rp(~) e ik~/2 + ~*(/~) e -ik~/2. (2.1a)
Function
K(x, z)
should be sought in the form
K(x,z)
= Ko(x,
z)e i k/2) x-z) + K (x, z)e
+ gl(x, z) e i(k/2)(x+z) -Jr K{'(x, z) e -i(k/2)(x+z).
(2.19)
In the formulae (2.18), (2.19) the functions
% ko, k 1 are varying on both arguments slowly
relative to exponents. Substituting (2.18) and (2.19)
458
V.E. Zakharov and E.A. Kuznetsov / Systems integrable by the inverse scattering transform
into (2.18) and rejecting integrals from high-
frequency oscillating function we finally get the
equations
K~(x, z) + ~(x + z)
+ fx~KO(X,S)ep(s+z)ds=O,
(2.20)
Ko(x,
z) + fx'Kl(x, s)ep*(s
+ z) ds = 0,
which coincide with the Marchenko equations for
the linear problem (2.15).
The given scheme of NSE derivation is applica-
ble (sometimes with small modifications) if we
take as an initial equation any other integrable
system, e.g. the Boussinesq or sine-Gordon equa-
tion. However, the integrability of these systems
was discovered later than the integrability of the
non-linear Schr~Sdinger equation.
Let us perform a separation of scales in the
system (3.2)-(3.4) considering solutions locally
close to (3.5). Introduce a parameter e, assuming
U 1 = a 1 -~- E2VI(x ,,
t'), u 2 = a 2 -Jr- g2V2(x t, tt),
w = £ei(~t+kx)~(X t, t'),
(3.6)
x' = ex, t' = - 3ke2t.
From (3.2), (3.3) as e ~ 0 we obtain
1~12 1~12
(3.7)
VI= 2a 1 , 112= 2a 2-
As e-~ 0 "in eq. (3.4) we demand the terms of
second order in e to vanish. This gives the follow-
ing condition:
a I + a 2 = k 2. (3.8)
3. Derivation of the nonlinear Sehr6dinger equation
from the matrix Korteweg-deVries equation
If we turn into zero the coefficient at e 3 in (3.4)
we obtain
It is easier to derive NSE from the matrix KdV
equation
u, + 3(uZ)x +
Uxx x
= 0. (3.1)
i+,, + +x'x' -
al~12~
= 0,
k 2 k 2
a-2ala2-2al(k2-aa)"
(3.9)
Here u is the complex N × N matrix. Let N = 2
[,,1 w]
and u Hermitian, u = w* u2 ' uL2 are real. Eq.
(3.1) is equivalent to the system
OU 1 03Ul --~
Ot t- Ox------T+3
(u?+ [w[2)=0,
Ou20----i- + -~x 3 3--~-a
(u220x
+
[wl2)=0,
Ow O3w
3-L(ua+.2)w=0.
Ot ~ Ox ------T + Ox
(3.2)
(3.3)
(3.4)
System (2.2)-(2.4) has an exact simple solution
U 1 -----a 1 , g2 = a2,
w=e i(kx+~t),
6o=k 3-3k(a 1+a2) ,
al, 2 are arbitrary constants.
(3.5)
So, we got the nonlinear Schr~Sdinger equation
again. It should be noted that the constant a
which characterizes the interaction, can have any
sign depending on the choice of k 2 and a v
From the scalar KdV equation we could only
get the equation with a > 0 (NSE with repulsion).
Let us observe the transformation of the
L-A
pair for the matrix KdV equation to a correspond-
ing pair for NSE. Let us start from the system
( u, w) is a matrix, and
(2.9), (2.10) in which u = w* u2
/
( ~1 ) is a two-component vector. • will be
4= ~2
found in the form
4 1 ~ ei(p+k/2)x+i(q+to/2)tXl '
(3 .lO)
4 2 -~ ei(p-k/2)x+i(q-~o/2)tx2"
V.E. Zakharoo and E.A. Kuznetsov / Systems integrable by the inverse scattering transform
459
Here p, q are the unknown constants,
X1,2
depend
on slow variables x', t'. Eqs. (2.9), (2.10) are con-
sidered in the X-plane near the point h = 2, 0 which
together with constants p and q are determined
from the requirement of vanishing in (2.9), (2.10)
the terms of zero order in e. From (2.9) we have
(3.11)
from which
al -- a 2 2a 1 - k 2
P =~= 2k '
h2=a1(a1 -k2) = 1
k 2 24 "
Now, by setting h---h 0 -#e/2X 0 and taking the
limit as e --* 0 in (2.9) we get a system
k] OX1
2 i( p + 1-) --~-- + q,×2 = ~×1,
k~ OX2
2i( p - )-) ~ + +*X1 =/~X2,
(3.12)
which represents one of the possible versions of
the L-operator for NSE (3.9) (compare with [4]).
Note that in the case of NSE with attraction
(a < 0) the point h 0 lies on the real axis. Similarly,
from the requirement of vanishing the terms of
zero order in e we find q = 2p 3. Simultaneously in
(2.10) the first order terms are cancelled. To com-
bine the terms proportional to ~ one can get the
second linear system integrating NSE (3.9). It is
not necessary to write it out here.
4. Other applications of the matrix KdV equation
The matrix KdV equation (3.1) represents a
suitable object for application of different multi-
scale expansions. Thus, from this equation one can
get the known system of "N-waves" (see, e.g. [11]).
Let in (3.1) u =
uij
be a Hermitian matrix,of order
N. Let us separate a diagonal part in it,
uij = uiSij + wij, wii
= 0, (4.1)
and introduce n real numbers a t (i= 1 .... , n).
Further we introduce parameter e, assuming
ui=a2+e2Vi(x',t'), x'=ex, t'=3et,
wij = ek ei(a'-aj)x-2i(a]-a~)t~ij(X' , t'),
(4.2)
Substitute (4.1), (4.2) into (3.1) and take the limit
as e---, 0. At i ~j there occurs a closed set of
equations for ~kij = ~k~i:
0 0~i k n
"~ik -I-
2aiak-- ~ +
2i(ai-
ak) E ~bij~jk = O.
)=1
(4.3)
The system (4.3) is a special case of the hyperbolic
nonlinear system for N-waves (N =
½n(n -
1)). In
order to obtain the
L-A
pair for this system let us
turn to the system (2.9), (2.10) in which ~/i repre-
sents now a column of n elements. Let us put for
them
Oi = ei(":-2"~°Xi(X',
t'), h 2 = ep. (4.4)
Substituting (4.1), (4.2), (4.4) into (2.9), (2.10) and
setting e ---, 0 we obtain
aXk
2iak--ff-~7 + ~ IlgkjXj "b ~Xk = O,
(4.5)
j~l
aXk 2a 2°qXk + ~ (ak--aj)~kjXj=O.
(4.6)
Ot" k OX"
j~l
The system (4.3) is the compatibility condition for
systems (4.5) and (4.6).
Now let us obtain a vector analogue of NSE
from the matrix KdV equation (see [5]). Let us set