CC BY
7
Izvestiya Instituía Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
2023. Volume 61. Pp. 137-155
MSC2020: 37K15
© A. B. Khasanov, U. A. Hoitmetov, Sh. Q. Sobirov
INTEGRATION OF THE MKDV EQUATION WITH NONSTATIONARY COEFFICIENTS AND ADDITIONAL TERMS IN THE CASE OF MOVING EIGENVALUES
In this paper, we consider the Cauchy problem for the non-stationary modified Korteweg-de Vries equation with an additional term and a self-consistent source in the case of moving eigenvalues. Also, the evolution of the scattering data of the Dirac operator is obtained, the potential of which is the solution of the loaded modified Korteweg-de Vries equation with a self-consistent source in the class of rapidly decreasing functions. Specific examples are given to illustrate the application of the obtained results.
Keywords: Gelfand-Levitan-Marchenko integral equation, system of Dirac equations, Jost solutions, scattering data.
DOI: 10.35634/2226-3594-2023-61-08
Introduction
When integrating nonlinear evolutionary equations of mathematical physics, the main and difficult problem is to obtain exact solutions to nonlinear equations, including nonlinear wave, soliton, etc. Over the past few decades, many efficient methods have been developed to find such solutions for many integrable equations, such as the Korteweg-de Vries (KdV) equation and its various generalizations, various types of nonlinear Schrodinger equations, etc.
One of such integrable nonlinear equations is the following modified Korteweg-de Vries (mKdV) equation [1]:
Ut ± 6u Ux + Uxxx = 0.
This equation can be applied in many areas, such as the propagation of ultrashort solitons with a small number of optical cycles in nonlinear media [2, 3], anharmonic lattices [4], Alfven waves [5], ion-acoustic solitons [6-8], lines transmission through the Schottky barrier [9], thin oceanic jets [10,11], internal waves [12,13], thermal impulses in solids [14], etc. To calculate the exact solutions of the mKdV equation, many methods have been created, for example, the bilinear approach of Hirota [15], the Wronskian technique [16-18] can be mentioned. There are also a lot of results about the mKdV equation [19-25] due to its simple expression and rich physical application. In recent works by Vanneeva [26], one can find exact solutions of the mKdV equation with variable coefficients
Ut + u2ux - g(t)uxxx + h(t)u = 0.
In this paper, we consider the following system of equations
2 N
ut + p(t)(6u2ux + uxxx) + q(t)ux = ^2 ®k(t)(/fcigfcl - Ik2gk2) ,
k= 1 ( )
L(t)/k = Ck /k, L(t)gk = Ck gk, k = 1, 2,..., 2 N,
where L{t) = i ( d,x , u(x^,t)\ ak{t) (k = 1,2N), are given continuously
\—u{x,t) — J
differentiable functions.
The system of equations (0.1) is considered under the initial condition
u(x, 0) = uo(x), x E R, (0.2)
where the initial function u0(x) (—to < x < to) has the following properties: 1)
prx
/ (1 + |x|)|u0(x)| dx < to; (0.3)
( — —UoixY
2) the operator L(0) = il dx, , has exactly 2N simple eigenvalues £i(0), £2(0),
\-u0{x)
..., 6N (0).
In the problem under consideration, fk = (/k1,/k2) is the eigenfunction of the operator L(t) corresponding to the eigenvalue £k, and gk = (gk1,gk2)T is the solution of the equation
Lgk = Cfc9k, for which
W{fk, 9k} = fki9k2 ~ fk29ki = Uk(t) t^ 0, k = 1, 2N, where uk (t) are the initially given continuous functions of t, satisfying the conditions
un(t) = -uk(t)min = -ik, Re jj uk{r) dr j > -lm{&(0)}, k = l,N, (0.4)
for all non-negative values of t. For definiteness, we assume that in the sum in the right-hand side of (0.1), the terms with Im£k > 0, k = 1, N, come first.
Let us assume that the function u(x, t) has the required smoothness and rather quickly tends to its limits at x ^ ±to, i. e.,
/+°°((1 + \x\)\u(x,t)\ + dx < 00 > k=l,2,3. (0.5)
The main purpose of this work is to obtain representations for the solution u{x,t), fk(x,t), gk(x, t), k = 1, 2N, of problem (0.1)—(0.5) in the framework of the inverse scattering method for the operator L(t).
§ 1. Preliminaries
Consider the following system of Dirac equations
Vlx + i&i = Uo(x)v2, (11)
V2x — i£,V 2 = —Uo(x)vi,
on the entire axis (—to < x < to), with the potential u0(x) satisfying the condition (0.3).
It can be seen that using operator L(0) and the vector function v = (v1; v2), the system (1.1) can be rewritten as
Lv = £v. (1.2)
t
The system of equations (1.1) has Jost solutions with the following asymptotics
Im£ = 0, x ^ —œ;
<P(x,£) ~ (0) ^
$(x,£) - (—J
$(x,£) - g)
(1.3)
Im £ = 0, x ^ œ.
For real £, pairs of vector functions and are pairs of linearly independent solutions
to the system of equations (1.1). Therefore, the following relations take place
> = a(£ )$ + b(£ ( ^ = -a(£ )p + b(£ )<p,
and < _ (1.4)
= -a(£ + b(£ U = a(£ V + b(£ )p,
where a(£) = , b(£) = . The following equalities are true
|a(£ )|2 + |b(£ )|2 = 1, a(£ ) = a(-£), b(£ ) = b(-£). The coefficients a(£) and b(£) are continuous for Im £ = 0 and satisfy the asymptotic equalities:
a(£ ) = 1 + O(£-1), b(£ ) = O(£-1), |£ The Unction tp(x, £) can be represented as follows (see [27, p. 33])
"X
^(x,£)=^0j + J K(x,s)Ssds, (1.5)
fK1(x s)\
where K(x, s) = ( k (x's w ' In representation (1.5), the kernel K(x, s) does not depend on £ and the equality
u(x) = -2Ki(x,x) (1.6)
holds. The function a(£) (a(£)) continues analytically to the upper (lower) half-plane and has a finite number of zeros £k (£fc) there, where £k (£fc) is an eigenvalue of the operator L(0), so that the following equality is true
<p(x, £k) = Ck $(x, £k) (p(x, £k) = Ck p(x, ak)), k =1, 2,...,N.
Definition 1.1. The set of values jr+(£) = £k, Ck, k = 1,2,..., iV j is called the scattering data for the operator L(0).
The components of the kernel K(x,y) in representation (1.5) for y > x are solutions to the Gelfand-Levitan-Marchenko system of integral equations
/><x
K2(x,y)+ Ki(x,s)F(s + y) ds = 0,
J x
/><x
-K1(x,y) + F(x + y) + / K2(x, s)F(s + y) ds = 0,
where
1 r ^ N
F(x) - j r+(£)e^ Qe«'*.
Note that the following vector functions
... / , n=l,N, (1.7)
are solutions to the equations L(0)hn = £nhn. Therefore, the following formula is valid:
aa(£n)
In addition, the functions hn (x) have the following asymptotics,
hn ~ — Cn ^^ ei?nx, x ^—to; hn ~ ^ e-i?nX, x ^ to. (1.8)
According to (1.8), we get the following equality
w{tpn,hn} = tpnihn2 - fn2hni = -Cn, n=l,N.
The following lemmas are true.
fyjx £)\ -
Lemma 1.1. If Y(x,£) = (y (x'£) 1 is a solution of equation (1.2), then Y(x,£) = y2(x, —£ )
( £^ satisfies the equation Lv = — £v.
Lemma 1.2. If vector functions Y = ( yi(x'£ ) | and Z = ( Zi(x'n) | are solutions of
\y2(x,£ y \Z2(x,n)J
equations LY = £Y and LZ = rqZ, then their components satisfy the equalities
d
^(yi^l + V2Z2) = + v)(yiZi - V2Z2), d
- V2Zl) = - rn){yiZ2 + V2Zl).
The validity of both lemmas is proved by a direct verification. The following theorem is true.
Theorem 1.1 (see [28, §6.2, p. 353]). The scattering data of the operator L uniquely determine L.
§ 2. Evolution of scattering data
Let potential u(x, t) in the system of equations LY = £Y be a solution to the equation
ut + p(t) (uxxx + 6u2ux) = G(x, t)' (2.1)
2 N
where G(x,t) = —q(t)ux(x,t) + ^ ak(t)(fki9ki — /k2gk2). The operator
k=1
A = (t)f —4i£3 + 2iu2£ 4u£2 + 2iux£ — 2u3 — uxx ^ (2 2)
A = p(t) I —4u£2 + 2iux £ + 2u3 + uxx 4i£3 — 2iu2£ ) (^2)
satisfies the following Lax relation
[L,A] ^ LA - AL = i ( _ ° . P(t)(-6u2ux - uxxx) ) (2.3)
V p(t)(-6u2Ux - Uxxx) 0 J
Therefore, equation (2.1) can be rewritten as
Lt + [L, A] = iR, (2.4)
where R = ^ G . Differentiating the equality
Ly = £y
with respect to t, we obtain
Lt y + Lyt = £yt, which, according to (2.4), can be rewritten in the form
(L - £)(yt - Ay) = -iRy.
Using the method of variation of constants, we can write
yt - Ay = B(x)j + D(x)<p. (2.5) Then, to determine B (x) and D(x), we obtain
MBx j + MDx y = -Ry, (2.6)
where M = ( ^ ^ .To solve equation (2.6), it is convenient to introduce the following
notation y = (y^J, t = (j^. According to (2.3) and the definition of the Wronskian, the following equalities are valid:
jT My = - Mj = a, jT Mj = My = 0.
Multiplying (2.6) by yT and jjT we obtain
B, = D, = (2.7)
aa
According to (2.2), as x ^ -to, we have
- - Ay ^ ( ^ )(1) ^ = f41'^) ^^,
therefore, based on (2.5), we obtain
D(x) ^ 4i£3p(t), B(x) ^ 0, x ^-to.
Hence, from (2.7) we can determine
1 rx ~ 1 fx
D(x) = — / ipTR<pdx + 4i£3p(t), B(x) = - ^Rydx. a Ja J
Thus, equality (2.5) has the following form:
<pt-A<p=(^J ^Updxy+^J + (2.8)
According to (1.4), equality (2.8) can be rewritten in the following form
ati + btip - A^aijj + = Q J t^Rpdx^jip + J tjjTR>^dx + 4i£3p(t)^j (aijj + hp
Passing in the last equality to the limit as x ^ and taking into account (2.2), we obtain
f x ~r at = — / V R^ dx,
J—x
1 /"x b i'x ~ bt = - ^Rf dx-- :tpTRtp dx + 8i£3p(t)b.
a J—x a </ —x
Therefore, at Im £ = 0 we have
= «iVOr* _ ^
x
= 8*£3p(i)r+ - - / + dr. (2.9)
Lemma 2.1. If vector-function p = ^i(x £)^ is a solution to the system of equations (1.1), then its components satisfy the following equality
x
I G{<p\ + <p%) dx = -Hamo E + Hq(t)amO- (2.10)
J-x k=i £ £k
Proof. Let the potential u(x, t) in the system of equations (0.1) be a solution to the equation
ut + p(t) (6u2ux + uxxx) = G(x, t) '
where G rather quickly tends to zero at x ^ ±to.
According to Lemma 1.1 and the first of the conditions (0.4), the right-hand side in the equation (0.1) can be rewritten in the form
2N N
(t)(fkigki — fk2gk^ = 2 ak(t)(fkigki — fk2gk^ .
k=i k=i,
Im £fc >0
According to Lemma 1.2, we have the following equality
ak (t)(/kigki — /k2gk2)(P? + ) = = «k (t)/kigki^2 + «k (t)/kigkip2 — «k (t)/k2gk2^? — «k (t)/k2gk2^2 = ak (t)
2
■ [(/fcl^l - fk2'-P2){gkl'-Pl + 9k2^2) + (fkl'-fil + fk2'-P2}{gkl'-Pl - 9k2f2)] +
+ [(/fcl¥>2 - fk2^Pl)(gk2^Pl + 9k\<f2) ~ (/fcl^2 + fk2^Pl)(gk2^Pl ~ Qkl^)] - ak^ d [UkWi + fk2^p2){gki^pi + ^2^2)] +
—2i(£ + £fc) dx
The following asymptotics are valid: for x ^ -to,
<P
gk
itkx
and, for x ^ +to,
y
gk
Uk (t)e
(ae %(-x I he)ix
1\ Uk{t) gkX
0 cke '
j -
jk j
be
ae%^x Ck 0
(2.11)
,-iik x
yk - Ck{ ^ e*kx
(2.12)
Integrating the last equation from -to to +to, and then using the asymptotics (2.11), (2.12), we obtain the following equalities:
®k (t)
d
[(fkiyi + fk2y2)(gkiyi + gk2y2)] dx
-2i(£ + £k) J—O dx -2i(£ + £k) V Ck ) 2i(£ + £k)
ak (t)
d
21 (£ - £k) J—O dx
-f [(<Plfk2 - <P2fkl)(<Pigk2 - f2gkl)] dx =
2i(£ - £k)
2i(£ - £k)
The following integral is calculated in the same way:
q(t)ux(y\ + y2) dx = -q(t) (yi + y2) du =
' — O J — O
/OO /»<O
^y2 + y2)' dx = 2q(t) (uyiyi + uy2y2) dx
-O J —O
/O
[( y2 + i£y2)yi + (yi + i£yi)y/2] dx =
O
= 2q(t) [ yiy2 + i£yiy2 + yiyZ + i£yiy2] dx = 2i£q(t) (yiy2dx =
= 2i£q(t) ]im (yiy2)
Consequently, we obtain the equality (2.10)
R
— R
2i£q(t)a(£ )b(£).
N
I + ^ d® = J] + ^mcm).
—O k=i £ £k
According to equalities (2.9) and (2.10), we have the following equation:
dr+ ~dt
£2 - £k
k=i
r+ (Im £ = 0).
Differentiating the equality yn = Cnjn with respect to t, we get the following relation
dy ~dt
dy
+
d£
? = ?r
d£n _ dCn , r dt dt%lJn+ n dt
+ Cn
? = ?n
dj d£
?=?r
d£n dt '
□
1
0
0
1
0
r^J
1
O
O
O
O
—O
—O
which, according to (1.7), can be rewritten in the form
d<Pn dCn &ipn d£n
w = ^n + cn— - a(£n)hn—, (2.13)
dyn dy
where
Similarly to the case of a continuous spectrum, taking into account (2.7), in the case of a discrete spectrum, we obtain the following equality:
9iPn -A<pn=(—^- [ dx) hn+i^- [ hTnRyn dx + 4i£3np(t)) <pn, (2.14)
dt \ Cn J — O J \Cn J —O /
where hn = (^j'2^ . According to (2.13), the last equality can be rewritten in the following form:
dcn , . n d%l)n .(c,d£n
x
dt ^ ' " dt ' dt 1
— / dx ) hn + ( I h^Rtpn dx + 4i£%p(t) ) Cnx\)n.
Cn J-tx I \ Cn
Passing in this equality to the limit as x ^ +to, taking into account (1.8) and (2.2), we obtain the following equalities:
Thus, we have the following identities
dC ( (O \
= (8 i£3p(t) ~ J GMnl + hn2rtpn2) dx) Cn,
d£n -J'Zo G(y2ni + yn2) dx
(2.15)
dt Cnà(£n)
It remains to note that, according to the identity
2i f+x à(£n) = -yr / <~Pniyn2dx, Cn J — x
the last equality can be rewritten as
d£n = IZo + dx
dt 2i f+tz yn\Vn2 dx
(2.16)
Lemma 2.2. If the vector-functions <yn(x,£n ) = ( ^ni (x'£n M , xUn (x,£n) = \ (x'£n )
\<fn2 (x,£ny r \^n2 (x,£n)
lhn (x £n)\
and hn (x, £n) = I h11 ( ' £ ) ) are the solutions of the equation Lv = £n v, then their components
\hn2 (x, £n) J
satisfy the equalities
/tx
G(hni^ni + hn2^n2) dx = ian(t)Pn(t)wn(t) + 2i£nq(t), (2.17)
•t
/oo px
G(yh\ + ^2) dx = —2^n(t)an(t) yni^n2 dx. (2.18)
-x J — x
Proof. Firstly, to prove the lemma, we write the following equality:
p+x p+x
/ G(hnx-0m + hn2-n2) dx = —q(t) / ux (x' t)(h ni -ni + hn2 —n2 ) dx +
N
r» + œ
(2.19)
+ 2^ ak(t) / (fki9ki - fk29k2)(h
k=i "-x At £k = £n, according to Lemma 1.2, we have
ak(/kigki — /k2gk2) (hni—ni + hn2—n2) = ak/"kigkihni—ni + ak/"kigkihn2—n2 —
- Ctkfk2gk2hniripni ~ akfk2gk2hn2i)n2 = 0./ffc , ^ t~ [(^i/fci + hn2fk2)(ijjnigki + Vv^fci)] +
—2i(£n + £k) dx
+ £ )~dx [(hnlfk2 ~ hn2fkl){^nigk2 - V^fcl)] •
Let's integrate the above equality over x from —to to +to:
^ y- [(^nl/fcl + hn2fk2)(lpnigkl + 1pn2gk2)] dx +
-2i(Ç„ + Çk ) J—œ dx
œ œ
+ s [ -J- [(^ni/fc2 - hn2fki)(tjjnigk2 - Ipn2gki)] dx =
2i(Çn — Çk^—œ dx
lim [(/ini/fci + hn2fk2)(tljnigki + ^2^2)] I^R +
. /. \ vrai,/ ki 1 <"n2j k2M Yniy ki 1 Yn2y k2 ; d
+ \ lim [(^«i/fc2 - hn2fki)(tjjnigk2 + Vv^fci)] =
2i(Çn - Çk) L R
iim C«tn-&)R _ ç ¿«tn-MRUkjt)c-j(
- £fc)
Therefore, for £k = £n we have
/x
(/kigki — /k2gk2)(hni-ni + hn2-n2) dx = 0.
■x
If £k = £n, then
Q-nit) (/ralfi'ral ~ fn29n2) {h-nl^nl + hn21pn2) ^^ ^
(h
igni + hn2gn2)(/ni—ni + /n2—n2^ —
[("0ni/n2 — r4)n2fni){hnign2 + hn2gni) + (ipnifn2 + ^n2fni){hnign2 — hn2gni)~\ =
-Cn^ni^n2an(t)
+ vn(t)gni g„2 - + Vn{t)gn2 J gn\
. / J. \ f'ni I '■'n^yyni I ¿/'/¿2 \ . / J. \
a(Çn) / WÇn)
= -Cn^ni^n2^y\an(t)un(t). (2.20)
Let us integrate equality (2.20) with respect to x:
/"œ fin (t)
/ (/niâ'ni - fn2gn2)ihnYtpni + hn2r(pn2) dx = --^—ran(t)ujn(t) / C^raiVv^ dx = J — œ <a(Çra) ./—œ
œ
ttt r-- / (Pni<fn2dx = --an(t)pn(t)un(t). (2.21)
<a(Çra) J — œ 2
-œ
œ
0
Let us calculate the following integral using equality (1.1) and asymptotics (1.3), (1.8):
/OO PO
ux(hm jm + hn2 jn2) dx = -q(t) (hm + hm jn2) du =
O —O
/O
(uh'n1 jn1 + uhm jln1) dx +
O
/O
(uh'n2 jn2 + uhn2 jn2) dx =
O
/O
{(-jn2 + i£njn2 )h'n1 + ( hn2 + i£n hn2 ) jnj dx +
O
O ( ) + q(t) (h'n2 (j,n1 + i£n ) + jn2 (hni + i£nhni ^ dx =
—O
O(
O O
i£nq(t) ((hni jn2)' + (hn2 jni)') dx = i£nq(t) (hm jn2 + hn2 jni)|°
—O
1
= i£nq(t)(e-*"x ■ e^x - (~Cne*"x • ^-e"*"*)) =
Using the last equality and equalities (2.21), we obtain identity (2.17). Now, we derive the equality (2.18). At £k = £n, according to Lemma 1.2, we have
/O
(fkigki - fk2gk2)(y2ni + yn2) dx =
O
ak^ ^ -y- [(fklfnl + tpn2fk2){tpnigkl + Pn29n2)] dx +
-2i(£n + £k) J—O dx
O
O
+ rw?fc^ N [ T~ \(fk2<fnl - tpn2fkl){tpnigk2 - Vn29nl)] dx =
2i(£n - £k) J—o dx
-ak (t) = ——-— lim
2i(£n + £k)
+ lim
2i(£n - £k)
(fkiyni + yn2fk2)(ynigki + yn2 gn2)
R n
—RJ
+
(fk2yni - yn2fki)(ynigk2 - yn2gki)
R i
— R-i
0.
If £k = £n, then
O
an(t) (fnigni - fn2gn2)(yni + yn2) dx
—O
O
an (t) f° d r
/ -J- [{jnlVnl + (Pn2jn2) \'~PnlQnl + <~Pn29n2) \ dx ~ 4i£n J — O dx
an(t)
2
an(t)
—O
O
2
an (t)
[(fniyn2 - ynifn2)(ynign2 + yn2gni)] dx -[(fn2yni + yn2fni)(ynign2 - yn2gni^ dx =
2yniyn2(fnign2 - fn2gni) dx = -an(t)Un(t) yniyn2 dx.
ni n2 ni n2 n2 ni n n
2 —O —O
So, we got this equality
/OO /»O
(fnigni - fn2gn2)(y2ni + yn2) dx = -an(t)un(t) yniyn2 dx. (2.22)
O —O
—O
O
In a similar way, it can be shown that
/+O
q(t)ux(y2n± + yn2 )dx = 0. (2.23)
O
Using equalities (2.22) and (2.23), we obtain
/OO po
G{tp2nl + y22) dx = -2uin(t)an(t) / tpni<fn2 dx. □
O —O
Substituting equality (2.17) into the right side of equality (2.15), we obtain the following expression:
dG
-7T = i8i£nP(t) + ian(t)/3n(t)ujn(t) - 2i£nq(t)]Cn, n=l,N. dt
According to equalities (2.16) and (2.18), we can calculate the evolution of the eigenvalue
= ian(t)un(t), n=l,N.
dt
Thus, we have proved the following theorem.
Theorem 2.1. If the functions u(x,t), fk(x,t), gk(x,t), k = 1,N, are a solution to the problem (0.1)-(0.5), then the scattering data of the operator L(t) with the potential u(x, t) satisfy the following differential equations
ian(t)u}n(t), n=l,N,
dr+ ~dt
dCn ~dt
d£n dt
N
/~v, ltl/,1, I t 1
r+ (lmf = 0),
- tT
k=i
= [HnP(t) + ianit)f3n{t)ujn{t) - 2i£nq(t)] Cn, n=l,N.
The obtained equalities completely determine the evolution of the scattering data, which makes it possible to apply the inverse scattering method to solve problem (0.1)-(0.5).
Example2.1. Consider the following Cauchy problem
27(6u2ux + uxxx) (3 - 3i(t2 + t)2)ux
Ut + (i + l)(2i3 + 3^ + 3)3 + (t+1)^ + 3^ + 3) = 2{t + 1){fn9n ~ fl29l2)>
L(t)fi = £ifi, L(t)gi = £igi,
u(x, 0) = —-—, Ui{t) = fngi2 - fngn = t. ch x
It is easy to find the scattering data for the operator L(0):
|r+(0) = 0, 6(0) = Cx( 0) = i}. According to Theorem 2.1, the evolution of scattering data is as follows
£i(t) = iY (t), r+ (t) = 0, Ci (t) = ie^,
where
t3 t2 1 7(t) = I+ 2" + 2' Mi) = 21n(t+1).
Applying the inverse problem method, we obtain the following relations
_6(2t3 + 3t2 + 3 )2{t + i^e-d+^+t2)*_
~ (213 + 312 + 6) (416 + 1215 + 9i4 + 12i3 + 18i2 + 9 + 9(t + i)4e-(2+2t2+ft3)*)
(36i3 + 54i2 + 54) (t + l)2e"(f+ir+2i2):c
f12{x,t) = e 2 _
(2i3 + 3i2 + 6) (4i6 + 12i5 + 9 i4 + 12i3 + 18i2 + 9 + 9(t +
r-x
Jo
. , 873(t)e27(t)x - 27(t)(t + 1)4e—27(t)x + 472(t)e2Y(t)x + (t + 1)4e—2Y(t)x
=-8fimF-x
x
o
| ¿(27(t) + l)(472(t) + {t + l)4e-4^x)
872(i)(i + l)2e-(27(i)+^)x -472(t)(t +1)2
u(x, t) =
where
A(x,t)
a(x, t)
(2y2(t) - 0.5)e2Y(t)x + (t + 1)2 ch(27(t)x - 2 ln(t + 1))
(613 + 9t2)(t + l)2 - (fi3 + t2 + 2) (213 + 3i2 + 2)2e^t3+2t2+2>
(213 + 3i2 + 2)2e(ti3+2i2+2)x + 9(t + l)2 -ef (2i3 + 3i2 + 6) ((2i3 + 3i2 + 3)2 + 9 (t + ife-^+2f2+2)x)
3(2 i3 + 3 i2 + 3)2 + 54(t + i)4e-(ft3+2t2+2)* § 3. Loaded mKdV equation with source
Consider the following equation:
2N
2
Ut + P(u(xo,t))(6u2Mx + Uxxx) + Q(u(xi,t))ux = ^Bk(u(X2,t))(/ki9ki - fk29k2), (3.1)
k=i
where P(y), Q(z) and Bk(s), k = 1' 2N, are polynomials in y, z and s, respectively. The equation (3.1) is not a particular case of the equation (0.1), because the coefficients in the equation (3.1) depend on the value of the solution on a manifold of lower dimension. Such equations are called loaded equations.
In the work [29], Nakhushev gave the most general definition of loaded equations and gave a detailed classification of various types of loaded equations. Among the works devoted to loaded equations, the works [30-38] should be singled out.
If in the problem (0.1)-(0.5) instead of the equation (0.1) we consider the equation (3.1), then the following theorem holds.
Theorem 3.1. If functions u(x, t), fk(x, t), gk(x,t), k = 1, N, are a solution to the problem (3.1), (0.2)-(0.5), in the class offunctions (0.5), then the scattering data of the operator L(t) with potential u(x' t) change according to t as follows
^- = iBn(u(x2,t))un(t), n=l,N, dt
dr+
It
8 iÇ*P(u(x0, t)) + 2* £ Mu(x^))uk(t) _ 2mu(xut))
k=i Ç2 - Çk
dCn _ i~o „•
~df
r+ (Im Ç = 0),
[8t£3P(u(x0, t)) + iBn(u(x2, t))pn(t)u)n(t) - 2t£nQ(u(x1,t))] Cn, n = TJf.
Example 3.1. Consider the following Cauchy problem
Ut + 6u2ux + uxxx + a(t)u(1,t)ux = 2p(t)u(0,t)(fngn — fngn),
Lfi = 6fi> Lgi = 6gi,
where
0) = —-L, Ui{t) = fugi2 - /12^11 = t, ch x
^ (t + 1)2 ((3t4 + 8t3 + 6t2 + 6)2e-10t + 36e10t) pit) =-
a(t) =
—2(3t4 + 8t3 + 6t2 + 6)2 (10 - 873(i) + z(i2 + i)2) (472(i)e-10i+2^) + e10i-2^)
-1673(i) t4 2t3 t2 1
As in Example 2.1, the scattering data of the operator L(0) have the form:
{r+(0) = 0, 6(0) = Cx( 0) = i}. According to Theorem 3.1, we have
£i(t) = iY (t), r+(t) = 0, Ci(t) = ie^(t),
where
p(t) = % f y 3(r) dr + if rp(r )u(0,T )u1(r ) dr + 2 / y (t )a(r )u(1,r) dr. (3.2) Jo ,/0 ,/0
Consequently, F(x,t) = e—7(t)x+M(t). Solving the system of integral equations of Gelfand-Levitan-Marchenko, one can obtain
4Y 2(t)e^(t)—Y(t)(x+y) = 472(t) + e2^t)-4j(t)x ■
Using the last equality and formula (1.6), we obtain the following:
-4Y 2(t)
u(xf) = 1 \
(272(i) - I)e-/i(t)+27(t)s + ch(27(i)x - pit))'
If we set x = 0 and x =1 in the last equality, then taking into account (3.2), we have the following problem
p [b) 07 [b) e2M(t)+472(t) e2M(t)+472(i)e4T(t) '
^(0) = 0.
Solution of this problem has the form p(t) = 10t. As a result, the solution of the considered problem is expressed as follows:
—4Y 2(t)
u(x, t)
fii(x,t)
(272(i) - I)e-10i+2-rW- + ch(27(i)x - 101)
87 2(t)e-(27(t)+0.5)x+10t
(2j(t) + 1)(4y 2(t) + e-4Y(t)x+20t)
47 (t)e-4Y(t)x-0.5x+20t
Z12^) e 2 (27(i) + l)(472(i) + e-47(i)-+20i)'
x
0
8Y3(t)e27(t)x-i0t — 2Y(t)e-2Y(i)x+i°i + 4Y2(t)e27(t)x-i0i + e-27(i)x+i0i
9l2{x>t) =-sW)-x
2
' fo A(s,t) ds / ,
Jo 872(i)e-(27(t) + |);c+101
where
A(Xt) = -21(t)-l +_tm_ arx t)=-ef(27(t) + l)(472(t) + e-^)^)
[X,l) >{l) i +4^(^47(^-201 + 1' 472(i) + 2e-47(i)x+20i
REFERENCES
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2. Leblond H., Sanchez F. Models for optical solitons in the two-cycle regime, Physical Review A, 2003, vol. 67, issue 1, 013804. https://doi.org/10.1103/PhysRevA.67.013804
3. Leblond H., Mihalache D. Models of few optical cycle solitons beyond the slowly varying envelope approximation, Physics Reports, 2013, vol. 523, issue 2, pp. 61-126. https://doi.org/10.1016/j.physrep.2012.10.006
4. Ono H. Soliton fission in anharmonic lattices with reflectionless inhomogeneity, Journal of the Physical Society of Japan, 1992, vol. 61, no. 12, pp. 4336-4343. https://doi.org/10.1143/JPSJ.61.4336
5. Kakutani T., Ono H. Weak non-linear hydromagnetic waves in a cold collision-free plasma, Journal of the Physical Society of Japan, 1969, vol. 26, no. 5, pp. 1305-1318.
https://doi.org/10.1143/JPSJ.26.1305
6. Konno K., Ichikawa Y. H. A modified Korteweg-de Vries equation for ion acoustic waves, Journal of the Physical Society of Japan, 1974, vol. 37, no. 6, pp. 1631-1636.
https://doi.org/10.1143/JPSJ.37.1631
7. Watanabe Sh. Ion acoustic soliton in plasma with negative ion, Journal of the Physical Society of Japan, 1984, vol. 53, no. 3, pp. 950-956. https://doi.org/10.1143/JPSJ.53.950
8. Lonngren K. E. Ion acoustic soliton experiment in a plasma, Optical and Quantum Electronics, 1998, vol. 30, pp. 615-630. https://doi.org/10.1023/A:1006910004292
9. Ziegler V., Dinkel J., Setzer C., Lonngren K. E. On the propagation of nonlinear solitary waves in a distributed Schottky barrier diode transmission line, Chaos, Solitons and Fractals, 2001, vol. 12, issue 9, pp. 1719-1728. https://doi.org/10.1016/S0960-0779(00)00137-5
10. Cushman-Roisin B., Pratt L., Ralph E. A general theory for equivalent barotropic thin jets, Journal of Physical Oceanography, 1993, vol. 23, issue 1, pp. 91-103. https://doi.org/10.1175/1520-0485(1993)023<0091:AGTFEB>2.0.C0;2
11. Ralph E.A., Pratt L. Predicting eddy detachment for an equivalent barotropic thin jet, Journal of Nonlinear Science, 1994, vol. 4, issue 1, pp. 355-374. https://doi.org/10.1007/BF02430638
12. Grimshaw R., Pelinovsky E., Talipova T., Kurkin A. Simulation of the transformation of internal solitary waves on oceanic shelves, Journal of Physical Oceanography, 2004, vol. 34, issue 12, pp. 2774-2791. https://doi.org/10.1175/JP02652.1
13. Grimshaw R. Internal solitary waves, Environmental stratified flows, New York: Springer, 2002, pp. 1-27. https://doi.org/10.1007/0-306-48024-7_1
14. Tappert F. D., Varma C. M. Asymptotic theory of self-trapping of heat pulses in solids, Physical Review Letters, 1970, vol. 25, issue 16, pp. 1108-1111. https://doi.org/10.1103/PhysRevLett.25.1108
15. Hirota R. Exact solution of the modified Korteweg-de Vries equation for multiple collisions of solitons, Journal of the Physical Society of Japan, 1972, vol. 33, no. 5, pp. 1456-1458. https://doi.org/10.1143/jpsj.33.1456
16. Satsuma J. A Wronskian representation of N-soliton solutions of nonlinear evolution equations, Journal of the Physical Society of Japan, 1979, vol. 46, no. 1, pp. 359-360.
https://doi.org/10.1143/JPSJ.46.359
17. Nimmo J. J. C., Freeman N. C. The use of Backlund transformations in obtaining N-soliton solutions in Wronskian form, Journal of Physics A: Mathematical and General, 1984, vol. 17, no. 7, pp. 1415-1424. https://doi.org/10.1088/0305-4470/17/7/009
18. Gesztesy F., Schweiger W. Rational KP and mKP-solutions in Wronskian form, Reports on Mathematical Physics, 1991, vol. 30, issue 2, pp. 205-222. https://doi.org/10.1016/0034-4877(91)90025-I
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23. Mamedov K. A. Integration of mKdV equation with a self-consistent source in the class of finite density functions in the case of moving eigenvalues, Russian Mathematics, 2020, vol. 64, issue 10, pp. 66-78. https://doi.org/10.3103/S1066369X20100072
24. Wu Jianping, Geng Xianguo. Inverse scattering transform and soliton classification of the coupled modified Korteweg-de Vries equation, Communications in Nonlinear Science and Numerical Simulation, 2017, vol. 53, pp. 83-93. https://doi.org/10.1016/j.cnsns.2017.03.022
25. Zhang Guoqiang, Yan Zhenya. Focusing and defocusing mKdV equations with nonzero boundary conditions: Inverse scattering transforms and soliton interactions, Physica D: Nonlinear Phenomena, 2020, vol. 410, 132521. https://doi.org/10.1016/j.physd.2020.132521
26. Vaneeva O. Lie symmetries and exact solutions of variable coefficient mKdV equations: An equivalence based approach, Communications in Nonlinear Science and Numerical Simulation, 2012, vol. 17, issue 2, pp. 611-618. https://doi.org/10.1016/j.cnsns.2011.06.038
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32. Hasanov A. B., Hoitmetov U. A. On integration of the loaded Korteweg-de Vries equation in the class of rapidly decreasing functions, Proceedings of the Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, 2021, vol. 47, no. 2, pp. 250-261. https://doi.org/10.30546/2409-4994.47.2.250
33. Hoitmetov U. A. Integration of the loaded KdV equation with a self-consistent source of integral type in the class of rapidly decreasing complex-valued functions, Siberian Advances in Mathematics, 2022, vol. 33, no. 2, pp. 102-114. https://doi.org/10.1134/S1055134422020043
34. Khasanov A. B., Hoitmetov U. A. Integration of the general loaded Korteweg-de Vries equation with an integral type source in the class of rapidly decreasing complex-valued functions, Russian Mathematics, 2021, vol. 65, issue 7, pp. 43-57. https://doi.org/10.3103/S1066369X21070069
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38. Hoitmetov U. A. Integration of the sine-Gordon equation with a source and an additional term, Reports on Mathematical Physics, 2022, vol. 90, issue 2, pp. 221-240. https://doi.org/10.1016/S0034-4877(22)00067-2
Received 30.12.2022 Accepted 17.04.2023
Aknazar Bekdurdiyevich Khasanov, Doctor of Physics and Mathematics, Professor, Department of Differential Equations, Samarkand State University, University boulvard, 15, Samarkand, 140104, Uzbekistan. ORCID: https://orcid.org/0000-0003-2571-5179 E-mail: ahasanov2002@mail.ru
Umid Azadovich Hoitmetov, Candidate of Physics and Mathematics, Department of Applied Mathematics and Mathematical Physics, Urgench State University, ul. Khamida Alimdjana, 14, Urgench, 220100, Uzbekistan.
ORCID: https://orcid.org/0000-0002-5974-6603 E-mail: x_umid@mail.ru
Shekhzod Quchqarboy ugli Sobirov, PhD student, Department of Applied Mathematics and Mathematical Physics, Urgench State University, ul. Khamida Alimdjana, 14, Urgench, 220100, Uzbekistan. ORCID: https://orcid.org/0000-0003-0405-3591 E-mail: shexzod1994@mail.ru
Citation: A. B. Khasanov, U. A. Hoitmetov, Sh. Q. Sobirov. Integration of the mKdV Equation with nonsta-tionary coefficients and additional terms in the case of moving eigenvalues, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2023, vol. 61, pp. 137-155.
А. Б. Хасанов, У. А. Хоитметов, Ш. К. Собиров
Интегрирование уравнения мКдФ с нестационарными коэффициентами и дополнительными членами в случае движущихся собственных значений
Ключевые слова: интегральное уравнение Гельфанда-Левитана-Марченко, система уравнений Дирака, решения Йоста, данные рассеяния.
УДК: 517.957
DOI: 10.35634/2226-3594-2023-61-08
В данной работе рассматривается задача Коши для нестационарного модифицированного уравнения Кортевега-де Фриза с дополнительным членом и с самосогласованным источником в случае движущихся собственных значений. Получена эволюция данных рассеяния оператора Дирака, потенциал которого является решением нагруженного модифицированного уравнения Кортевега-де Фриза с самосогласованным источником в классе быстроубывающих функций. Приведены конкретные примеры, иллюстрирующие применение полученных результатов.
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2. Leblond H., Sanchez F. Models for optical solitons in the two-cycle regime // Physical Review A. 2003. Vol. 67. Issue 1. 013804. https://doi.org/10.1103/PhysRevA.67.013804
3. Leblond H., Mihalache D. Models of few optical cycle solitons beyond the slowly varying envelope approximation //Physics Reports. 2013. Vol. 523. Issue 2. P. 61-126. https://doi.org/10.1016/j.physrep.2012.10.006
4. Ono H. Soliton fission in anharmonic lattices with reflectionless inhomogeneity // Journal of the Physical Society of Japan. 1992. Vol. 61. No. 12. P. 4336-4343. https://doi.org/10.1143/JPSJ.61.4336
5. Kakutani T., Ono H. Weak non-linear hydromagnetic waves in a cold collision-free plasma // Journal of the Physical Society of Japan. 1969. Vol. 26. No. 5. P. 1305-1318.
https://doi.org/10.1143/JPSJ.26.1305
6. Konno K., Ichikawa Y. H. A modified Korteweg-de Vries equation for ion acoustic waves // Journal of the Physical Society of Japan. 1974. Vol. 37. No. 6. P. 1631-1636. https://doi.org/10.1143/JPSJ.37.1631
7. Watanabe S. Ion acoustic soliton in plasma with negative ion // Journal of the Physical Society of Japan. 1984. Vol. 53. No. 3. P. 950-956. https://doi.org/10.1143/JPSJ.53.950
8. Lonngren K. E. Ion acoustic soliton experiment in a plasma // Optical and Quantum Electronics. 1998. Vol. 30. P. 615-630. https://doi.org/10.1023/A:1006910004292
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26. Vaneeva O. Lie symmetries and exact solutions of variable coefficient mKdV equations: An equivalence based approach // Communications in Nonlinear Science and Numerical Simulation. 2012. Vol. 17. Issue 2. P. 611-618. https://doi.org/10.1016/j.cnsns.2011.06.038
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Поступила в редакцию 30.12.2022
Принята к публикации 17.04.2023
Хасанов Акназар Бекдурдиевич, д. ф.-м. н., профессор, кафедра дифференциальных уравнений, Самаркандский государственный университет, 140104, Узбекистан, г. Самарканд, Университетский бульвар, 15.
ORCID: https://orcid.org/0000-0003-2571-5179 E-mail: ahasanov2002@mail.ru
Хоитметов Умид Азадович, к. ф.-м. н., доцент, кафедра прикладной математики и математической физики, Ургенчский государственный университет, 220100, Узбекистан, г. Ургенч, ул. Х. Алимджа-на, 14.
ORCID: https://orcid.org/0000-0002-5974-6603 E-mail: x_umid@mail.ru
Собиров Шехзод Кучкарбой угли, аспирант, кафедра прикладной математики и математической физики, Ургенчский государственный университет, 220100, Узбекистан, г. Ургенч, ул. Х. Алимджа-на, 14.
ORCID: https://orcid.org/0000-0003-0405-3591 E-mail: shexzod1994@mail.ru
Цитирование: А. Б. Хасанов, У. А. Хоитметов, Ш. К. Собиров. Интегрирование уравнения мКдФ с нестационарными коэффициентами и дополнительными членами в случае движущихся собственных значений // Известия Института математики и информатики Удмуртского государственного университета. 2023. Т. 61. С. 137-155.
