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Bose-Einstein condensation (BEC) is a phenomenon that condensation into a single quantum state occurs for identical bosons when the inter-particle separation is less than the thermal de Broglie wavelength of the particles. In this work report, the modulational instability and intrinsic localized mode of spin wave in atomic chain of spinor BEC are investigated. Compared with the solid-state spin systems in which the Heisenberg-like short-range exchange interaction plays the dominate role, the atomic spin chain where the long-range dipole-dipole interaction between different spin sites is dominant. As a result, an analogy of atomic chains of spinor BECs can be made to the spin chains in the solid magnetic systems. However a key difference of the atomic spin chain from solid-state one is that we are facing a completely new spin system in atomic physics where the long-range interactions play a dominant role in the spin dynamics.