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In this article we are concerned with the existence of traveling wave solutions of a general class of nonlocal wave equations: utt − a 2 uxx = (β * u p)xx, p > 1. Members of the class arise as mathematical models for the propagation of waves in a wide variety of situations. We assume that the kernel β is a bell-shaped function satisfying some mild differentiability and growth conditions. Taking advantage of growth properties of bell-shaped functions, we give a simple proof for the existence of bell-shaped traveling wave solutions.

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We study the concentrated NLS on R n , with power non-linearities, driven by the fractional Laplacian, (−∆) s , s > n 2 . We construct the solitary waves explicitly, in an optimal range of the parameters, so that they belong to the natural energy space H s . Next, we provide a complete classification of their spectral stability. Finally, we show that the waves are non-degenerate and consequently orbitally stable, whenever they are spectrally stable.Incidentally, our construction shows that the soliton profiles for the concentrated NLS are in fact exact minimizers of the Sobolev embedding H s (R n ) → L ∞ (R n ), which provides an alternative calculation and justification of the sharp constants in these inequalities.

In this paper, we consider the degenerate semi-linear Schrödinger and Korteweg-deVries equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of NLS and traveling waves, which turn out to have compact support, compactons. We show that the compactons are unique bell-shaped solutions of the corresponding PDE's and for appropriate variational problems as well. We provide a complete spectral characterization of such waves, for all values of p. Namely, we show that all waves are spectrally stable for 2 < p ≤ 8, while a single mode instability occurs for p > 8. This extends previous work of Germain, Harrop-Griffits and Marzuola, [3], who have previously established orbital stability for some specific waves, in the range p < 8.