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The question of whether features and behaviors that are characteristic tocompletely integrable systems persist in the transition to non-integrablesettings is a central one in the field of nonlinear dispersive equations. Inthis work, we investigate this topic in the context of focusing nonlinearSchr\"odinger (NLS) equations. In particular, we consider non-integrablecounterparts of the (integrable) focusing cubic NLS equation, which aredistinct generalizations of cubic NLS and involve a broad class ofnonlinearities, with the cases of power and saturable nonlinearities serving asillustrative examples. This is a notably different direction from the oneexplored in other works, where the non-integrable models considered are onlysmall perturbations of the integrable one. We study the Cauchy problem on thereal line for both vanishing and non-vanishing boundary conditions at infinityand quantify the proximity of solutions between the integrable andnon-integrable models via estimates in appropriate metrics as well aspointwise. These results establish that the distance of solutions grows at mostlinearly with respect to time, while the growth rate of each solution ischiefly controlled by the size of the initial data and the nonlinearityparameters. A major implication of these closeness estimates is that integrabledynamics emerging from small initial conditions may persist in thenon-integrable setting for significantly long times. In the case of zeroboundary conditions at infinity, this persistence includes soliton and solitoncollision dynamics, while in the case of nonzero boundary conditions atinfinity, it establishes the nonlinear behavior of the non-integrable models atthe early stages of the ubiquitous phenomenon of modulational instability. Forthis latter and more challenging type of boundary conditions, ... (fullabstract in article)

On the proximity between the wave dynamics of the integrable focusing nonlinear Schr\"odinger equation and its non-integrable generalizations