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

TL;DR

This paper investigates the pole dynamics of two-soliton solutions of the modified KdV equation, revealing classes of smooth, complex solutions that can blow up in finite time despite decaying exponentially at infinity.

Contribution

It provides a detailed analysis of pole behavior and demonstrates the existence of finite-time blowup solutions in the complex-valued setting of the modified KdV equation.

Findings

Pole dynamics of two-soliton solutions are characterized.

Existence of smooth, complex solutions that blow up in finite time.

Solutions decay exponentially at infinity before blowup.

Abstract

The dynamics of the poles of the two--soliton solutions of the modified Korteweg--de Vries equation $$ u_t + 6u^2u_x + u_{xxx} = 0 $$ 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.