Soliton solution for the Landau-Lifshitz equation of a one-dimensional bicomponent magnonic crystal

We investigate nonlinear localized magnetic excitations in a one-dimensional bicomponent magnonic crystal under a periodic magnetic field of spatially varying strength. The governing Landau-Lifshitz equation is transformed into a variable coefficient nonlinear Schrödinger (VCNLS) equation using stereographic projection. In general, the VCNLS equation is nonintegrable and by using Painlevé analysis, we obtain necessary conditions for the VCNLS equation to pass the Weiss-Tabor-Carnevale Painlevé test. A sufficient integrability condition is obtained by further exploring a transformation, which can map the VCNLS equation into the well-known standard nonlinear Schrödinger equation. The transformation builds a systematic connection between the solution of the standard nonlinear Schrödinger equation and VCNLS equation. The results show that the excitation of magnetization in the form of a soliton exists on the oscillatory background with a structure similar to the form of spin Bloch waves. Such a solution exists only when certain conditions on the coefficient of the VCNLS equation are satisfied. To corroborate the analytical results, we performed the numerical simulation by solving the governing VCNLS equation with integrability conditions using the split step Fourier method and the result agrees well with analytical results, and it suggests 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 depends on the ferromagnetic materials which form the bicomponent magnonic crystal.