Summary_ (Исследование некоторых типов дифференциальных уравнений с сильной нелинейностью), страница 4

(96)-(97) in the case of many-particle sources for which the quasi-classicalapproximation can be applied and the field sources have the formJ 1,  J  , J 2,  J 3,  0,(98)where J  is the classical 4-vector of the current density.Then when the potentials A2,  A3,  0 , the Yang-Mills equations coincide with theMaxwell equations for the potentials A1, . That is why they can be regarded as a nonlineargeneralization of the Maxwell equations.Consider nontrivial solutions to the Yang-Mills equations (96)-(97) with the sources (98)in the stationary spherically symmetric case:J 1,0  c (r ), J 1,l  0, l  1, 2,3,J 2,  J 3,  0, r  (( x1 )2  ( x 2 )2  ( x3 )2 )1/ 2 ,(99)where  (r ) is the source charge density, x1, x 2 , x3 are Cartesian coordinates with the zeropoint at the center of the source and r is the distance between a point of the source and itscenter.In this case, solutions of these equations can be sought in the formAk ,0  0, A1,l  xl [u (r ) x0  u0 (r )], k , l  1, 2,3,A2,l  xl [v(r ) x0  v0 (r )], A3,l  xl [w(r ) x 0  w0 (r )], x 0  ct ,where u, u0 , v, v0 , w, w0 are some functions of r and t is time.Then from formula (97) for the field strengths, we find(100)17F 1,0l  xl u (r ), F 2,0l  xl v(r ), F 3,0l  xl w(r ),F k ,ml  0, k , m, l  1, 2,3.(101)Formulas (101) imply that the strengths of this Yang-Mills field are stationary and at thesame time, nonstationary potentials of form (100) correspond to them.Substituting expressions (100) and (101) into the Yang-Mills equations (96), we obtainthe following system of equations:ru  3u  gr 2 (wv0  vw0 )  4 ,(102)rv  3v  gr 2 (uw0  wu0 )  0, rw  3w  gr 2 (vu0  uv0 )  0.(103)Expressing w0 and v0 in Eqs.

(103) and substituting them into Eq. (102), we come to theequationr (u 2  v2  w2 )  6(u 2  v2  w2 )  8 u,(104)which is the only differential relation to be satisfied by the three functionsu(r ), v(r ) and w(r ) .The functions satisfying this equation can be represented in the following form:u   R (q ) cos  (q ) / r 3 ,v   R(q)sin  (q) cos  (q) / r 3 , w   R(q)sin  (q)sin (q) / r 3 ,qr00R(q)   cos  (q)dq, q  4  r 2 (r )dr ,(105)(106)where q  q(r ) presents the charge of the spherical region bounded by the radius r and (q) and  (q) are arbitrary differentiable functions.In order to choose a unique solution of the studied equations, let us consider thefollowing components, besides the components of the source 4-vectors of current densitiesJ k , :I k ,  J k ,  ( gc / 4 ) klm F l , Am ,(107)which, as follows from (96), satisfy the differential equation of conservation  I k ,  0 andthat is why can be identified with components of the full 4-vectors of current densities.

Usingthem, we can add the following relativistic invariant condition to the Yang-Mills equations:I k , I k ,  J k , J k , ,(108)expressing the conservation of the intrinsic energy in a source.The application of condition (108) and also the condition of equivalence of the axes withk=2 and k=3 in the gauge space allows one to find concrete expressions for the functions (q) and  (q) which acquire the form (q)  q / K0 , (q)   / 4, K0  const.3Ryder L. Quantum Field Theory.

– Мoscow: Mir, 1987.(109)18Then for the nonzero field strength components F 1,l 0 we findF 1,l 0  K sin(q / K ) x l / r 3 , K  K 0 / 2, l  1, 2, 3,(110)where q  q(r ) is determined by the formula given in (106).2.2. A solution to the Yang-Mills equations for nonstationary sphericallysymmetric sourcesIn the dissertation an investigation is carried out for the Yang-Mills equations (96)-(97)with the sources of the form(4 / c) J 1,0  j 0 (t , r ), (4 / c) J 1,l  x l j (t , r ), l  1,2,3,J 2,  J 3,  0, t  x 0 / c, r 2  ( x1 ) 2  ( x 2 ) 2  ( x 3 ) 2 ,(111)where t is time and r is the distance from the source center.The field potentials are sought in the formA1,0   0 (t , r ), A 2,0   0 (t , r ), A3,0   0 (t , r ),A1,l  x l  (t , r ), A 2,l  x l  (t , r ), A3,l  x l  (t , r ).(112)Then from (97) we obtain the following expressions for the field strengths:F k ,ml  0, k , m, l  1,2,3,F 1,0l  x l u (t , r ), F 2,0l  x l v(t , r ), F 3,0l  x l w(t , r ),(113)whereu  (1 / c) / t  (1 / r ) 0 / r  g ( 0    0 ),v  (1 / c) / t  (1 / r ) 0 / r  g ( 0   0 ),(114)w  (1 / c) / t  (1 / r ) 0 / r  g (  0   0  ).The Yang-Mills equation (96) results in the following system of equations:ru / r  3u  gr 2 ( w  v )   j 0 ,rv / r  3v  gr 2 (u  w )  0,(115)rw / r  3w  gr 2 (v  u )  0,(1 / c)u / t  g (v 0  w 0 )  j ,(1 / c)v / t  g ( w 0  u 0 )  0,(116)(1 / c)w / t  g (u  v )  0.00As is known, the relations D J k ,  0, where D is the Yang-Mills covariant derivative,are a consequence of the Yang-Mills equations.

This gives the following equations:(1 / c)j 0 / t  rj / r  3 j  0,j 0  0  r 2 j  0, j 0 0  r 2 j  0.(117)The considered field sources are invariant under the gauge rotation about the first axis.That is why the following condition can be fulfilled by the choice of a gauge:19j 0 0  r 2 j  0 .(118)Let us multiply the equations in (115) by j and add them to the corresponding equationsin (116) multiplied by j 0 . Then using relations (117) and (118) for the potentials, we cometo the following equations:(1 / c) j 0 u / t  j (ru / r  3u )  0,(1 / c) j 0 v / t  j (rv / r  3v)  0,(119)(1 / c) j 0 w / t  j (rw / r  3w)  0.Multiplying the three equations in (115) by u, v, w , respectively, and then adding them,we findu(ru / r  3u)  v(rv / r  3v)  w(rw / r  3w)   j 0 u .(120)The application of Eqs.

(107)-(108) gives the relation(ru / r  3u ) 2  (rv / r  3v) 2  (rw / r  3w) 2  (r / c) 2 [(u / t ) 2  (v / t ) 2  (w / t ) 2 ]  ( j 0 ) 2  r 2 j 2 .(121)An analysis of Eqs. (119) shows that they have the exact solution of the formru  P(q) / r 3 , v  Q(q) / r 3 , w  S (q) / r 3 , q   r 2 j 0 (t , r )dr ,0(122)where P, Q, S are differentiable functions of the argument q which is the charge of thespherical region of the radius r at the moment t .The substitution of (122) into Eqs. (120) and (121) givesPdP / dq  QdQ / dq  SdS / dq  P .(dP / dq) 2  (dQ / dq) 2  (dS / dq) 2  1 .(123)(124)For Eqs. (123) and (124) we find the following solution taking into account theequivalence of the second and third axes in the gauge space:P(q)  K sin(q / K ), Q(q)  S (q)  2 1/ 2 K[1  cos(q / K )] ,(125)where K is the constant introduced in the previous section.The obtained formulas allow one to generalize the stationary solution with sphericalsymmetry found above for the nonstationary spherically symmetric case.2.3.

Axially symmetric wave solutions to the Yang-Mills equationsLet us turn to axially symmetric wave solutions of the Yang-Mills equations (96)(97) in regions outside field sources where J k ,  0 . They can be sought in the formA k ,0   k ,0 ,A k ,1  ( k ,1 x   k , 2 y ) /  ,A k , 2  ( k ,1 y   k , 2 x) /  , A k ,3   k ,3 , k ,   k , ( ,  , z),   x 0 ,   ( x 2  y 2 )1/2 ,x  x1 , y  x 2 , z  x 3 .(126)20Then for the field strengths we findF k ,01  ( fF k ,12  fk ,1k ,4x fk ,2y ) /  , F k ,02  ( f, F k ,13  ( ffk ,qk ,5 fx fk ,qk ,6k ,1y fk ,2x) /  , F k ,03  fy ) /  , F k , 23  ( fk ,5y fk ,6k ,3,x) /  ,(127)( ,  , z ), q  1, 2, ..., 6,wherefk ,1  k ,1   k ,0  g klm l ,0 m,1 , ffk ,3  k ,3   zk ,0  g klm l ,0 m,3 , ffk ,5  zk ,1   k ,3  g klm l ,1 m,3 , fk ,2k ,4k ,6  k , 2  g klm l .0 m, 2 ,  k , 2   k , 2 /   g klm l ,1 m, 2 ,  zk , 2  g klm l , 2 m,3 ,(128)k ,   k , /  ,  k ,   k , /  ,  zk ,   k , / z.In the considered case J k ,  0 , the Yang-Mills equation (96) givesf k ,1  fk ,1/   f zk ,3  g klm ( f l ,1 m,1  f l , 2 m, 2  f l ,3 m,3 )  0,fk ,1  f zk ,5  g klm ( f l ,1 m,0  f l , 4 m, 2  f l ,5 m,3 )  0,fk , 2  f k , 4  f zk ,6  g klm ( f l , 2 m,0  f l , 4 m,1  f l ,6 m,3 )  0,fk ,3  f k ,5  fk ,5(129)/   g klm ( f l ,5 m,1  f l ,6 m, 2  f l ,3 m,0 ).The potential components  k , satisfying the system of equations (129) are sought in theform 1,0  0,  2,0  P( ,  ),  3,0  Q( ,  ),     z, 1, 2   (  ) / g ,  2,2   3,2  0,  k ,1  0,  k ,3   k ,0 ,(130)where P( ,  ), Q( ,  ) and  (  ) are some differentiable functions.Then from (128) we obtainf 1,1  0,f 2,1  P ,f 3,1  Q ,f 1, 2  0,f 2, 2  Q,f k ,3  0, f 1, 4  (    /  ) / g , f 2, 4  f 3, 4  0, f k ,5   ff 3, 2  P,k ,1, f k ,6   f k , 2 .(131)Substituting (130) and (131) into Eqs.

(129), we findP  P /    2 P  0, Q  Q /    2 Q  0, (    /  )  0.(132)Equations (132) have the following solutions that tend to zero as    :  b /  , P  G(  z ) /  b , Q  H (  z) /  b ,   x 0 ,(133)where G and H are arbitrary differentiable functions and b is an arbitrary nonzero constant.As a result, we obtain a class of axially symmetric solutions to the Yang-Mills equationsdescribing by formulas (130), (131) and (133).2.4. Non-Abelian expanding waves21In the dissertation, non-Abelian expanding waves radiated from cosmic sources of theYang-Mills fields in the case of a N-parametric gauge group are studied. Outside theirsources, the Yang-Mills fields are described by the equations4  F a,  f abc Ab F c,  0 ,(134)F a,    Aa,   Aa,  f abc Ab, Ac, ,(135)Aa, and F a, are potentials and strengths of awhere  ,  0,1,2,3, a, b, c  1,2,..., N ,Yang-Mills field, respectively, f abc are the structure constants of an N-parametric gaugegroup and     / x  , where x  are orthogonal space-time coordinates of the MInkowskigeometry.Wave solutions to Eqs.

(134)-(135) are thought in the formA a ,0  u a ( y 0 , y1 , y 2 , y3 ),A a ,l  ( x l / r ) A a , 0 ,y 0  x 0  r , yl  x l ,l  1, 2, 3, a  1,2,..., N , r  ( x1 ) 2  ( x 2 ) 2  ( x 3 ) 2 ,(136)where u a are some functions of the wave phase y0  x 0  r and spatial coordinates yl  x l .Further the most interesting case for physical applications is studied in which we have agauge group with compact semi-simple Lie algebra corresponding to completelyantisymmetric structure constants f abc . Then after the substitution of expressions (136) intoformula (135) for the strengths of a Yang-Mills field, we findF a,0n  u a / yn , F a,in  (1 / r )( yi u a / yn  y n u a / yi ), i, n  1,2,3. (137)Let us substitute formulas (136) and (137) into Eq.

(134) and introduce the notations3u a 2u aap   yi, q .2yii 1i 1 y i3a(138)Then after calculations, we come to the following equations:1  p aq   f abc u b p c , r r  y 0ayny12  y 22  y32 ,(139)p a p a 0, n  1,2,3.r 2 y n(140)p a  s a ( y0 ) / r ,(141)Equations (140) have the solutionwhere s a are arbitrary differentiable functions of the argument y 0 .From (138), (139) and (141) we come to the following equations:3 yii 14u a s a ( y 0 ),yir(142)Slavnov А.А., Faddeev L.D.