Singularity formation and blowup of complex-valued solutions of the modified KdV equation

Abstract

The dynamics of the poles of the two--soliton solutions of the modified Korteweg--de Vries equation $$u_t+6u^2{u}_x+u_{xxx}=0$$ are determined. A consequence of this study is the existence of classes of smooth, complex--valued solutions of this equation, defined for $-\infty <x<\infty$, exponentially decreasing to zero as $|x|\to \infty$, that blow up in finite time.