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In this dissertation we prove local existence and uniqueness of solutions of the focusing modified Korteweg - de Vries equation u_t + u^2u_x + u_{xxx} = 0 in classes of unbounded functions that admit an asymptotic expansion at infinity in decreasing powers of $x$. We show that an asymptotic solution differs from a genuine solution by a smooth function that is of Schwartz class with respect to $x$ and that solves a generalized version of the focusing mKdV equation. The latter equation is solved by discretization methods. The text is written for researchers of partial differential equations but all proofs are given with full details and the text should be accessible to graduate students of mathematics.