The estimates of the ill-posedness index of the (deformed-) continuous Heisenberg spin equation

Although the exact treatment of the continuous Heisenberg spin is already known, the exact solution of the deformed system is not found in the literature. In this paper, some traveling wave solutions of the deformed (indicated by the coefficient α) continuous Heisenberg spin equation are obtained. Based on the exact solution being constructed here, the ill-posedness results are proved by the estimation of the Fourier integral in Ḣs. If α = 0, the range of the mild ill-posedness index s is (1,32), which is consistent with the result of the formal analysis of the solution. Moreover, the upper bound of the strong ill-posedness index s jumps at α = 0: If α = 0, the upper bound is 2; if α = 0, then the upper bound jumps to 32.