Solutions to the complex Korteweg-de Vries equation: Blow-up solutions and non-singular solutions

TL;DR

This paper derives and analyzes blow-up and non-singular solutions for the complex Korteweg-de Vries equation, utilizing transformations to connect it with the modified Korteweg-de Vries equation and illustrating the dynamics of these solutions.

Contribution

It introduces a method to generate solutions for the complex Korteweg-de Vries equation using a complex Miura transformation from the modified equation.

Findings

Derived blow-up solutions from 1-soliton and double-pole solutions.

Obtained solitons, breathers, and rational solutions via transformation.

Illustrated the dynamics of the solutions.

Abstract

In the paper two kinds of solutions are derived for the complex Korteweg-de Vries equation, including blow-up solutions and non-singular solutions. We derive blow-up solutions from known 1-soliton solution and a double-pole solution. There is a complex Miura transformation between the complex Korteweg-de Vries equation and a modified Korteweg-de Vries equation. Using the transformation, solitons, breathers and rational solutions to the complex Korteweg-de Vries equation are obtained from those of the modified Korteweg-de Vries equation. Dynamics of the obtained solutions are illustrated.