By numerically solving the Gross-Pita evskii equation,we advocate explorative studies of the formation and evolution o f matter-wave characterized by Aderson localization (AL),which is the issue of the considerable researches on Bose-Einstein condensate (BEC).The random speckl e potential is simulated with a random function produced by numeral methods.Star ting from the quasi-one-dimensional Gross-Pitaevskii equation,we study the localization of a weakly interacting Bose-Einstein condensate trapped in a rand om speckle potential in logarithmic coordinates,under the condition of weak diso rder.The stability and dilatation of localized states are investigated analytica lly by means of the split-step Fourier method,and the weak atomic interaction i s modulated by controlling s-wave scattering length of Bose condensate atoms.Me anwhile,we also study the effects of the nonlinear spatial modulation on the sha pe,energy and localization length of the density envelope.The results show that there exists Anderson localization in the presence of disorder and adiabatic con dition due to the exponential tail.Remarkably,it is found that the nonlinear spa tial modulation has important influence on the energy,the central and tail regio n,and localization length of the localized states.Our results demonstrate some n ovel phenomena,opening up new means of matter-wave manipulation.