Analytical criteria for magnetization reversal in $\varphi_0$ Josephson junction

The $\varphi_0$ Josephson junctions formed by ordinary superconductors and a magnetic non-centrosymmetric interlayer are studied. We derive an analytical solution for the magnetization dynamics induced by an arbitrary current pulse and formulate the criteria for magnetization reversal. Using the obtained results, the form and duration of the current pulse are optimized. The agreement between analytical and numerical investigations is reached in the case of a large product of the ratio Josephson to magnetic energy, strength of spin-orbit interaction and a minimal value of the current pulse. The obtained results allow one to predict magnetization reversal at the chosen system parameters.