Riemann-Hilbert problem for the modified Landau-Lifshitz equation with nonzero boundary conditions

We study a matrix Riemann–Hilbert (RH) problem for the modified Landau–Lifshitz (mLL) equation with nonzero boundary conditions at infinity. In contrast to the case of zero boundary conditions, multivalued functions arise during direct scattering. To formulate the RH problem, we introduce an affine transformation converting the Riemann surface into the complex plane. In the direct scattering problem, we study the analyticity, symmetries, and asymptotic behavior of Jost functions and the scattering matrix in detail. In addition, we find the discrete spectrum, residue conditions, trace formulas, and theta conditions in two cases: with simple poles and with second-order poles present in the spectrum. We solve the inverse problems using the RH problem formulated in terms of Jost functions and scattering coefficients. For further studying the structure of the soliton waves, we consider the dynamical behavior of soliton solutions for the mLL equation with reflectionless potentials. We graphically analyze some remarkable characteristics of these soliton solutions. Based on the analytic solutions, we discuss the influence of each parameter on the dynamics of the soliton waves and breather waves and propose a method for controlling such nonlinear phenomena.