Quaternion-Valued Solutions to the KdV Equation: Non-Uniqueness and Tail Behavior of 1-Soliton Solutions

Hilliard, Zachary Thomas

Quaternion-Valued Solutions to the KdV Equation: Non-Uniqueness and Tail Behavior of 1-Soliton Solutions

Authors

Hilliard, Zachary Thomas

Abstract

Both the Korteweg-de Vries (KdV) equation and the quaternions are very well understood, but the combination of the two still remains somewhat of a mystery. Previous research has shown that with the choice of two quaternionic coefficients and a complex number, one can produce solutions to the non-commutative analog of the KdV equation using Darboux transformations. In order to understand the different aspects of 2-soliton interactions, we must first understand which choices of parameters yield the same solution. We have been able to derive of a set of algebraic conditions that must be satisfied in order for two choices of parameters to lead to the same solution. In addition, it has been noted that there are periodic functions in the tails of breather 1-soliton solutions. We claim to have found the equations for these functions and explain how we can use them to answer some more interesting questions.