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We address the linear-mode analysis performed near an equilibrium configuration in the fluid rest frame with a dynamical magnetic field perturbed on a constant configuration. We develop a simple and general algorithm for an analytic solution search that works on an order-by-order basis in the derivative expansion. This method can be applied to general sets of hydrodynamic equations. Applying our method to the first-order relativistic magnetohydrodynamics, we demonstrate that the method finds a complete set of solutions. We obtain two sets of analytic solutions for the four and two coupled modes with seven dissipative transport coefficients. The former set has been missing in the literature for a long time due to the difficulties originating from coupled degrees of freedom and strong anisotropy provided by a magnetic field. The newly developed method resolves these difficulties. We also find that the small-momentum expansions of the solutions break down when the momentum direction is nearly perpendicular to an equilibrium magnetic field due to the presence of another small quantity, that is, a trigonometric function representing the anisotropy. We elaborate on the angle dependence of the solutions and provide alternative series representations that work near the right angle. This identifies the origin of a discrepancy found in recent works. Finally, we discuss the issues of causality and stability based on our analytic solutions and recent developments in the literature.
Fang et al. (Thu,) studied this question.