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

**P. D. Miller and S. R. Clarke Department of Mathematics and Statistics
Monash University
Clayton, VIC 3168 Australia
**

### Abstract:

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 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.