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We use multiscale perturbation theory in conjunction with the inverse scattering transform to study the interaction of a number of solitons of the cubic nonlinear Schroedinger equation under the influence of a small correction to the nonlinear potential. We assume that the solitons are all moving with the same velocity at the initial instant; this maximizes the effect each soliton has on the others as a consequence of the perturbation. Over the long time scales that we consider, the amplitudes of the solitons remain fixed, while their center of mass coordinates obey Newton's equations with a force law for which we present an integral formula. For the interaction of two solitons with a quintic perturbation term we present more details since symmetries -- one related to the form of the perturbation and one related to the small number of particles involved -- allow the problem to be reduced to a one-dimensional one with a single parameter, an effective mass. The main results include calculations of the binding energy and oscillation frequency of nearby solitons in the stable case when the perturbation is an attractive correction to the potential and of the asymptotic "ejection" velocity in the unstable case. Numerical experiments illustrate the accuracy of the perturbative calculations and indicate their range of validity., 28 pages, 7 figures, Submitted to Phys Rev E Revised: 21 pages, 6 figures, To appear in Phys Rev E (many displayed equations moved inline to shorten manuscript)

Fully nonlinear modulation equations are obtained for plane waves in a discrete system with quadratic nonlinearity, in the limit when the modulational scales are long compared to the wavelength and period of the modulated wave. The discrete system we study is a model for second-harmonic generation in nonlinear optical waveguide arrays and also for exciton waves at the interface between two crystals near Fermi resonance. The modulation equations predict their own breakdown by changing type from hyperbolic to elliptic. Modulational stability (hyperbolicity of the modulation equations) is explicitly shown to be implied by linear stability but not vice versa. When the plane-wave parameters vary slowly in regions of linear stability, the modulation equations are hyperbolic and accurately describe the macroscopic behavior of the system whose microscopic dynamics is locally given by plane waves. We show how the existence of Riemann invariants allows one to test modulated wave initial data to see whether the modulating wave will avoid all linear instabilities and ultimately resolve into simple disturbances that satisfy the Hopf or inviscid Burgers equation. We apply our general results to several important limiting cases of the microscopic model in question.

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.

The bound state solutions of the linear Schrodinger equation describing steady propagation of a signal in a planar waveguide are explicitly obtained for the case of a refractive index profile created by the interaction of N arbitrary dark solitons in a defocusing Kerr medium. Analysis of these solutions shows that linear signals confined to the arms of the waveguide prior to the collision do not interfere with each other at all as they pass through the interaction region of the dark solitons. Each signal emerges from the collision with exactly the same power and angle as it had at the input, thus behaving as though all solitons other than the one in which it was initially confined were altogether absent.