Analytical soliton solution for the Landau-Lifshitz equation of one dimensional magnonic crystal

Nonlinear localized magnetic excitations in one dimensional magnonic crystal is investigated under periodic magntic field. The governing Landau-Lifshitz equation is transformed into variable coefficient nonlinear Schrodinger equation(VCNLS) using sterographic projection. The VCNLS equation is in general nonintegrable, by using painleve analysis necessary conditions for the VCNLS equation to pass Weiss-Tabor-Carnevale (WTC) Painleve test are obtained. A sufficient integrability condition is obtained by further exploring a transformation, which can map the VCNLS equation into the well-known standard nonlinear Schrodinger equation. The transformation built a systematic connection between the solution of the standard nonlinear Schrodinger equation and VC-NLS equation. The results shows the excitation of magnetization in the form of soliton has spatialperiod exists on the background of spin Bloch waves. Such solution exisits only certain constrain conditions on the coefficient of the VCNLS equation are satisfied. The analytical results suggest a way to control the dynamics of magnetization in the form of solitons by an appropriate spatial modulation of the nonlinearity coefficient in the governing VCNLS equation which is determined by the ferromagnetic materials which forms the magnonic crystal.