Steady States and Travelling Wave Solutions of the Heisenberg and M − I Spin Systems

Abstract

We obtain general expressions for the equilibrium states and traveling wave solutions of the Heisenberg and Myrzakulov–I continuum spin systems, expressed as 1+1 and 1+2 PDEs respectively in the form ~St = ~S ×~Sxx,and~St=(~S×~Sy + u~S)x, ux = −(~S, ~Sx ×~Sy), ~S=(S1, S2, S3), S21 + S22 + S23 = 1. We reduce these equations to ODEs which can be solved analytically, since the original systems are known to be completely integrable by IST and, therefore, all their reductions are also expected to be integrable. Indeed, in some cases, the reduced equations are explicitly solved by trigonometric functions, while in others, we use the fact that they possess the Painlev´e property. We expect that our results will be useful in terms of their continuation and stability properties when one studies small non - integrable perturbations of the above integrable spin models.