CC BY
21
DOI: 10.17516/1997-1397-2022-15-2-176-185 УДК 517.95
Inverse Scattering and Loaded Modified Korteweg-de Vries Equation
Michal FeCkan*
Mathematical Institute of the Slovak Academy of Sciences
Bratislava, Slovakia
Gayrat UrazboeV
Urgench state university Urgench, Uzbekistan
Institute of Mathematics, Khorezm Branch, Uzbekistan Academy of Sciences
Urgench, Uzbekistan
Iroda Baltaeva*
Urgench state university Urgench, Uzbekistan Khorezm Mamun Academy Khorezm region, Khiva, Uzbekistan
Received 11.09.2021, received in revised form 19.11.2021, accepted 10.02.2022 Abstract. The Cauchy problem for the loaded modified Korteweg-de Vries equation in the class of "rapidly decreasing" functions is considered in this paper. The main result of this work is a theorem on the evolution of the scattering data of the Dirac operator. Potential of the operator is the solution to the loaded modified Korteweg-de Vries equation. The obtained equalities allow one to apply the method of the inverse scattering transform to solve the Cauchy problem for the loaded modified Korteweg-de Vries equation.
Keywords: loaded modified KdV equation, inverse scattering method, "rapidly decreasing" functions, soliton, evolution of the scattering data.
Citation: M. Feckan, G.Urazboev, I. Baltaeva, Inverse Scattering and Loaded Modified Korteweg-de Vries Equation, J. Sib. Fed. Univ. Math. Phys., 2022, 15(2), 176-185. DOI: 10.17516/1997-1397-2022-15-2-176-185.
Introduction
The study of non-linear waves in the viscoelastic tube is of interest, since system of such tubes is a model of the vessels of the blood circulatory system. Understanding the wave processes in the blood circulatory system can help predict the development of diseases [5].
In arterial mechanics, a widely used model assumes that artery is a thin-walled pre-stressed elastic tube with a variable radius (or with a stenosis) and blood is considered as an ideal fluid [3].
* [email protected] https://orcid.org/0000-0002-7385-6737 [email protected] https://orcid.org/0000-0002-7420-2516
[email protected] https://orcid.org/0000-0001-5624-7529
© Siberian Federal University. All rights reserved
The governing equation that models weakly non-linear waves in such fluid-filled elastic tubes is the modified Korteweg-de Vries equation
ut — 6u2ux + uxxx — h(t)ux
0,
where t is a scaled coordinate along the axis of the vessel after static deformation characterizing axisymmetric stenosis on the surface of the arterial wall; x is a variable that depends on time and coordinates along the axis of the vessel; h(t) is a form of stenosis and u(x, t) characterizes the average axial velocity of the fluid.
Let us assume that form of stenosis h(t) is proportional to u(0,t) and consider the following loaded modified Korteweg-de Vries equation
ut — 6u2ux + uxxx — j(t)u(0, t)ux = 0,
(1)
where u = u(x,t) is unknown real value function (x e R, t > 0), and j(t) is arbitrary continuous function. Equation (1) is considered with initial condition
u\t=0 = uo(x),
where real value function u0(x) has the following properties:
oo
1. I (1 + \x\) \u0(x)\ dx < to.
— oo
d
(2)
2. The equation L (0) y =
dx
uo . d
\ u0 —idX
yi y2
t
yi y2
x G R1 has N simple
eigenvalues and it does not have spectral singularities.
Let us assume that function u(x,t) is sufficiently smooth, tends to its limit rapidly enough when x , and it satisfies the condition
(1 + \x\) \u(x,t)\ + ]T
j=i
dju(x, t)
dxj
dx < t ^ 0.
(3)
Note that completed integrability of the modified Korteweg-de Vries (mKdV) equation was established in the class of "rapidly decreasing" functions using the method of the inverse scattering problem [11]. The evolution equations for non-linear waves which differ by small terms from equations soluble by the inverse scattering method (KdV, NSE, mKdV) were considered [4]. A perturbation theory scheme was formulated. It is based on the inverse scattering method. The term "loaded equation" was introduced by A. M. Nakhushev [7]. The most general definition of a loaded equation was given and various loaded equations were classified in detail. Loaded differential equations, the loaded part of which contains only the value of the desired solution at fixed points of the domain were considered [2,9,12].
The goal of this paper is to study the integration of the loaded mKdV equation in the class of "rapidly decreasing" functions in terms of inverse scattering problem.
1. Uniqueness of the solution
In this part we use the method given in [6]. Theorem 1. If problem (1)-(2) has solution then it is unique.
oo
— oo
Proof. Let v(x,t) be another solution of (1)-(2). Let us introduce w(x,t) = (u(x,t) — v(x,t))x. Then we obtain
Wt = 6[(u — v)(uX + vX) + (u + v)(ux + Vx )w]+
+ 3[(u — v)(u + v)(u + v)xx + (u2 + v2)Wx] — Wxxx+ (4)
+ ((u(0, t) + v(0, t)Wx + (u(0, t) — v(0, t)(u + v)xx ) . Multiplying (4) by w and integrating with respect to x over (—to, to), we have
1 d fw fw fw
-— w2 dx = 6 (u — v)(u2x + v2x)wdx + 6 (u + v)(ux + vx)w2dx+
2 dt J —w J—w J—w
/pw
wOQ (u — v)(u + v)(u + v)xxwdx + 3 (u2 + v2)wxwdx — wxxxwdx+ (5)
J —w J —w
y (t) Cw Y(t) Cw
+ Y—-)(u(0,t) — v(0,t)) (u + v)xxwdx + Y—-)(u(0,t) + v(0,t)) wxwdx.
2 J — oo 2 J — oo
- (u(0,t) — v(0,t)) J (u + v)xxwdx + —(u(0,t) + v(0,t)) I wx^
Let us denote max(v,x + vx) by m, max |(u + v)(ux + vx)\ by n, max |(u + v)(u + v)xx\ by k, max |(u2 + v2)x\ by l, max |u(0, t) + v(0, t)| by p and max |(u + v)xx\ by q. Using the Cauchy-Schwarz inequality we obtain from (5) the following inequality
cw / /*w / /*w /*w
—dt I w2dx ^ 6m^ I (u — v)2dx^l I w2dx + 6n j w2dx+
/ w / w
J / (u — v)2dxi w2dx-
/ fw / rw 31 rw i fw
+ 3kJ (u — v)2dxi w2dx + — w2dx + ~ / (w2x)xdx+
—w —w - —w - —w
qn(t) , , I iw 2J i(t) Ifw 2J Ifw 2J +--——max ^ — v| y j wax +--^Py J wxdx^ j wax.
Here it was taken into account that w and its derivatives tend to zero as x ^ ± to. There are constants mi, k\ > 0 such that [8]
/ {' w / {' w / {' w
t / (u — v)2dx ^ mit (u — v)xdx, max [u — v| ^ k\ a (u — v)xdx.
Then we derive
d w w w w
—— w2dx ^ 6mmi* w2dx* w2dx + 6n w2dx+
— dt J—r^ \ ./—^1 \/ ./—^1 ./—^1
w w 3l w + 3kmiJ J w2dxJ J w2dx + — J w2 dx+
ww
+ q ) ; i w2dxil I w2dx.
w
2
Let us denote f w2dx by E(t) and (1—mm\ + 1—n + 6km1 + 3l + qk1Y(t)) by C(t),
—w
dE (t)
dt
< C(t)E(t).
This differential inequality yields
E(t) < E(0) exp / C(s)ds, J 0
which implies that if E(0) = 0 then E(t) = 0 and thereby
w(x, t) = (u(x, t) — v(x,t))x =0
u(x, t) — v(x, t) = C. Assuming t = 0, we obtain C = 0. Theorem 1 is proved.
2. Scattering problem
Let us consider the following system of equation
Lv =
( d
i — u
dx d
\ u —i —
dx
vi=Hv;)-- œ <6>
with real value function u(x) that satisfies the condition of "rapid decrease"
/œ
(1 + |x|) |u(x)|dx < œ. (7)
-œ
The present section contains well known information on the direct and inverse scattering problem for problem (6)-(7) that is required for further consideration [1]. Condition (7) implies that system of equation (6) has the Jost solutions f(x,£) and ^(x,£) with asymptotic relations
f V 0 ) exp("^x),
^ x as x ^ —œ, (8)
f ^ —1 ^ exp(i£x), ^ ~ ^ 0 ^ exp(i£x),
, 1 . as x ^ œ. (9)
i> ~ ( 0 j exp(—i£x),
For real pairs {f,f} and are pairs of linearly independent solutions of equation (6).
Therefore,
f f = a(£)$ + b(№, (10)
\f = —a(£)^ + m^p. )
The following equality holds
a(£) = W = — f; (11)
and for all real £
a(0a(0 + b(e)b(£) = 1.
Function a(k) admits analytic continuation into the upper half-plane Im k > 0. In Im k ^ 0 function a(k) has asymptotic behavior a(£) = 1 + 0\ 1 ). Function a(k) can have a finite
WZU
number of zeroes £k, k = 1, 2, N in the upper half-plane Im k > 0. Zeros £k of function a(k) correspond to the eigenvalues of operator L in the upper half-plane. Let us note that operator L can have spectral singularities which are in the continuous spectrum.
We suppose that operator L does not have spectral singularities and zeros of function a(k) are simple:
V>(x, £k) = Ck t(x, &), k = 1, 2,...,N. (12)
The set ir+(£) = ^^, £k, Ck, k = 1,2, 3,..n \ is called scattering data for system of equal a(e) J
tions (6).
The following representation for the solution 0(x,£) is valid
^ =(0) eiix + Jx K (X'S) eiiSds, (13)
where K (x, s) = ( K <*■ '> ) does not depend on variable { and it is related to the potential
V K (x, s) J
function u(x) as follows
u(x) = 2iK1 (x,x). (14)
The components of kernel K(x, y) for y > x are solutions of the following Gelfand-Levitan-Marchenko (GLM) system of equation
K2(x,y)+ Ki(x,s)F(s + y)ds = 0,
Jx ^
-Ki(x,y) + F (x + y)+ / K2(x,s)F (s + y)ds = 0,
x
3 6(e) w
(15)
where F (x) = — e*xd£ — i V Cj e* x.
K ' a(£) ^ ^ j
j=i
3. Evolution of scattering data
It is easy to verify that functions
d (g(x,£) — Bnf (x,£)) _ hn(x) = -—-^, n = 1, 2, 3,...,N (16)
are solutions of the system of equations Ly = £ny. Using (11) for Im £ > 0, we define the following asymptotic relations
0 ~ a(£) ^ 0 ^ exp(i£x) as x ^ —œ, V ~ a(£) ( 1 j exp(—i£x) as x ^ œ.
Using these asymptotic relations, we obtain asymptotic relations for solutions hn(x)
0
hn(x)--Cn ^ 0 ^ exp(«£„x)
x) as x —to,
0
(17)
and
hn(x) ~ ( i ) exp(—i£nx) as x ^ to,
W [pn, hn} = Pnihn2 — Pn2Ki = —Cn, n = 1, 2, 3,..., N,
where Pn = P(x,£n).
Let function u(x,t) in (6) be a solution of the mKdV equation
Ut — 6u2ux + Uxxx = G(x, t),
(18)
where function G(x,t) is sufficiently smooth and G(x,t) = o(1) when x ^ t ^ 0. Equation (18) is considered with initial condition (2). According to [10], the following Theorem is valid.
Theorem 2. If function u(x,t) is a solution of equation (18) in the class of functions (3) then the scattering data of system (6) with function u(x, t) depend on t as follows
dr + i C
_ = 8i£3 r+ — a2 J G(p2 + p2)dx, Ime = 0, 8i£,n — i j G (hni^ni + hn2^n2) dx^ Cn,
CO
I G(p2n1 + P2n2)dx
dC ~dt
de,n _ —œ
~dt =
■ = 1, 2, ..., N.
2 J PniPn2dx
Here pn(x,t) are normalized eigenfunctions which correspond to the eigenvalue £n of system of equations (6).
Let us apply the result of Theorem 2 when
G(x,t) = j(t)u(0,t)ux.
According to Theorem 2, we have the following representation
= 8if r+ — ii(t)u(0, t)
dt a2
/CO
ux(p2 + P )dx, Im k = 0.
-co
By virtue of system of equations (6) and asymptotic relations (9), we have
ux(pi + P2 )dx = —2
c CO
Pix iP2x + eP2} + P2x —iPlx + ePl
dx
/00
e(Pi ■ P2)xdx = —2ea(e)b(e).
-co
Consequently, for Im k = 0 we obtain
dr+
— = (8i? + 2i£i(t)u(0,t)) r+.
(19)
CO
CO
From relations
din dt
j(t)u(0,t) f ux(f2n 1 + ¥2n2)dx
— C
CO
2 / <£n 1 <Pn2dx
n = 1, 2, 3,...,N,
and
/CO fC
ux ivh + V2n2)dx == -2 [f'nl (iV'n2 + in^nl) + v'nl (-iV'n1 + £n¥>nl)] dx :
C -C
= -2£n (^nl • =0,
we have
d£n dt
0
1, 2, 3,...,N.
(20)
Using system of equations (6) and asymptotic relations (17), we have
/CO /»C _ t
Ux (hnl0nl + hn20n2) dx = —£n (hnl0n2)x + (hn20nl )
-oo J—co
dx £n
Taking into account the last expression, we obtain
dCn ~dt
= (8iin + iY(t)u(0, t)£n) Cri, n =1, 2, 3,..., N.
(21)
Considering relations (19), (20) and (21), we arrive to the following theorem.
Theorem 3. If function u(x,t) is a solution of problem (1)-(3) then the scattering data of system of equations (6) with function u(x,t) depend on t as follows
^j- = (8i£3 + 2i£7(t)u(0,t)) r+ for Im £ = 0,
dCn ~dt d£n dt
= (8iiT + ij(t)u(0, t)in) Cn = 0 , n = 1, 2, 3,...,N.
The obtained relations completely determine the evolution of the scattering data for system of equations (6). It allows us to find the solution of problem (1)-(3) by using the method of inverse scattering problem.
Example. Let us consider the following Cauchy problem:
ut - 6u2ux + uxxx - Y(t)u(0,t)ux = 0,
u |t=0
2
sh 2x'
where Y(t) = - -
1
: sh(-8t + arcsh t).
AVT+t2
To find the general solution of this problem we use the method of inverse scattering problem. First of all, let us find a solution of the direct problem for the following system of equations
L(0)y =
d
i— uo dx
d
\ u0 -idi
y1
y2
=i
y1
y2
n
x
In this case, we have the following scattering data
r+(0) = 0, N =1, £(0) = i, C1 = 2. According to Theorem 3, we find the evolution of scattering data depending on t :
r+(£,t) = 0, £i(t)= i, Ci(t) = 2exp S(t),
where
S(t) = 8t - y(t)u(0, t)dx. Jo
Then using this scattering data, we find a solution of inverse scattering problem . Solving the GLM system of equations with F(x) = -2i exp (-x + S(t)), we obtain
K ( ) = 2i exp(-x - y + S(t)) Kl(x,y) 1 - exp(-4x + 2S(t)) .
Applying equality (14), we obtain
2
u(x,t) = -
t
sh(2x - 8t + f y(t)u(0, t)dT)
o
t
Putting x = 0 and introducing f (t) = f y(t)u(0, t)dx, we obtain the following Cauchy problem
o
f'(t) _ 2
Y(t) sh(f (t) - 8t) ' f (0)=0.
Solving this problem with Y(t) =--, 1 sh(-8t + arcsht), we have
+12
f (t) = arcsh t.
As a result, the solution of problem under consideration is expressed as follows
2
u(x,t) = -;-- .
v ' sh(2x - 8t + arcsh t)
It is well known that solution of the modified Korteweg-de Vries equation
ut - 6u2 ux + uxxx = 0,
that satisfies the same initial condition has the form
2
u(x,t) = —;--.
v ' 7 sh(2x - 8t)
The difference between solutions of the loaded modified Korteweg-de Vries equation and the modified Korteweg-de Vries equation is shown in Fig. 1.
40 -200-20 --40 -
Conclusion
The method of inverse scattering problem can be used to obtain solutions of the Cauchy problem for the loaded modified Korteweg-de Vries equations in the class of "rapidly decreasing" functions. Function u(0,t) that appears in equations of Theorem 3 is unknown in contrast to function u(x, 0). If the scattering data is used to find potential u(x,t) then function u(0,t) is included in the solution. Therefore, we have a functional equation relating u(x, t) to u(x, 0) which is reduced to the Cauchy problem for an ordinary differential equation of the first order. For some y(t) the Cauchy problem for ODE can be solved exactly and we obtain a solution of the Cauchy problem for the loaded mKdV equation.
References
[1] M.J.Ablowitz, H.Segur, Solitons And The Inverse Scattering Transform, Philadelphia, PA, SIAM, 1981.
[2] U.I.Baltaeva, Solvability of the analogs of the problem Tricomi for the mixed type loaded equations with parabolic-hyperbolic operators, Boundary Value Problems, 211(2014), 2-12. DOI: 10.1186/s13661-014-0211-6
[3] H.Demiray, Variable coefficient modified KdV equation in fluid-filled elastic tubes with stenosis: Solitary waves, Chaos Soliton Fract, 42(2009), 358-364.
DOI: 10.1016/j.chaos.2008.12.014
[4] V.I.Karpman, E.M.Maslov, Perturbation theory for solitons, Zh. Eksp. Teor. Fiz., 73(1977), 537-559.
[5] N.A.Kudryashov, I.L.Chernyavskii, Nonlinear waves in fluid flow through a viscoelastic tube, Fluid Dynamics, 41(2006), 49-62.
[6] P.D.Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21(1968), 467-490.
[7] A.M.Nakhushev, Loaded equations and their applications, Differ. Uravn., 19(1983), 86-94.
[8] A.Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
[9] G.U.Urazboev, I.I.Baltaeva, On integration of the general loaded Korteweg-de vries equation with a self-consistent source, Instruments and Systems: Monitoring, Control, and Diagnostics. Scientific journal, 10(2019), 7-10 (in Russian). DOI: 10.25791/pribor.10.2019.941
[10] G.U.Urazboev, A.B.Khasanov, The integration of the mKdV equation with self-consistent source, In: Proceedings of the 2nd International Conference "Function Spaces. Differential operators. Problems of mathematical education" Dedicated to the 80th anniversary of L. D. Kudryavtsev, Moscow, FIZMATLIT, 2003, 340-349.
[11] M.Wadati, The exact solution of the modified Korteweg-de Vries equation, J. Phys. Soc. Japan, 32(1972), 16-81.
[12] A.B.Yakhshimuratov, M.M.Matyokubov, Integration of loaded Korteweg-de Vries equation in a class of periodic functions, Russian Mathematics (Izv. VUZ.), 2(2016), no. 2, 72-76. DOI: 10.3103/S1066369X16020110
Метод обратной задачи рассеяния и нагруженное модифицированное уравнение Кортевега-де Фриза
Михал Фечкан
Математический институт Словацкой академии наук
Братислава, Словакия
Гайрат Уразбоев
Ургенчский государственный университет Ургенч, Узбекистан
Институт математики Хорезмского отделения Академии наук Узбекистана
Ургенч, Узбекистан
Ирода Балтаева
Ургенчский государственный университет Ургенч, Узбекистан Хорезмская Академия Мамуна Хорезмская область, Хива, Узбекистан
Аннотация. В данной статье мы рассматриваем задачу Коши для нагруженного модифицированного уравнения Кортевега-де Фриза в классе «быстроубывающих» функций. Основной результат настоящей работы представляет собой теорему об эволюции данных рассеяния оператора Дирака, потенциал которого является решением нагруженного модифицированного уравнения Кортевега-де Фриза. Полученные равенства позволяют применить метод обратной задачи рассеяния для решения задачи Коши для нагруженного модифицированного уравнения Кортевега-де Фриза.
Ключевые слова: нагруженное модифицированное уравнение КдФ, метод обратной задачи рассеяния, "быстроубывающие" функции, солитонное решение, эволюция данных рассеяния.
