An Exactly Solvable Model for the Interaction of Linear Waves with Korteweg-de Vries Solitons

Under certain mode-matching conditions, small-amplitude waves can be trapped by coupling to solitons of nonlinear fields. We present a model for this phenomenon, consisting of a linear equation coupled to the Korteweg–de Vries (KdV) equation. The model has one parameter, a coupling constant κ. For one value of the coupling constant, κ = 1, the linear equation becomes the linearized KdV equation, for which the linear waves can indeed be trapped by solitons and, moreover, for which the initial value problem for the linear waves has been solved exactly by Sachs [S83] in terms of quadratic forms in the Jost eigenfunctions of the associated Schrödinger operator. We consider in detail a different case of weaker coupling, κ = 1/2. We show that in this case linear waves may again be trapped by solitons, and like the stronger coupling case κ = 1, the initial value problem for the linear waves can also be solved exactly, this time in terms of linear forms in the Jost eigenfunctions. We present a family of exact solutions, and we develop the completeness relation for this family of exact solutions, finally giving the solution formula for the initial value problem. For κ = 1/2, the scattering theory of linear waves trapped by solitons is developed. We show that there exists an explicit increasing sequence of bifurcation values of the coupling constant, κ = 1/2, 1, 5/3, . . ., for which some linear waves may become trapped by solitons. By studying a third-order eigenvalue equation, we show that for κ < 1/2 all linear waves are scattered by solitons, and that for 1/2 < κ < 1, as well as for κ > 1, some linear waves are amplified by solitons.