Multi-Soliton Solutions to the Modified Korteweg-de Vries Equation

The modified Korteweg-de Vries (mKdV) equation is a nonlinear partial differential equation used to model the behavior of internal waves in stratified fluids, traffic flow and congestion along highways, and the propagation of Alfven waves (electromagnetic waves occurring in ionized gases). Certain solutions to the mKdV equation appear as solitary waves (multi-solitons), which are waves containing one or more individual solitons, each maintaining a constant shape and speed when they are far apart but interacting nonlinearly when they collide. Using a recently developed method, our goal is to describe all multi-soliton solutions to the mKdV equation mathematically. This is done by formulating an n-soliton solution in terms of matrices A, B, and C with respective sizes n×n, n×1, and 1×n, as well as an auxiliary matrix P satisfying the matrix equation AP+PA=BC. In our research we verify that the n-soliton solution formula indeed satisfies the mKdV equation and investigate the effect of the matrices A and C on the physical properties (shape, speed, and position) of the multi-soliton solution. We also use Mathematica to animate multi-soliton solutions and to better understand their properties.