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The nonlinear propagation of different types of DANW acoustic dust nonlinear waves has been investigated in a magnetized dusty plasma consisting of negatively charged dust particles, Maxwellian electrons, and ions. Application of the standard reductive perturbation theory is used to derive the corresponding three-dimensional nonlinear a complex Ginzburg–Landau (3D-CGLE) equation which governs the dynamics of the dust-acoustic wave packets. The stationary analytical solutions of the CGLE are numerically analysed where the effect of the physical parameters of the dusty plasma model on the wave’s propagation is taken into account. It has been found that there can be a relationship between the appearance of soliton waves and electromagnetic waves, as well as between shock-like waves and periodic travelling waves. Expression of the importance of these findings is the cornerstone of explaining the true relationship between the propagation of nonlinear waves in the physics of space, for example, the Earth’s magnetic field.

The crystal structure and magnetic state of the (1 − x)BiFeO$_3$-(x)BiMnO$_3$ solid solution has been analyzed by X-ray diffraction using lab-based and synchrotron radiation facilities, magnetization measurements, differential thermal analysis, and differential scanning calorimetry. Dopant concentration increases lead to the room-temperature structural transitions from the polar-active rhombohedral phase to the antipolar orthorhombic phase, and then to the monoclinic phase accompanied by the formation of two-phase regions consisting of the adjacent structural phases in the concentration ranges 0.25 < x$_1$ < 0.30 and 0.50 ≤ x$_2$ < 0.65, respectively. The accompanied changes in the magnetic structure refer to the magnetic transitions from the modulated antiferromagnetic structure to the non-colinear antiferromagnetic structure, and then to the orbitally ordered ferromagnetic structure. The compounds with a two-phase structural state at room temperature are characterized by irreversible temperature-driven structural transitions, which favor the stabilization of high-temperature structural phases. The magnetic structure of the compounds also exhibits an irreversible temperature-induced transition, resulting in an increase of the contribution from the magnetic phase associated with the high-temperature structural phase. The relationship between the structural parameters and the magnetic state of the compounds with a metastable structure is studied and discussed depending on the chemical composition and heating prehistory.