| Summary: | A feature of immeasurable interest in nonlinear systems is that of spatially localized traveling pulses, or solitary waves - states which persist indefinitely in time, focus energy, and facilitate its transfer. Furthermore, in many lattice systems, discreteness effects are important and play a key role in these dynamics.
In this thesis, we construct the multiple families of solitary standing (time-periodic) waves of the discrete, focusing cubically nonlinear Schrödinger equation (DNLS). These states are related to the so-called Peierls-Nabarro energy barrier, which refers to the energy difference between these distinct states and is thought to be responsible for the absence of indefinitely traveling, non-deforming solitary (spatially localized) waves of arbitrary velocity in many (non-dissipative) discrete systems. Instead, one observes that traveling waves of many discrete equations radiate energy and deform until they eventually cease to propagate and settle to a stationary time-periodic standing wave centered at a vertex.
We address two specific cases of DNLS: (1) nearest-neighbor coupling on a cubic lattice in dimensions d = 1,2,3, and (2) long-range site coupling in dimension d = 1. These states are obtained via a bifurcation analysis about the continuum nonlinear Schrödinger equation (NLS) limit, with respect to a natural small parameter. Depending on the spatial dimension, these may be vertex-, bond-, cell-, or face-centered. In the first case of nearest-neighbor coupling, we construct an explicit asymptotic expansion. In the second case of one-dimensional long-range coupling when the decay of the site coupling with respect to distance is sufficiently slow, the continuum limiting NLS equation has Laplacian of fractional power. Finally, we show that the energy difference among distinct states of the same frequency is exponentially small with respect to the small parameter beyond all polynomial orders. This provides a rigorous bound for the Peierls-Nabarro barrier.
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