We study systematically a matrix Riemann-Hilbert problem for the modified Landau-Lifshitz (mLL) equation with nonzero boundary conditions at infinity. Unlike the zero boundary conditions case, there occur double-valued functions during the process of the direct scattering. In order to establish the Riemann-Hilbert (RH) problem, it is necessary to make appropriate modification, that is, to introduce an affine transformation that can convert the Riemann surface into a complex plane. In the direct scattering problem, the analyticity, symmetries, asymptotic behaviors of Jost functions and scattering matrix are presented in detail. Furthermore, the discrete spectrum, residual conditions, trace foumulae and theta conditions are established with simple and double poles. The inverse problems are solved via a matrix RH problem formulated by Jost function and scattering coefficients. Finally, the dynamic behavior of some typical soliton solutions of the mLL equation with reflection-less potentials are given to further study the structure of the soliton waves. In addition, some remarkable characteristics of these soliton solutions are analyzed graphically. According to analytic solutions, the influences of each parameters on dynamics of the soliton waves and breather waves are discussed, and the method of how to control such nonlinear phenomena are suggested.

中文翻译：

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具有非零边界条件的修正 Landau-Lifshitz 方程的 Riemann-Hilbert 问题

我们系统地研究了具有无穷远非零边界条件的修正 Landau-Lifshitz (mLL) 方程的矩阵 Riemann-Hilbert 问题。与零边界条件的情况不同，在直接散射过程中会出现双值函数。为了建立黎曼-希尔伯特（RH）问题，需要进行适当的修改，即引入一种可以将黎曼曲面转化为复平面的仿射变换。在直接散射问题中，详细介绍了Jost函数和散射矩阵的解析性、对称性、渐近行为。此外，离散谱、残差条件、痕量分子式和θ条件是用单极和双极建立的。逆问题通过由约斯特函数和散射系数公式化的矩阵 RH 问题解决。最后，给出了具有无反射势的mLL方程的一些典型孤子解的动力学行为，以进一步研究孤子波的结构。此外，还对这些孤子解的一些显着特征进行了图形分析。根据解析解，讨论了各参数对孤子波和呼吸波动力学的影响，并提出了控制这种非线性现象的方法。以图形方式分析了这些孤子解的一些显着特征。根据解析解，讨论了各参数对孤子波和呼吸波动力学的影响，并提出了控制这种非线性现象的方法。以图形方式分析了这些孤子解决方案的一些显着特征。根据解析解，讨论了各参数对孤子波和呼吸波动力学的影响，并提出了控制这种非线性现象的方法。