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First, linear stability analysis examines modulation instability and identifies modulation-stable and unstable parameter regimes across different parameter planes. Subsequently, we establish Lax pairs and (n,N\u2212n)-fold generalized Darboux transformation to construct exact analytical solutions for diverse breathers and rogue waves featuring periodic phase distributions and periodic backgrounds. Based on these solutions, rigorous analysis reveals that SOC induces spatiotemporal periodic modulation on the phase and spin-density distributions of breathers and rogue waves, while Raman coupling generates periodic modulation in their backgrounds. By adjusting Raman coupling strength and initial plane-wave amplitudes and wavenumbers, backgrounds exhibiting double-periodic, single-periodic, or non-periodic modulation can be excited. Additionally, we discover higher-order breathers with curved trajectories and dual-rogue-wave structures. Nonlinear interactions among breathers and rogue waves of distinct structures and orders are systematically investigated. Finally, the stability of the analytical solutions for breathers was verified through numerical simulations. This study may facilitate a deeper understanding of periodic modulation mechanisms for localized waves under SOC and Raman coupling.<\/jats:p>","DOI":"10.1063\/5.0295201","type":"journal-article","created":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T13:12:27Z","timestamp":1768396347000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":10,"title":["Periodic modulation on phase and backgrounds for breathers and rogue waves in two-component Bose\u2013Einstein condensates"],"prefix":"10.1063","volume":"36","author":[{"given":"Fei-Yan","family":"Liu","sequence":"first","affiliation":[{"name":"Research Group of Nonlinear Optical Science and Quantum Technology, School of Microelectronics, Wuhan Textile University 1 , Wuhan 430200,","place":["China"]},{"name":"Research Center of Nonlinear Science, School of Mathematics and Statistics, Wuhan Textile University 2 , Wuhan 430200,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5334-7188","authenticated-orcid":false,"given":"Qin","family":"Zhou","sequence":"additional","affiliation":[{"name":"Research Group of Nonlinear Optical Science and Quantum Technology, School of Microelectronics, Wuhan Textile University 1 , Wuhan 430200,","place":["China"]},{"name":"Research Center of Nonlinear Science, School of Mathematics and Statistics, Wuhan Textile University 2 , Wuhan 430200,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,1,14]]},"reference":[{"issue":"8","key":"2026011408085835800_c1","doi-asserted-by":"publisher","first-page":"1402","DOI":"10.1103\/PhysRevLett.86.1402","article-title":"Bose-Einstein condensates in standing waves: The cubic nonlinear Schr\u00f6dinger equation with a periodic potential","volume":"86","year":"2001","journal-title":"Phys. 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