Symmetries, solitons and integrable models

Abstract

We study symmetries and soliton solutions of some integrable models. We investigate symmetries of a generalized Heisenberg magnet model and derive its local and nonlocal conserved quantities and show that the local conserved quantities Poisson commute with themselves and with the nonlocal conserved quantities. We apply the dressing method of Zakharov and Shabat on the generalized coupled dispersionless integrable system and compute its multi-soliton solutions. The dressing method has been related with the standard Darboux transformation. We also related the dressed solutions with quasideterminant multi-soliton solutions. We also present a noncommutative generalization of the coupled dispersionless system and apply Darboux transformation in order to compute explicit multisoliton solutions in terms of quasideterminants. In a similar pattern a Darboux transformation of a noncommutative sine-Gordon equation has been investigated and explicit multi-kink solutions have been computed. All these noncommutative generalizations have been compared with the commutative integrable systems. The role of the quasideterminand Darboux matrix for the noncommutative systems has been explored by explicitly computing the noncommutative soliton solutions of the noncommutative models.