4 and γ in Fig. 7 of Ref. [15], where the condensate life time lasts several hundred milliseconds and c$_{2}$ changes sign. On the other hand, the spin domain emerges in a time scale typically shorter than 100 ms [33]. Therefore the suppression of the domain formation in $^{87}$Rb conden- sate is experimentally feasible. In summary, we propose to localize the spin mixing dynamics in a spin-1 condensate by temporally modulat- ing the spin exchange interaction. For condensed atoms, the modulation can be facilitated with the technique of optical Feshbach resonance [13, 15]. We demonstrate the suppression of the intrinsic instability thus the inhibition of spontaneous spin domain formation in a ferromagnet- ically interacting spin-1 Bose condensate, such as $^{87}$Rb condensate in the F = 1 manifold. In addition, the ef- fective freezing of spin mixing dynamics due to spin ex- change interaction provides a cleaner approach to inves- tigate magnetic spin dipolar interaction effect in a $^{87}$Rb Bose condensate [11, 12]. W. Z. acknowledges support by the 973 Program Grant No. 2009CB929300, the National Natural Science Foun- dation of China Grant No. 10904017, and the Program for New Century Excellent Talents in University. with ω$_{x}$ = ω$_{y}$ = (2 π )240 Hz and ω$_{z}$ = (2 π )24 Hz. The modulation function is c$_{2}$ ( t ) = d cos(Ω t ) with d = c$_{2}$ , and Ω = ω$_{z}$ which is about 3 times larger than the free spin evolution frequency 2 π/T$_{0}$ . Two cases will be con- sidered: (1) The modulation is applied immediately ( ρ$_{0}$ oscillates around ρ$_{u}$ ); (2) The modulation is turned on at t = 6 (1 /ω$_{z}$ ) ( ρ$_{0}$ oscillates around ρ$_{d}$ ). The results from numerical simulations are shown in Fig. 5. With modulation, the spin dynamics is clearly localized as ρ$_{+}$ (same for ρ$_{−}$ ) oscillates with a smaller amplitude around its initial value, in contrast to the large amplitude unmodulated result [panel (a)]. In addition, the unmodulated dynamics shows domain structures af- ter t ≈ 300, due to the intrinsic dynamical instability. While for both modulated cases, no spin domains are observed. Thus temporal modulation of spin exchange interaction does suppress the intrinsic dynamical insta- bility and prohibit spontaneous domain formation. In our extensive numerical simulations, we find that when addi- tional white noises are added, spin domains are found to arise quicker for the unmodulated case; yet for the two cases with modulations, almost the same behaviors are observed as if no noise were added. Although the life time of the condensate at opti- cal Feshbach resonance is reduced dramatically in $^{87}$Rb gases [15], we notice that there exists at least one magic window of relative frequency, e.g., between resonance β FIG. 5: (Color online) (a) Localization of the spin dynam- ics for a trapped spin-1 condensate in 3 cases: (i) mod- ulation starts at t = 0 (red dashed line); (ii) no modu- lation (black dash-dotted line); and (iii) modulation starts at t = 6 (blue solid line). Spatial distribution m ( z ) = ∫ dr 2 πr ( | Φ + ( r, z ) | 2 − | Φ$_{−}$( r, z ) | $^{2}$) at different times for the above three cases: (b) — case (i); (c) — case (iii); (d) — case (ii). a z = √ ℏ /Mω z . Trivial and flat m ( z ) at early times ( t < 250) has been omitted. Dynamical instability induced spontaneous domain formation is prohibited by the modu- lated spin exchange interaction. [1] L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, Oxford, 1960). [2] G. Horvath, R. Thompson, and P. Knight, Contemporary Physics 38 , 25 (1997). [3] H. Saito and M. Ueda, Phys. Rev. Lett. 90 , 040403 (2003). [4] F. Abdullaev and R. Kraenkel, Phys. Lett. A 272 , 395 (2000). [5] F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, Phys. Rev. A 67 , 013605 (2003). [6] G. D. Montesinos, V. M. P´ erez-Garc´ıa, and H. Michinel, Phys. Rev. Lett. 92 , 133901 (2004). [7] W. Zhang, D. L. Zhou, M.-S. Chang, M. S. Chapman, and L. You, Phys. Rev. Lett. 95 , 180403 (2005). [8] M. Chang, Q. Qin, W. Zhang, L. You, and M. Chapman, Nature Phys. 1 , 111 (2005). [9] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, Nature (London) 443 , 312 (2006). [10] M. Vengalattore, J. M. Higbie, S. R. Leslie, J. Guzman, L. E. Sadler, and D. M. Stamper-Kurn, Phys. Rev. Lett. 98 , 200801 (2007). [11] S. Yi and H. Pu, Phys. Rev. Lett. 97 , 020401 (2006). [12] M. Vengalattore, S. R. Leslie, J. Guzman, and D. M. Stamper-Kurn, Phys. Rev. Lett. 100 , 170403 (2008). [13] F. K. Fatemi, K. M. Jones, and P. D. Lett, Phys. Rev. Lett. 85 , 4462 (2000). [14] J. Cheng, H. Jing, and Y. Yan, Phys. Rev. A 77 , 061604 (2008). [15] C. D. Hamley, E. M. Bookjans, G. Behin-Aein, P. Ah- madi, and M. S. Chapman, Phys. Rev. A 79 , 023401 (2009). [16] H. Jing, J. Fu, Z. Geng, and W.-M. Liu, Phys. Rev. A