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11ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ
THE EXACT RESONANCE TYPE SOLUTION OF SINE-GORDON
EQUATION.
Khusainova Galina Vladimirovna
Candidate of Science, associate professor of the Ural State Architecture and Art Academy, Ekaterinburg Khusainov Damir Zinnurovich
Candidate of Science, associate professor of the Ural State Architecture and Art Academy, Ekaterinburg Sagaradze Igor Victorovich Candidate of Science, associate professor of the Ural State University of Architecture and Art, Ekaterinburg,
Abstract
The exact rational-exponential soliton solution of sine-Gordon equation is obtained by procedure based on Hirota method. The simplest rational-exponential solution of sine-Gordon equation is obtained as limit case of two-soliton solution. It has shown that the solution is described the bound state of two solitons types of kinks.
Keywords: soliton, rational-exponential solution, Hirota method
Хусаинова Галина Владимировна
канд. физ.-мат. наук, доцент Уральского государственного архитектурно-художественного университета, 620075 Россия, Екатеринбург, ул. Карла Либкнехта, 23.
Хусаинов Дамир Зиннурович
канд. физ.-мат. наук, доцент Уральского государственного архитектурно-художественного университета, 620075 Россия, Екатеринбург, ул. Карла Либкнехта, 23.
Сагарадзе Игорь Викторович
канд. физ.-мат. наук, доцент Уральского государственного архитектурно-художественного университета , 620075 Россия, Екатеринбург, ул. Карла Либкнехта, 23.
ТОЧНОЕ РЕШЕНИЕ РЕЗОНАНСНОГО ТИПА В УРАВНЕНИИ
СИНУС-ГОРДОН.
Аннотация
Получено точное полиномиально-экспоненциальное решение для уравнения синус-Гордон с помощью процедуры, основанной на методе Хироты. Простейшее полиномиально-экспоненциальное решение получено как предельный случай двух-солитонного решения. Показано, что это решение описывает связанное состояние двух кинков.
Ключевые слова: солитон, полиномиально-экспоненциальное решение, метод Хироты.
Let us consider the sine-Gordon (sG) equation
(dx -d2)u = sinu (1)
(the symbol д x denotes partial derivative with respect to x).It is known that eq.(1) has multi-soliton solutions (exponential solutions) [1].
In this work, we shall construct the simplest exact solution of another type for sG equation, namely, rational-exponential solution (afterwards referred to as RE solution).This solution is rational function of polynomials multiplied by exponents. It should be mentioned that RE solution for eq.(1) were found by Poppe [3] using the Fredholm determinant method.
We suggest a new approach to the construction of RE solutions. The proposed procedure is based on the formal perturbation theory for nonlinear partial differential equations in Hirota's bilinear form [2] with another choice of starting solution. In order to determine this starting solution, we study the simplest RE solution. We can obtained RE simplest solution from the two-exponential solution by choosing the phase constants as definite singular functions of physical parameters( such as the soliton amplitude and speed) and performing an appropriate limiting procedure.
Let us discuss the two-soliton solution of sG equation:
u(x, t) = 4arctang(xt), (2)
V 7 f(x,t) V 7
g(x,y) = e11 + e12 , (3)
f(x,y) = 1 + a(1,2)el1+12 , (4)
where ii = Pix - Qit +i0, Pi2 - Q2 = 1, Pi, Qi, i0 are arbitrary finite real
constants (i = 1,2) and a(1,2) = (P - P2)2 - (Q1 - Q2)2
(P1 + P2)2 -(Q1 +Q2)2
Introducing the the parametric dependence p = ch^, ^ = sh^ (i = 1,2), we can transformed the expression for a(1,2):
sh
a(1,2) =
2
2
ch
2
2
Note, that for finite real-phase constants (i = 1,2) in the limit p ^ p and Q1 ^ Q2 the expression (2)-(4) gives one-soliton solution. However, we can find a nontrivial solution taking definite complex values of (i = 1,2) .If we take
exP(rli) = - exP( ^2) =
1
a(1,2)
then after transformations two-soliton solution tends to the form:
g(x,t)
f(x,t)
ch
9 sh 2
sh
2
sh- ch
22
9 -92
sh
2
ch
ll + l 2
ln2
(5)
As for x^0 shx~x, chxtherefore in the limit p ^p and Qj ^Q2 (^ we find from expressions (2) and (5) the simplest RE solu-
tion:
u(x,t) = 4arctan
(sh^1x - ch^1t )e
ri
1 + le
4
l1
or
u(x,t) = 4arctan
where r1 = P1x -Q1t, P1 -Qj = 1 .
(Q1x - P1t )• e
i1
1 + le
4
(6)
l1
Note, that limit p ^ p and Qj ^ Q2 correspond to degenerate (resonance) case since two solitons have equal parameters.To find higher RE solutions in a similar way is a complicated problem. Here we proposed the method permits to obtained RE
solutions directly in an explicit form. This method is based on Hirota's formal perturbation theory[4].
Through the transformation [2]:
g(x,t)
u(x,t) = 4arctan- , ( ' ) f(x,t) '
the equation (1) is transformed into bilinear form:
(D2 - D2) g • f =g • f (8)
(DX -Dt2)(g • g - f • f)= 0 . (9)
Here D are the operators of Hirota [2] which are given by
DXDmf(x,t)g(x,t) =
d 1 n i- -—1
Kd x d x' y Kd t d t' ,
f(x,t)g(x',t')
x=x' t=t'
(7)
(10)
n=0,1,2...; m=0,1,2....
Expanding functions f(x,t) and g(x,t) in a power series in a parameter s :
g = £gl + s g3 + s g5 + •••• , (11)
f = 1 + s2f2 +s4f4 + •••• (12) and substituting (11),(12) into (8),(9), one obtains the set of equation
(DX - D2)gx =gx , (13a)
(d2 - D2 )(g2 - 2ft )= 0 , (136)
The analysis of eq.(6) shows that we can choosing the starting solution g1 in the form
N
(14)
g1 = S Q^1 ,
i=1
Here Qj = Q;x - pt + Q, ^ = P.x - t + , p2 - Of = 1, p,Q;,Q rf are
arbitrary constants (i = 1,2,...N).
Taking the starting solution g1 for N=1 :
g1 = Q1 e^1 (15)
we find from the system (13) the simplest RE solution of eq. (1):
m
g(x,t) = Qje \
f(x,y) = 1 +
~4~
or
u(x,t) = 4arctan
Qi
ch(^ - ln2)
(16)
(17)
(18)
Note, that the solution (6) coincides with solution (18) for C = 0, ^0 = 0. To get more complicated RE solutions we must start with function (14).The solution (18) is shown in Fig.1. One can see, that eq.(18) describes the bound state of 2n -kinks. For t = 0 the interaction of kinks have complicated nature. The evolution of this pair show that for large x,t kinks are moving in opposite directions.
Fig.l Solution u(x,t) given by the eq.(6), (P12 = 1,25, Qf = 0,25, C = 0, л0 = 0).
References
1. Косевич А.М., Иванов Б.А., Ковалев А.С. Нелинейные волны намагниченности. Динамические и топологические солитоны. - Киев: Наук. Думка, 1983 -192с.
2. Hirota R. Exact solution of the Sine-Gordon equation for multiple collisions of solitons//J.Phys.Soc.Jap. -1972-Vol.33,№5-P.1459-1464.
3. Poppe C. Construction of solutions of the sine - Gordon equation by means of Fredholm determinants//Physica D -1983 -Vol.9 -P.103 - 139.
4. Bezmaternih G.V. (Khusainova G.V.), Borisov A.B. Rational - Exponential Solutions of Nonlinear Equations// Lett.Math.Physics -1989 -Vol.18-P.1 - 8.
