An exact solution formula for the half-line Korteweg-de Vries equation

The half-line Korteweg-de Vries (KdV) equation, u_t+u_x-6uu_x+u_xxx=0, is an integrable, nonlinear partial differential equation used to model surface waves in shallow, narrow canals and acoustic waves in plasmas (ionized gases). Using a recent method we develop a solution formula for a large class of solutions to the half-line KdV equation, which includes the so-called multi-soliton solutions.  The formula uses a triplet of constant matrices A,B, and C with respective sizes n x n, n x 1, and 1 x n, for any positive integer n.  The solution formula uses matrix exponentials and involves an n x n matrix P satisfying the auxiliary matrix equation AP+PA=BC.  We analyze the unique solvability of this matrix equation and provide the existence and uniqueness based on the eigenvalues of the matrix A. We then prove that our solution formula satisfies the half-line KdV equation when the auxiliary matrix equation has a unique solution. We also investigate various physical properties of our solutions and relate those physical properties to the eigenvalues of the matrix A.