Modulation instability is a crucial phenomenon in the study of nonlinear dynamics, where an unstable system results in the destruction of its original states, accompanied by the rapid growth of small perturbation instabilities. The Bose-Einstein condensate serves as an ideal platform for exploring modulation instability due to its precise experimental control over the system's nonlinear dynamics. Therefore, studying modulation instabilities holds profound significance in comprehending the nature of Bose-Einstein condensate systems. In this paper, we reveal that spin-orbit coupling can always introduce modulation instability into a kind of specific state. We call it spin-orbit-coupling-induced modulation instability. The states are specific as they are zero-quasimomentum states. We find that there exist four different zero-quasimomentum states, and we classify them as no-current-carrying states and current-carrying states according to whether the states carry current or not. In literature, modulation instability of the no-current-carrying states has been investigated. The current-carrying states are unique due to their current originating from spin-orbit coupling, and their existence is unstable due to nonlinearity. We find that all these zero-quasimomentum states are modulationally unstable in all parameter regimes. The consequence of such modulation instability is the formation of complex wave structures.

The properties of modulation instability and the corresponding nonlinear dynamic images are primarily investigated using Bogoliubov de Gennes (BdG) instability analysis and the split step Fourier method. BdG instability analysis is a widely employed technique for analyzing instability in the study of superfluidity and Bose-Einstein condensates. It primarily examines the system's stability and its response to perturbations by solving nonlinear eigenvalue equations. By diagonalizing the BdG Hamiltonian matrix, the eigenvalues can be obtained. The eigenvalues of the matrix may be complex due to the non-Hermitian nature of the BdG Hamiltonian. If one or more complex numbers exist in the eigenvalues, the state becomes unstable. Consequently, any imposed disturbance experiences exponential growth, leading to the instability of the state. In addition, the split step Fourier method is commonly used for handling time evolution. The underlying principle of this method is to separate the terms of the system Hamiltonian and process them individually. The key step involves employing distinct treatments for the nonlinear and linear terms of the equation to be solved.

Initially, we investigate the case of g>g₁₂ and observe that the system exhibits a four-band modulation instability image in Fig. 1. Among these bands, the two branches positioned near the lower quasi-momentum region are referred to as the primary modulation instability band, while the two branches near the higher quasi-momentum region are known as the secondary modulation instability band. Notably, it is determined that identical chemical potentials of the two current-carrying states yield the same modulation instability image. Furthermore, we perform calculations to ascertain the nonlinear dynamic images (Figs. 2 and 3). The investigation reveals that the density evolution of the two components follows similar ways, exhibiting trends of movement in both positive and negative directions along the x-axis. As time progresses, both components undergo chaotic oscillations. In the quasi-momentum space, distinct motion trends and reversal symmetry are observed between the two components. After a certain period of evolution, significant separation occurs. This phenomenon arises from the modulation instability being predominantly influenced by different modulation instability bands at various stages. Initially, the primary modulation instability band dominates, while in later stages, the secondary modulation instability band takes control. Ultimately, the system tends to approach the quasi-momentum space of the secondary modulation instability band, leading to chaotic propagation. Simultaneously, we also examine the scenario where g<g₁₂ and observe that the system's modulation instability image consists of only two bands (Fig. 4): the primary modulation instability band. This disappearance of the secondary modulation instability band occurs as the repulsive interaction between the components intensifies, causing the two unstable branches to merge. Following a nonlinear dynamic analysis (Figs. 5 and 6), we observe that the motion trends become less pronounced due to the absence of the secondary modulation instability band. Nevertheless, in this case, the two components still exhibit distinct motion patterns and maintain reverse symmetry. The reason behind this phenomenon remains consistent with the previous situation. However, since there are only two branches of modulation instability, the system consistently resides near the quasi-momentum space of the main modulation instability band once the wave function enters chaotic oscillation.

We delve into the examination of modulation instability and its consequential dynamic patterns in one-dimensional two-component Bose-Einstein condensates with spin-orbit coupling. The study reveals the existence of four distinct zero momentum states within the system, where two of them carry currents while the remaining two do not under specific conditions. It should be noted that the generation of these four states is not solely determined by spin-orbit coupling; however, the presence of spin-orbit coupling does impact the modulation instability of the system. Previous research predominantly focuses on the zero quasi-momentum state without current carrying, neglecting the investigation of the zero quasi-momentum state with current carrying. We specifically explore the modulation instability of current-carrying zero momentum states. The findings indicate that in the presence of Rabi coupling, when the intra-component interaction surpasses the inter-component interaction, the modulation instability image manifests four branches, consisting of two main modulation instability bands and two secondary modulation instability bands. Conversely, when the intra-component interaction is lower than the inter-component interaction, the modulation instability image presents only two branches. We also establish a correlation between modulation instability and the nonlinear dynamic evolution of the system. Additionally, the presence of modulation instability can trigger the emergence of intricate patterns.