Summary_ (Исследование некоторых типов дифференциальных уравнений с сильной нелинейностью), страница 5

An Introduction to Quantum Theory of Gauge Fields. – Мoscow: Nauka, 1988.22 2u a1 ads ab cas(y)fus(y),s(y).0abc002dy 0r2i 1 y i3(143)As can be readily verified, Eq. (142) has the solutionu a  s a ( y0 ) / r  g a ( y0 , 1 ,  2 ,  3 ),  i  yi / r,(144)where g a are arbitrary differentiable functions.Using (144), Eqs.

(143) can be represented in the form2 ag a  32g a2  g12 s a ( y 0 )  f abc g b s c ( y 0 ), (145)  i  2ii k i  i 1 i  ki 1 i3ikwhere  i  yi / r satisfy the relation12   22   32  1.(146)Taking into account this relation, the arguments  i of the functions g a can be expressedin terms of two independent arguments. As such arguments, it is convenient to choose thefollowing values  and  :g a ( y0 , 1 ,  2 ,  3 )  h a ( y 0 , ,  ),   1  1  1 ,   arctg 2 .

(147)ln 2  1  1  3 Then, as calculations show, Eqs. (145) acquire the form 2ha  2ha 1  th 2 s a ( y0 )  f abc h b s c ( y0 ) .22(148)h a  v a ( y0 , ,  )   ( y0 )s a ( y0 ) ln(ch )  d a ( y0 )(149)Let us putand choose (N+1) functions  ( y0 ) and d a ( y0 ) so that the following N equalities aresatisfied:s a ( y0 )   ( y0 )s a ( y0 )  f abc d b ( y0 )s c ( y0 )  0 .(150)Multiplying these equalities by s a and summing over the index a , we findN  s / s, s 2   ( s a ) 2 , s  ds / dy 0 .(151)a 1From Eqs.

(148)-(150) we obtain 2v a  2v a 1  th 2 f abc v b ( y0 ) s c ( y0 ), v a  v a ( y0 , ,  ).(152)22Consider these equations. We will seek their solutions in the form of the real part of thefollowing sum:Mv a ( y 0 , ,  )  ReVna ( y 0 , ) exp  n(  i )  ,n 0(153)23where Vna ( y0 , ) are some complex functions and n, M are nonnegative integers. Thensubstituting (153) into Eq.

(152), we come to the equations 2Vna 2nVna 1  th 2 f abcVnb s c ( y 0 ) . 2Let us represent the solution of Eqs. (154) in the formVna  Vna ( y0 , ),   12 (1  th ) .(154)(155)Then from Eqs. (154) we obtain (  1) 2Vna 2Vna (n  1  2 ) f abcVnb s c ( y 0 )  0 .(156)We will seek the solution of Eqs. (156) in the form of the following power series in  :Vna   aj ,n ( y 0 ) j ,(157)j 0where aj ,n ( y 0 ) are some complex functions.Then we find that the functions aj,n ( y 0 ) satisfy the recurrence relationsaj 1, nj ( j  1)aj ,n  f abcbj ,n s c( j  1)( j  1  n),j  0, 1, 2,...

,(158)Where the complex functions a0,n  a0,n ( y0 ) can be chosen arbitrarily.Using the recurrence relation (158), it is easy to verify that the consequence aj ,n ( y 0 ) isbounded for any y 0 . This shows the absolute convergence of series (157) when  1 (  ) .As a result of applying formulas (153), (155), (157) and (158) we find the functionsv a ( y0 , ,  ) which allow one to also determine the functions u a ( y0 , y1 , y 2 , y3 ), wherey0  x 0  r , yl  x l , l  1, 2, 3 .

After that, formulas (136) and (137) give the soughtexpressions for the potentials and strengths of non-Abelian wave solutions. They aredetermined by N arbitrary complex functions a0,n ( y 0 ) .When s a ( y0 )  0 , the non-Abelian wave solutions have longitudinal components. Thiscircumstance can be used to seek cosmic sources of the Yang-Mills fields.Let us now substitute series (157) into formula (153) and take into account expressions(147) and (155) for  ,  and  . Then we come to the formulaMv a  Re  aj ,n ( x 0  r )m 0 j 0where aj.n satisfy the recurrence relation (158).(r  x1 ) j ( x 2  ix 3 ) n,( 2r ) j ( r  x 1 ) n(159)243.

Investigation of nonlinear equations of relativistic movements of particlesunder the action of Yukawa’s and Coulomb’s potentials3.1. Classical equations of movements of relativistic particles under the action of ascalar and vectorial fieldsLet a relativistic uncharged particle with rest mass m0 move under the action of a scalarfield with potential  . We will consider it in the classical approximation and use theprinciple of least action to describe it. For this purpose, let us apply the following expressionfor the action S:t2S  m0 c  ( )ds, (0)  0 ,(160)t1wherexnx 0  ct , x1 , x 2 , x 3 are spatial coordinates,are space-time coordinates,ds 2  dxn dx n , t is time, c is the speed of light, ( ) is some function of the potential  andt1 , t 2 are moments of time corresponding to fixed values of the particle coordinates.

Action(160) is relativistic invariant and when   0 coincides with the well-known classicalexpression5.Taking the variation of (160) with respect to the trajectories x n (s) , from the principle ofleast action: S  0 , we find that in an inertial frame of reference ( )x ndx d  ( ) n   0 .ds ds (161)These equations can be represented in the considered inertial frame as( )d 2 xnds2 d dx n   ( ) ds ds x n 0,(162)where d / ds   / x n dx n / ds .Consider the nonrelativistic case.

Then Eqs. (162) gived 2 xk  k , k  1, 2, 3,   c 2 ln ( ) , dx k / dt  c .(163)2dtxOn the other hand, the classical nonrelativistic equations for an uncharged particlemoving under the action of the potential  should have the formd 2 xk k .(164)2dtxFrom (163) and (164) we find that the function  should be identified with the potential:5Landau L. D., Lifshitz Е.М. Theory of Fields. – Мoscow.: Nauka, 1967.25  c 2 ln ( )   .(165)Since (0)  1 , as indicated in (160), from (165) we find  exp  / c 2 .(166)Consider now the movement of a particle with charge q under the action of a scalar fieldwith potential  and a vectorial electromagnetic field with potentials An , n  0, 1, 2, 3 .

Inthis case, instead of action (160), we should have the actiont2tq 2S  m0 c  ( )ds   An dx n ,ctt1( )  exp( / c 2 ) .(167)1Taking the variation of action S with respect to the trajectories x n (s) and using theprinciple of least action, we come to the following equations in an inertial frame of reference: d 2 x n d dx n m0 exp( / c 2 ) c 22dsdsxndsdx m  qF nm 0,ds(168)where Fnm  Am / x n  An / x m is the tensor of strengths of the electromagnetic field.Let us multiply the left-hand side of (168) by dx n / ds and take the sum over n.

Then, ascan be readily verified, we obtain the identical zero. This implies that the first equation (n=0)in the system of the dynamic equation (168) is a consequence of the other three equations(n=1,2,3).3.2. Movement of a relativistic particle in Yukawa’s and Coulomb’s fieldsLet us turn to the movement of a relativistic particle with rest mass m0 and charge qunder the action of Yukawa’s potential  and Coulomb’s potential A0 generated by animmovable spherical source with its center at the zero-point of spatial coordinates.For these potentials we havefexp( r ),   0,rA0 h,rA1  A2  A3  0 .(169)Here f , h,   const and r is the distance from the zero-point of the spatial coordinates.Let a particle move in the plane x 3  0 .

We will choose the polar coordinates r and  :x1  r cos  , x 2  r sin  , x 3  0.(170)Then the left-hand side of the last equation in (168) (n=3) is identically equal to zero, as wellas its right-hand side, and when n=1,2, taking into account (169), Eqs. (168) acquire thefollowing form, as is shown by calculations:f exp( r )hq22 m0 exp( / c 2 ) r  r 2  2(1r)(rc)1  (r 2  r 2 2 ) / c 2 , (171)22cr r26f exp(  r )r 2r  r  2 (1  r )  0 ,rc(172)where r  dr / d ,   d / d and  is the intrinsic time of the moving particle:d  ds / c .As stated above, the first equation in (168) (n=0) is a consequence of the other threeequations (n=1,2,3) and hence of Eqs.

(171) and (172).From Eq. (172) we find the first integral f  (1  r )D  2 exp  2 exp(r)dr,2r c r rD  const .As calculations show, this formula can be represented in the formDf  2 exp   (r ) / c 2 ,  (r )   exp( r ) .rrConsider the function r  r ( ) and put 1.r(173)(174)(175)Then using the first integral (174), after a number of calculations, we obtain that Eq. (171) acquiresthe form    f(1   /  ) exp(   /   2 / c 2 ) 2Dhq 2exp( / c 2 ) 1  ( D 2 / c 2 ) exp( 2 / c 2 )(  2   2 )  0,D mp(176)where    ( ),    d / d ,    d 2 / d 2 and    f exp(  /  ) .Let us choose a dimensionless form of this equation by puttingUfhqf, a, b, g 2.22Dm0 Dc(177)Then from Eq. (176) we obtain the following equation for the function U  U ( ) :U   U  b exp(  gU exp( 1 / U ))  1  ( g / a)(U  2  U 2 ) exp( 2 gU exp( 1 / U )) (178) a(1  1 / U ) exp( 1 / U  2 gU exp( 1 / U )),3.3.

Numerical investigation of periodic orbits of relativistic particles in Yukawa’s andCoulomb’s fieldsFor Eq. (178) a series of numerical computations are performed by the Runge-Kuttamethod of the fourth order. They are carried out for various values of the parameters a, b, g27and U 0  U (0) . In them we put U (0)  0 . Then the angle   0 corresponds to an extremeof the function r ( ) . These computations show that for certain values of the parametersrelativistic particles can move in closed orbits in the fields of Yukawa’s and Coulomb’sforces.Below some of the obtained numerical results are presented.In Figs.

1-3 plots are given which show the dependence of the dimensionless radius of theorbit of a particle r  1 / U on the polar angle  in the following three cases:1) U 0  0.1 , a  0.015 , b  0.1 and g  0.9 .2)U 0  0.2 , a  0.05 , b  0.15 and g  0.1 .3)U 0  0.3 , a  0.015 , b  0.1 and g  0.9 .Fig. 1. Dependence of the radial coordinate of a particle on the polar angle in case 128Fig. 2.