Abstract
Our goal in this paper is to give a proof of local existence of solutions for the modified Korteweg-de-Vries (mKdV) in a weighted Sobolev space with initial data. A parabolic regularization is suggested, discussed and implemented for the approximation of solutions for the mKdV equation with an appropriate class of nonlineariy. The study for the mKdV equation and computations of blow-up without weighted was treated by Bona,J.L and Dougalis,V.A.. Further results can be found in Bona,J.L.-Souganidis,P.E. and Strauss,W.A. .
 Connections with biophysics are illustrated in Cortez, H et al [27]-[29].
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