On breathers in affine Toda theories
Oscillating solitonic solutions, the breathers, of affine Toda theory are studied. These breather solutions are constructed from two solitons of the same mass with velocity opposite of each other; by analytically continuing its velocity or rapidity to a complex value, the resulting solution becomes a periodic solution. Generally, the parameters in the soliton solutions are restricted to a certain range of definition. In particular, it is shown for a(^(1))(_n) and d(^(1))(_4) cases, these restrictions can be calculated explicitly. To some cases of a(^(a))(_n) theories, one can show that there are sine-Gordon embedded solitons which give rise to a sine-Gordon breather. Furthermore, these breather solutions carry topological charges. These topological charges are calculated and it is found that they are exactly the same as the topological charges of some single soliton cases. Moreover, for the non-zero topological charges, one can show they belong to the irreducible fundamental representation component of the tensor product of two fundamental representations associated with the constituent solitons. This Clebsch-Gordan decomposition property is in agreement with the fusing rule of soliton which in turn is similar to the fusing rule of the fundamental Toda particles. One can also make a conjecture that the zero topological charge is always carried by a breather whose constituent solitons are associated with either conjugate or self-conjugate fundamental representations. Although it is not possible to know the individual topological charge carried by the constituent solitons in a breather, nevertheless using the crossing symmetry similar to that of the crossing symmetry of the S'-matrix, one can perform a superficial calculation to determine the constituent soliton's topological charges. Attempts to understand the exact scattering matrices of the sine-Gordon solitons and breathers from a root space point of . view is also discussed. This study tries to mimic the exact S-matrix construction of the real coupling regime affine Toda theory from the root space by Dorey. In this study, one replaces the ordinary Coxeter element, which plays an important role in the real coupling regime, with other transformations to incorporate the infinite product nature of the sine-Gordon soliton scattering matrix. However, the desired consistent construction seems to elude the author in this study.