Bäcklund transformations and local conservation laws for self-dual Yang-Mills fields with arbitrary gauge group

1988 ◽  
Vol 127 (3) ◽  
pp. 167-170 ◽  
Author(s):  
C.J. Papachristou ◽  
B.Kent Harrison
Keyword(s):  
Conservation Laws ◽  
Gauge Group ◽  
Bäcklund Transformations ◽  
Local Conservation ◽  
Yang Mills ◽  
Backlund Transformations ◽  
Arbitrary Gauge

Bäcklund transformations and local conservation laws for principal chiral fields

1980 ◽  
Vol 91 (3-4) ◽  
pp. 387-391 ◽  
Author(s):  
A.T. Ogielski ◽  
M.K. Prasad ◽  
A. Sinha ◽  
Ling-Lie Chau Wang
Keyword(s):  
Conservation Laws ◽  
Bäcklund Transformations ◽  
Local Conservation ◽  
Backlund Transformations

scholarly journals Bäcklund transformations and the Atiyah–Ward ansatz for non-commutative anti-self-dual Yang–Mills equations

2009 ◽  
Vol 465 (2108) ◽  
pp. 2613-2632 ◽  
Author(s):  
Claire R. Gilson ◽  
Masashi Hamanaka ◽  
Jonathan J. C. Nimmo
Keyword(s):  
Exact Solutions ◽  
Gauge Group ◽  
Bäcklund Transformations ◽  
Twistor Theory ◽  
Yang Mills ◽  
Seed Solution ◽  
Backlund Transformations

We present Bäcklund transformations for the non-commutative anti-self-dual Yang–Mills equation where the gauge group is G = G L (2), and use it to generate a series of exact solutions from a simple seed solution. The solutions generated by this approach are represented in terms of quasi-determinants. We also explain the origins of all the ingredients of the Bäcklund transformations within the framework of non-commutative twistor theory. In particular, we show that the generated solutions belong to a non-commutative version of the Atiyah–Ward ansatz.

scholarly journals BÄCKLUND TRANSFORMATIONS FOR NON-COMMUTATIVE ANTI-SELF-DUAL YANG–MILLS EQUATIONS

2009 ◽  
Vol 51 (A) ◽  
pp. 83-93 ◽  
Author(s):  
CLAIRE R. GILSON ◽  
MASASHI HAMANAKA ◽  
JONATHAN J. C. NIMMO
Keyword(s):  
Exact Solutions ◽  
Gauge Group ◽  
Bäcklund Transformations ◽  
Yang Mills ◽  
Seed Solution ◽  
Backlund Transformations

AbstractWe present Bäcklund transformations for the non-commutative anti-self-dual Yang–Mills equations where the gauge group is G = GL(2) and use it to generate a series of exact solutions from a simple seed solution. The solutions generated by this approach are represented in terms of quasi-determinants and belong to a non-commutative version of the Atiyah–Ward ansatz. In the commutative limit, our results coincide with those by Corrigan, Fairlie, Yates and Goddard.

A new hierarchy of Korteweg–de Vries equations

1976 ◽  
Vol 351 (1666) ◽  
pp. 407-422 ◽  
Keyword(s):  
Conservation Laws ◽  
Boussinesq Equation ◽  
Soliton Solutions ◽  
Boussinesq Equations ◽  
Bäcklund Transformations ◽  
Multiple Soliton Solutions ◽  
Oscillating Solutions ◽  
Backlund Transformations ◽  
De Vries ◽  
Rank 2

We have found new hierarchies of Korteweg–de Vries and Boussinesq equations which have multiple soliton solutions. In contrast to the stan­dard hierarchy of K. de V. equations found by Lax, these equations do not appear to fit the present inverse formalism or possess the various pro­perties associated with it such as Bäcklund transformations. The most interesting of the new K. de V. equations is ( u nx ≡ ∂ n u /∂ x n ) ( u 4 x + 30 uu 2 x + 60 u 3 ) x + u t = 0. We have proved that this equation has N -soliton solutions but we have been able to find only two soliton solutions for the rest of this hierarchy. The above equation has higher conservation laws of rank 3, 4, 6 and 7 but none of rank 2, 5 and 8 and hence it would seem that an unusual series of conservation laws exists with every third one missing. Apart from the Boussinesq equation itself, which has N -soliton solutions, ( u xx + 6 u 2 ) xx + u xx – u tt = 0 we have found only two-soliton solutions to the rest of this second class. The new equations have bounded oscillating solutions which do not occur for the K. de V. equation itself.

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