On finite-zone method of construction of n-solitonic solutions of vectorial non-linear Schrödinger equation Текст научной статьи по специальности «Математика»

ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ

ON FINITE-ZONE METHOD OF CONSTRUCTION OF N-SOLITONIC SOLUTIONS OF VECTORIAL NON-LINEAR SCHRyDINGER EQUATION

Asghari-Larimi Mohammad

PhD Candidate Muminov Khikmat

PhD, DSc, Professor, S.U. Umarov Physical-Technical Institute, Academy of Sciences of the Republic of Tajikistan ABSTRACT

In this paper new types of three-soliton solutions of the vector non-linear Schrodinger equation with mixed boundary condition by use of the algebraic geometric method of delinearization are constructed. Asymptotes of these solutions satisfy to the mixed boundary conditions.

Keywords: Vectorial non-linear Schrodinger equation, Three-soliton solutions, N-soliton potential.

1. Introduction

Often in physics at investigation of non-linear wave processes the problem reduces to the systems of the differential equations modeling interaction the finite number of waves and wave packages. One of the most well-known equations is the vector non-linear Schrodinger equation (VNSE).The scalar non-linear Schrodinger equation is enough thoroughly studied. The major class of multi-soliton solutions of this equation is found. It should be noted that this equation generally has been integrated by the inverse scattering method. In the present work we will study the vector non-linear Schrodinger equation by the so-called delinearization, or algebraic geometric method, which has been offered for the first time in the paper [4] and developed further in papers [1-3,5]. This method is more simple, but the new class before unknown multisoliton solutions come out. By use of this method of integrating.most thoroughly this method is stated in the review [5]. Further shortly without proofs we will stop on high lights of the present method.

2. Preliminaries

We consider the problem is solving on eigenvalues of Schrodinger operator with a zero eigenvalue

iyt-£;+U(x,x))ip(x,t;k) = 0 (1)

In order to obtain multi-soliton solutions it is necessary to require, the function^ (x, t; k)to be a meromorphic function in a finite plane with the first order poles in some points = 1,2,...,N.

The solution of the Schrodinger equation could be represented as follows

W(x, t; k) =

QN(x,t; k)eikx+ik2t

(2)

Where

QN(x,t;k) = kN + ai(x,t)kN-i + - + QN(x,t) (3)

is a k-th degree polynomial and points are singular points in a complex plane.

As one can see the function W(x, t; k)is analytical by kin all points of the plane except singular points Hj.

Thus, we have introduced an essential limitation which allows us to use a theory of function of complex variable as the meromorphic function can be uniquely determined, knowing its residue in singular points.

As shown in paper [5], function W(x, t; k) in the point Pj could be expressedin terms of residues as follows

N

W(x, t;k) = —^ Cij res W(x, t; k); i = 1,2,..., N, (4)

i=i

whereQ, is a constant matrix of N x N dimension.

In the review [5] following theorems have been formulated, we give them without proofs:

Theorem 1. Let the parameters nN(Cij) which determine the function W(x, t; k) of the form (3) with the conditions (4) satisfy the following requirements:

a) The matrix Ctj is a skew-hermitian Ctj = C^

b) If the points are numerated in such a way that

Im^i > 0,i = l.p'Im^i < 0,i = p + 1,N.

Then the hermitian matrix

(1Ckl),1<k,l<p

must be positively defined one, and

(jCkl),p + 1<k,l<N.

must be negatively defined (these matrices can be nonnegative as well).

If those conditions on the parameters are fulfilled, the function

V(x, t; k) is the smooth function of real x, t for all k ± and

satisfies the equation L = idt — + U (x, t) with a real

smooth potential^ (x, t).

For these functions we have

det M(x,t;k) .. ,,2t

W(x, t; k) =—-—-±elkx+lk

detM (x,t)

U(x,t) = 2dj;ln detM(x,t),

where

eí{o)l-o>j)

Míj(x, о = Су + -, = ßi(x + ßit), i,j

ßi ßj = 1,2,..., N,

Щ = Mu.i.j = 1,2.....N,M00 = l,Mi0 = eimt,

Moi=T-,i,j = l,2.....N. (S)

k — ßi

Let's enter following labels:

Wj = resW(x, t; k),0j(x, t) = bijW(x, t,^j),j =

1,2,..., N. (6)

Where functionOjand Wj satisfy to following equations:

i0jt - 0jxx + U(x, t)4>j = 0,j = 1,2.....N,

W - Wjxx + U(x, t)Wj = 0,j = 1,2.....N. (7)

Where indices xand mean partial derivatives by the according variable.

Theorem 2. Let the

functions <Pj(x, t),Wj(x, t)correspond to the set of data ^1,...,^N, Cij and to the rational function which has one of the following from

Zbf

(8)

Then takes place following requirements of the coordination:

U+^eib'22 + C2

Í = 1

n

= ^£11Ф1(Х,1)12

=1 n

— ^ Vi(x,t)Ei]V](x,t),(9)

Where

í J=1

Eij = Cij (Е(р.{) — E(H¡)).

(1О)

Definition 1. We shall call the integrable potential U (x, t)which is given by our construction with N parameters ii,...,iN and N x N matrix Cy the N-soliton potential.

3. Three-soliton solution

In the section we will consider three-soliton solutions of the system of three non-linearSchrodinger equations that there corresponds toexistence of three poles ij and dimensionality 3 x 3of matrix Cy .Accordinglythe dimensionality matrix Eyis 3x3. Assumingn = 1 and C2 = 0 E¿j = -£2fr Yjwe can redefine functions $and Wj

Pi = $,92= Y1V1+72^2,93 = YiW +Y2W2+ Y3W3 (11)

Then from the formulas (10) we will obtain the potential as follows

U = 2{£i\(pi\2 + eM2 + £3\<P3\2}, (12) and the system of equations (7) will take the form lPit-Pixx + U(x,t)pi = 0, ÍP2t - P2XX + U(x, t)p2 = 0, iP3t - P3XX + U(x, t)p3 = 0. (13)

Again as a result of this construction functions constructed through residue Wj, have decreasing boundary conditions, and function $ has oscillating asymptotic. The formula (5) for residue at N = 3 can be rewrite

r es detMn(x,t)

V¡ = V(x, t; k) =

1 k = ßj detM(x,t)

(14)

Where Моы = Мы, к, I = 1,2,3; M00 = О, Mol = S^.Mko =

,1Шк

Function W(x, t; k) could be found through the residues

Ze- J

Ш-)eik(x+kt)

к — U;

=1

k — ßj

(1S)

As it follows from (11), the matrix Ci , take the form

c = £2yiYj ij EJd—Eißj)'

(1б)

Calculating determinant (16), we obtain functions W1 and W2and W3 in the explicit form

A = yi6iW1+P1 + 2P2 + 2P3 - y2eiW3+P3 - y^ei(W1-W2 + W3)-P1+P2+P3 + y^eiW1+P1 + 2P3 - y^eiW1+P1 + 2P2 + y6eiW2+P2 + y7ei(W1 + W2-W3)-P1 + P2 + P3 + ys6iW1 + P1 + 2P2 + y9eM1 + P1 - yw6i(W1 + W2-W3) + P1-P2 + P3 - yiieiW1-P1 + 2P2 + 2P3 - yi2eiW2-P2 + 2P3 + y^3eiW1-P1 + 2P3 - y^eiW3-P3 + 2P2 + yit.eiW1-P1 + 2P2

y16eiW1 + P1 + 2P3 + yi7ei-W1-W2 + W3)+P1 + P2-P3 + yweiW1 + P1 + yi9eiW3-P3 - y2oeiW3-P3 + y^eiW1-P1

y 22e

y24e

i W2-P2

iW1-P1 + y23eiW2-P2

B = z,eiw3-p3 + z?eiw2-p2 — z,eiwl-pl — z.eiw3-p3 — z„eiw2-p2 + z,eiwl-pl

+z1ei(-Wl+W2+W3)+Pl+P2-P3 — z8elWl+Pl — z9eiw2+2Pl+P2 + z eíw3+2pl-p3 + z eíw2+2pl+p2 + Zl2eiw2+P2

+ Z13el(-Wl + W2 + W3) + Pl-P2 + P3 — Z14elW3 + P3 + zlseí(Wl + W2-W3) + Pl-p2 + P3 + Zl6eíw2 + 2pl-p2 + 2p3 — Z17el(Wl + W2-W3)-Pl + P2 + P3 + ZigeLW2 + P2 + 2p3 + z19eíW2 + P2 — z2QeÍW2-P2 + 2P3 — Z21eíWl-Pl + 2P3 + Z22elw2 + P2 + 2P3 — z eiW2 + 2pl-p2 + z eiW2 + 2pl-p2 + 2p3

n

з

C = eiw2 + P2 - hgiW3 + 2p2 + P3 + h,,eÍW3 + P3 - hgí(w1-W2 + W3)-p1 + P2 + P3 + he™2 + 2Pl + P2 + hgÍW3 + 2Pl + P3 + h-jC^i^i

D = m1e2(p2+P3)

- h8elWi-pi + h9elW3-p3 + h10

h13el(-wi+w2+w3)+Pi-P2+P3 + h hweiW3+2vi-p3 - h19e + h24eíW3+2p2+P3

- m2e2(Pi+p2+P3) + m3e

-m6e

10

eí(Wi-W2 + W3) + pi + p2-p3 + hííeí(-Wi + W2 + W3)-Pi + P2 + P3 + h

PLW3 + P3 — h pÍW3 + 2pi + p3 — h pLWi + pi — h

14c h15c h16c h17

ÍW3 + 2P2-P3 _ U pÍW3 + 2pi + 2p2-p3

h20e

Í(W3-W2)+P2 + P3

h16elWi+pi - h17e h21eÍW2-p2 + h22eÍWi-pi + h23

■7*

PIW3+Pi+2P2-P3 12е ÍW3-P3

ÍWi + 2p2-pi

e-

mAe

í(W3-W2) + 2pi + p2 + P3 + m^gí(W2-W3)+p2 + P3

Í(W2-W3) + 2PI + P2 + P3 + m7gí(Wi-W2) + Pi + P2 + 2p3 + m gí(w3-w i) + Pi + 2p2 + P3 + m¡¡e2(pi + P2 + P3)

mi2eí(-W3-Wi) + Pi + 2p2 + P3 + m^e2(Pi + P2 + P3)

- minei(wi-w3)+pi+p3 - mííeí(w2-w3)+2pi+P2+P3 -

+ m14e2(Pi+P3) + m15e

- m18ei(wi-w3)+pi+2p2+p3 + + m22el(w2-wi)+pi+p2+2P3 -

- m26ei(wi-w2)+pi+p2 + m27e

- m4?e2(Pi+P2) + m^e2pi - m

32

33

Í(wi-W3) + Pi + 2P2 + P3

19

e2(Pi + P2-

2(Pi+P3) + m. 35 + 36 - "l37

í(W2-Wi)+pi + p2 + 2p3

m17e

13

2(Pi + P2 + P3)

m16e

m,Qei(w3-wi)+pi+p3 + m20ei(w3-^2)+2Pi+P2+P3 - m21e m23e2(pi+p2+p3) - m24ei(w2-wi)+pi+p2 + m25e2(Pi+r>2)

Í(Wi-W2)+PI + P2 + 2P3

2Рз + 34

т2ве + m29e

+ + m3f, - тЯ7 - m00 + m

2(Р2 + РЗ) + m e2(Pi + P2 + P3)

m31e

2P2

39

(17)

Hence

_A _B _C

Vi-D,V2-D,V3~D-

5. Conclusion

Using formulas (6), (12), (17), we write solutions of the equation (14) as follows

Ф1 = b( 1 +

e

itài(x,t)

e

i^2(x,t)

-W1(x,t)+--V2(x,t)

к- к -^2

e

i^3(x,t)

+ —-W3(x,t) )eik(x+kt),

к ^3

<P2 = Yl?1 + Y2^2

<P3 = Y1V1 + Y2V2 + Y3V3. (18)

One can easily show, that e 0,ф3 ^ 0 at x ^ ±œ

ik(x+kt)

<P2

References

1. K. Abdulloev, A. Maksodov, K. Muminov. New type of two-soliton solution of the vector non-linear Schrodinger equation with the mixed boundary conditions. Journal of Technical Physics.1993. vol.63 (3), P.180-184.

2. I.V. Cherednik. Functional Analysis and Applications,Russian journal. 1972. vol.12 (3), p.45.

3. B.A. Dubrovin, I.M. Krichever, T.G. Malanyuk, V.G. Makhankov. Exact solutions of a time dependent Schrodinger equation with selfconsistent potential. Elementary Particles and Atom. Nuclear. 1988. vol.19 (3), P.579.

4. I.M. Krichever. Functional Analysis and Applications, Russian journal. 1986. vol.20 (3), p.42.

5. S.P. Novikov. Soliton Theory. Inverse Scattering Method, Moscow, Nauka (in Russian).1979. p.213.

РАСЧЕТ НАПРЯЖЕННО-ДЕФОРМИРОВАННОГО СОСТОЯНИЯ АНИЗОТРОПНЫХ ПЛАСТИН С КОНЦЕНТРАТОРАМИ НАПРЯЖЕНИЙ РАЗЛИЧНЫХ ТИПОВ

Астапов Юрий Владимирович

Тульский Государственный Университет, магистрант, г.Тула

Соколова Марина Юрьевна

доктор физико-математических наук, профессор, Тульский Государственный Университет, г. Тула

В статье рассматривается деформирование пластин из ортотропного материала, нагружаемых растягивающей силой и ослабленных концентраторами напряжений различных типов. Исследуется распределение напряжений вблизи концентратора напряжений с целью определения влияния формы и размеров концентраторов и анизотропии свойств материала пластины на это распределение. Задача решается методом конечных элементов с построением матрицы граничной жесткости [1].

Основные уравнения, описывающие напряженно-деформированное состояние пластин, получены при условии неизменности температуры на основе приведенной в работах [2, 3] постановки связанной краевой термомеханической задачи нелинейной термоупругости в от-счетной конфигурации. К основным уравнениям конечного изотермического деформирования относятся:

1) определяющие соотношения анизотропной термоупругости, предложенные в работе [4] и связывающие «повернутый» обобщенный тензор напряжений

2

dV

R

dV

R • S • R 1 ( S - тензор истинных напряже-

0

ний Коши) с энергетически сопряженной с ним неголо-номной мерой деформаций М. Мера М была введена в работе [5] и определяется как решение дифференциального уравнения

м =

1

сМ

dt

WR = R • W • R

-1

(1)

где W =— (Уу + уУ) - тензор деформации скорости,

R - ортогональный тензор, входящий в полярное разложение аффинора деформации

Ф = U • ^ U = иТ, R 1 = RT, V

- тензор искаже-

ния.

Для неогуковского материала определяющие соотношения представляют собой тензорно-линейную