Abstract
The linear Schr\odinger equation is explicitly solved for the case of a refractive index profile created by the interaction of $N$ bright solitons in a Kerr medium. The bound states give exact solutions to the problem of determining how a beam of light is split as it passes through the collision of $N$ solitons; the main result of this work is an analytical formula for the power transmission matrix of this linear problem. As specific examples, the cases of the two-soliton and three-soliton linear couplers are considered, and the power transmission characteristics for bound states are explicitly calculated in both cases. It is further shown that the behavior of linear waves propagating through an arbitrary collision of $N$ solitons can be completely understood in terms of the scattering of linear waves from a soliton X junction. The unbound states of an $N$-soliton waveguide are also explicitly computable and describe the evolution of radiation modes.
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