Linear and Nonlinear Dispersion Relations for Torsional Alfvén Waves in Solar Magnetic Flux Tubes

We provide a thorough analytical derivation of both linear and nonlinear dispersion laws for torsional Alfv\'{e}n waves propagating in solar atmosphere zero-beta magnetic flux tubes. We use perturbation theory to methodically deduce the following from the ideal magnetohydrodynamic equations under the zero-beta approximation: (1) the linear dispersion relation $\omega = k v_A$, (2) the nonlinear dispersion relation $\omega = k v_A + (k/v_A)|a|^2$, (3) group and phase velocities at both orders, and (4) amplitude-dependent frequency shifts. According to our multiple-scale research, the connection between second harmonic production and mean axial flows is the source of the nonlinearity. For specific wavenumbers, modulational instability results from the resultant nonlinear Schrödinger equation's focused cubic nonlinearity. With numerical estimates pertinent to solar spicules, we give precise formulas for frequency changes as functions of wave amplitude and background plasma characteristics. Detailed graphical analysis that demonstrate frequency changes of up to 25\% in recorded wave amplitudes, indicating considerable nonlinear effects in the solar atmosphere, corroborate these findings. The findings give a theoretical foundation for interpreting high-resolution observations of torsional movements in magnetic bright spots and shed fresh light on wave-driven mass transport processes in the solar chromosphere.