Application of inverse scattering transform to the problems of generalized amplitude modulation of waves
The notion of the generalized amplitude modulation of oscillations and waves is introduced. The inverse Scattering Transform Method is used to investigate the problem of generalized amplitude modulation for the Korteweg – de Vries equation. Some theorems on these problems are presented
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Інститут математики НАН України
2001
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| Цитувати: | Application of inverse scattering transform to the problems of generalized amplitude modulation of waves / I.P. Syroid // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 399-404. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-1746972021-01-28T01:26:39Z Application of inverse scattering transform to the problems of generalized amplitude modulation of waves Syroid, I.P. The notion of the generalized amplitude modulation of oscillations and waves is introduced. The inverse Scattering Transform Method is used to investigate the problem of generalized amplitude modulation for the Korteweg – de Vries equation. Some theorems on these problems are presented 2001 Article Application of inverse scattering transform to the problems of generalized amplitude modulation of waves / I.P. Syroid // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 399-404. — Бібліогр.: 9 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/174697 en Нелінійні коливання Інститут математики НАН України |
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The notion of the generalized amplitude modulation of oscillations and waves is introduced. The
inverse Scattering Transform Method is used to investigate the problem of generalized amplitude modulation for the Korteweg – de Vries equation. Some theorems on these problems are
presented |
| format |
Article |
| author |
Syroid, I.P. |
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Syroid, I.P. Application of inverse scattering transform to the problems of generalized amplitude modulation of waves Нелінійні коливання |
| author_facet |
Syroid, I.P. |
| author_sort |
Syroid, I.P. |
| title |
Application of inverse scattering transform to the problems of generalized amplitude modulation of waves |
| title_short |
Application of inverse scattering transform to the problems of generalized amplitude modulation of waves |
| title_full |
Application of inverse scattering transform to the problems of generalized amplitude modulation of waves |
| title_fullStr |
Application of inverse scattering transform to the problems of generalized amplitude modulation of waves |
| title_full_unstemmed |
Application of inverse scattering transform to the problems of generalized amplitude modulation of waves |
| title_sort |
application of inverse scattering transform to the problems of generalized amplitude modulation of waves |
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Інститут математики НАН України |
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2001 |
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http://dspace.nbuv.gov.ua/handle/123456789/174697 |
| citation_txt |
Application of inverse scattering transform to the problems of generalized amplitude modulation of waves / I.P. Syroid // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 399-404. — Бібліогр.: 9 назв. — англ. |
| series |
Нелінійні коливання |
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AT syroidip applicationofinversescatteringtransformtotheproblemsofgeneralizedamplitudemodulationofwaves |
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2025-07-15T11:44:53Z |
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2025-07-15T11:44:53Z |
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1837713215121784832 |
| fulltext |
Nonlinear Oscillations, Vol. 4, No. 3, 2001
APPLICATION OF INVERSE SCATTERING TRANSFORM
TO THE PROBLEMS OF GENERALIZED AMPLITUDE
MODULATION OF WAVES
I.-P. P. Syroid
Pidstrygach Institute of Applied Problems of Mechanics and Mathematics,
NAS of Ukraine
Naukova St., 3B, L’viv, 79053, Ukraine
e-mail: igor@iapmm.lviv.ua
The notion of the generalized amplitude modulation of oscillations and waves is introduced. The
inverse Scattering Transform Method is used to investigate the problem of generalized ampli-
tude modulation for the Korteweg – de Vries equation. Some theorems on these problems are
presented.
AMS Subject Classification: 34, 35Q58
1. Introduction and Definitions
Some applications of the Inverse Scattering Transform (IST) are represented. The IST was
discovered in the papers of the outstanding scientists: M.D. Kruskal, C.S. Gardner, J.M. Greene,
R.M. Miura, P.D. Lax, V.E. Zakharov, A.B. Shabat, S.P. Novikov, V.O. Marchenko, L.D.Faddeev,
and others (see [1 – 8]).
The Korteweg – de Vries (KdV) equation has been shown to describe the asimptotic
development of small — but finite amplitude shallow — water waves, hydromagnetic waves
in a cold plasma, ion-acoustic waves, acoustic waves in an anharmonic crystal (see [1, 2]). In
this paper, the author describes a new field of applications for KdV-equation using the IST.
These new applications lie in some problems formulated by analogy with the problems of radio
communications, mathematical theory of communication, electronics. Our basic assumptions
are connected with the notion of a generalized amplitude modulation for the KdV equation in
spaces with weak dispertion.
In this paper we deal with application of the IST to finding a solution of a Cauchy problem
for the KdV equation with initial conditions that satisfy the condition of generalized amplitude
modulation.
The notion of amplitude modulation of oscillations and waves has been introduced in electri-
cal engineering, radio communication, electronics, mathematical theory of communication. The
simplest case deals with the notion of amplitude modulation for a harmonic oscillation.
Definition 1. Let s(x) = A sin(ωx+ϕ) be a harmonic oscillation with constant amplitudeA.
The amplitude modulation of the oscillation s(x) with a function m(x) ≥ 0 consists of forming
the product sm(x),
sm(x) = Am(x) sin(ωx+ ϕ), (1)
where the amplitude function Am(x) for sm(x) is already a variable and is specified depending
on the applications.
c© I.-P. P. Syroid, 2001 399
400 I.-P.P. SYROID
Existence of periodic solutions of the Korteweg – de Vries equation has been proved in 1974
by S.P.Novikov, V.O. Marchenko, and P.D. Lax, see [1, 3].
Apart from periodic and oscillation solutions, there are soliton and N -soliton solutions of
the Korteweg – de Vries equation. Therefore, the generalized amplitude modulation using soli-
tons and other decreasing on ±∞ functions (waves) is a vital and actual problem.
Definition 2. Let s(x) be a bounded and Stepanov oscillating solution of some differential
equation on the axis −∞ < x < ∞, and let f(x) be a function decreasing on ±∞ and satisfying
the following condition:
∞∫
−∞
(1 + |x|)|f(x)|dx < ∞. (2)
A generalized amplitude modulation of s(x) using the function f(x), by definition, is the
product
sf (x) = f(x)s(x).
2. Statement of the Problem
Let
vt = 6vvx − vxxx (3)
be the Korteweg – de Vries (KdV) equation. Let the initial condition function, v(x, 0), be given
by product
v(x, 0) = f(x)s(x). (4)
The aim of this paper is to solve the Cauchy problem (3), (4) by using the Inverse Scattering
Transform (IST) Method. Our basic assumption is that the initial function v(x, 0) is a generali-
zed amplitude modulation as in Definition 2.
Our remark to this problem is the following. V.E.Zakharov and A.B.Shabat (see [7]) have
solved the nonlinear Schrödinger equation by using the IST. The soliton they have constructed
is an envelope of waves in nonlinear optics (of course, we are interested only in stable amplitude
modulations for the nonlinear Schrödinger equation).
We shall be working under the following convention. Let v(x, t) be a solution of the initial
Cauchy problem (3), (4). We shall say that the problem of generalized amplitude modulation
has a positive solution, if v(x, t) satisfies condition (2) or (5) for fixed finite t, 0 ≤ t < ∞.
3. New Results Obtained by the Author (a Lemma and Theorems)
Let L = −d2/dx2 + q(x) and L : L2(−∞,∞) → L2(−∞,∞). We will suppose that the
domain of definition of the operator L is sufficient for L to be selfadjoint, L = L∗.
APPLICATION OF INVERSE SCATTERING TRANSFORM TO THE PROBLEMS OF GENERALIZED.. . 401
Lemma 1. Let the following assumptions 1) – 3) hold.
1) The function q(x) has three derivatives and let the condition
∞∫
−∞
(1 + |x|)
∣∣∣q(k)(x)
∣∣∣ dx < ∞, k = 0, 1, 2, 3, (5)
hold;
2) The operator L = −d2/dx2 + q(x), L : L2(−∞,∞) → L2(−∞,∞) has an eigenfunction
f1(x) corresponding to an eigenvalue λ1, Lf1(x) = λ1f1(x), where f1 ∈ L2(−∞,∞) and λ1 <
0;
3) Let a number λ2 > 0 be an eigenvalue of the continuous spectrum of the operator L with
a corresponding eigenfunction f2(x) and let λ2 = d|λ1|, where d >> 1.
Then the function v(x) = f1(x)f2(x) is a solution of the equation
−ϕ′′′(x) + 2
(
q(x)
d
dx
+
d
dx
q(x)
)
ϕ(x) = 4µϕ′(x), (6)
where the number µ 6= 0 is obtained via λ1 and λ2, and v ∈ L2(−∞,∞).
Theorem 1. Let the initial function
v(x, 0) = f1(x)f2(x) (7)
in Cauchy problem (3), (4) be a generalized amplitude modulation (see Definition 2) and let
v(x, 0) = f1(x)f2(x) be a solution of equation (6) under the conditions of Lemma. Then there
exists a solution of the Cauchy problem (3), (4), v(x, t), and satisfies the following conditions:
max
0≤t≤T
∞∫
−∞
(1 + |x|)
∣∣∣v(i)(x, t)∣∣∣ dx < ∞, i = 0, 1, 2, 3, (8)
for all T > 0, T 6= ∞.
As a consequence of Theorem 1, we have a positive solution of the problem of generalized
amplitude modulation with the condition
∞∫
−∞
(1 + |x|) |v(x, t)| dx < ∞ (9)
for all t > 0, t 6= ∞, with the initial generalized amplitude modulation v(x, 0), t = 0, and
v(x, t), t ≥ 0, t 6= ∞.
Remark 1. The manifold of KdV-solutions has a certain parametric property. We can now
formulate this parametric property using invariance of the Korteweg – de Vries (KdV) equati-
on (3) under the Galilei transformation (for the Galilei transformation for the KdV equation,
see [2, 5]). If v(x, t) is a solution of the KdV equation (3), then
u(x, t, τ) = v(x− 6τt, t)− τ, τ is a parameter, (10)
402 I.-P.P. SYROID
is also a solution of equation (3). If τ 6= 0, the solution u(x, t, τ) does not satisfy the property (8).
But the first term v(x − 6τt, t) does satisfy (8). We conclude also that the KdV model has the
property of the phase-parametric modulation of the solution u(x, t, τ) in the variable x via the
phase function ϕ(t, τ) = −6τt for evolution in time t and τ 6= 0.
Now suppose that the conditions in the following Definition 3 hold, for Theorem 2 and
Theorem 3, instead of the previous Definition 2.
Definition 3. Let s(x) be a periodic three times differentiable function defined on the whole
real axis and let f(x) be a three times differentiable function satisfying the condition
∞∫
−∞
(1 + |x|)
∣∣∣f (i)(x)
∣∣∣ dx < ∞, i = 0, 1, 2, 3. (11)
Let sf (x) be defined by sf (x) = f(x)s(x). The function sf (x) is said to be a generalized ampli-
tude modulation of s(x) by using the function f(x).
In the following Theorem 2 we will touch only the case of the oscillating function f(x) =
ϕ(x) with the compact support.
Theorem 2. Let the following assumptions 1) – 3) hold.
1) Let s(x) be a periodic three times differentiable function defined on the whole real axis;
2) Let ϕ(x) be a three times differentiable oscillating function with the compact support on
the real axis;
3) Let the frequency of oscillation, ωϕ, of ϕ(x) on its support and the frequency ωs of the
function s(x) be so that the inequality ωϕ << ωs holds.
Then there exists a solution v(x, t) of the Cauchy problem (3), (4) satisfying the following
condition (8),
max
0≤t≤T
∞∫
−∞
(1 + |x|)
∣∣∣v(i)(x, t)∣∣∣ dx < ∞, i = 0, 1, 2, 3,
for all T > 0, T 6= ∞.
Remark 2. The condition ωϕ << ωs in 3) of Theorem 2 is essential. This condition
guarantees a nonresonant generalized amplitude modulation of oscillations. Theorem 2 is very
important in applications. The following example can be useful in applications. Let ϕ0(x)
belong to C∞0 (R) and let f(x) be a periodic three times differentiable function on the whole
real axis. Let
ϕ(x) = ϕ0(x)f(x)
satisfy condition 3) of Theorem 2. Then for the initial conditions
v(x, 0) = ϕ0(x)f(x)s(x) = ϕ(x)s(x),
the conclusion of Theorem 2 is true.
APPLICATION OF INVERSE SCATTERING TRANSFORM TO THE PROBLEMS OF GENERALIZED.. . 403
Theorem 3. Let s(x) be a periodic three times differentiable function on the whole real axis.
Let f(x) be a three times differentiable function and let f(x) satisfy the assumption (11) (see
Definition 3). Then, for the initial data v(x, 0) = f(x)s(x), the solution v(x, t) of the Cauchy
problem (3), (4) exists and satisfies the condition (8) for all T > 0, T 6= ∞.
Let conditions of Theorem 3 hold. Then for the solution v(x, t) of the KdV-equation above,
Remark 1 holds true.
The main result of paper [2] as well as previous to [2] numerical computations are that
the solution of an initial-value problem for the KdV-equation (3) may be an N -soliton soluti-
on. In the following Theorem 4, we shall be using the factorization condition of reflectionless
potentials instead of the condition of the generalized amplitude modulation. For the definition
of reflectionless potentionals, see [1 – 4, 9].
Theorem 4. Let V (x, 0) = f1(x)f2(x) be a reflectionless potential of a selfadjoint Schrödinger
operator L : L2(R) → L2(R). Then
1) the function V (x, 0) satisfies the condition
∞∫
−∞
eε|x||V (x, 0)|dx < ∞ for some ε > 0;
2) the solution V (x, t) of the Cauchy problem (3), (4) corresponding to the initial function
V (x, 0) = f1(x)f2(x) is an N -soliton solution with the following representation:
V (x, t) = −2
∂2
∂x2
ln ∆(x, t),
∆(x, t) = Det
(
δkl +m2
k(0)
e−(κk+κl)xe8κ
3
kt
κk + κl
)
,
where {κl > 0,mk > 0} is reflectionless scattering data of the Schrödinger operatorL : L2(R) →
L2(R) with the potential V (x, 0);
3) V (x, t) satisfies the condition
∞∫
−∞
eε|x||V (x, t)|dx < ∞, ε > 0
for all fixed t ∈ [0,∞);
4) the solution V (x, t) of the initial Cauchy problem (3), (4) has the representation:
V (x, t) = −2
[
∆(x, t)−1x
n∑
l=1
∆̃l(x, t) + ∆(x, t)−1
n∑
l=1
∆̃l(x, t)
′
x
]
,
where ∆̃l(x, t) = ∆l(x, t)exp{−κlx}, by the definition, is the determinant obtained from the
determinant ∆(x, t) by substituting the derivative of l-column instead of the l-column.
404 I.-P.P. SYROID
Remark 3. The proof of Theorem 4 is given as a consequence of [3] and [4, 9].
Remark 4. This paper belongs to the field of mathematics. The obtained results could have
very interesting application in problems of media with weak dispersion. Phenomena of stable
behaviour of the solution of the Korteweg – de Vries equation (3) with generalized amplitude
modulation was mathematically substantiated in this article. On the problem of stability of
solitons, also see paper [8].
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[in Russian], Nauka, Moscow (1980).
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equation,” Phys. Rev. Lett., 19, No. 19, 1095 – 1097 (1967).
3. Marchenko V.O. Sturm – Liouville Operators and Its Applications [in Russian], Naukova Dumka, Kyiv (1977).
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absent,” Sib. Mat. Zh., 22, No. 1, 151 – 157 (1981).
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problem,” Ukr. Mat. Zh., 42, No. 2, 223 – 230 (1990).
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64, No. 5, 1627 – 1739 (1973).
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Received 29.12.2001
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