Quaternion-Valued Breather Soliton, Rational, and Periodic KdV Solutions

TL;DR

This paper introduces quaternion-valued solutions to the non-commutative KdV equation, including solitons, rational, and periodic solutions, and explores their properties, interactions, and non-singularity conditions.

Contribution

It provides a complete characterization of parameters for non-singular solutions and introduces a non-linear superposition principle for multi-soliton interactions.

Findings

Non-singular solutions are always non-singular in the non-commutative case.

A formula for phase shift in 2-soliton interactions is derived.

Solutions include breather, rational, and periodic types, with hybrid solutions also constructed.

Abstract

Quaternion-valued solutions to the non-commutative KdV equation are produced using determinants. The solutions produced in this way are (breather) soliton solutions, rational solutions, spatially periodic solutions and hybrids of these three basic types. A complete characterization of the parameters that lead to non-singular 1-soliton and periodic solutions is given. Surprisingly, it is shown that such solutions are never singular when the solution is essentially non-commutative. When a 1-soliton solution is combined with another solution through an iterated Darboux transformation, the result behaves asymptotically like a combination of different solutions. This ``non-linear superposition principle'' is used to find a formula for the phase shift in the general 2-soliton interaction. A concluding section compares these results with other research on non-commutative soliton equations and lists some open questions.