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Fourier spectral methods achieve exponential accuracy both on the approximation level and for solving partial differential equations (PDEs), if the solution is analytic. If the solution is discontinuous but piecewise analytic up to the discontinuities, Fourier spectral methods produce poor pointwise accuracy, but still maintains exponential accuracy after post-processing. In earlier work, an extended technique is provided to recover exponential accuracy for functions which have end-point singularities, from the knowledge of point values on standard collocation points. In this paper, we develop a technique to recover exponential accuracy from the first $N$ Fourier coefficients of functions which are analytic in the open interval but have unbounded derivative singularities at end points. With this post-processing method, we are able to obtain exponential accuracy of spectral methods applied to linear transport equations involving such functions.

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We prove that there is a unique way to construct a geometric scale of Hilbert spaces interpolating between two given spaces. We investigate what properties of operators, other than boundedness, are preserved by interpolation. We show that self-adjointness is, but subnormality and Krein subnormality are not. On the way to this last result, we establish a representation theorem for cyclic Krein subnormal operators.

Solitary waves bifurcated from edges of Bloch bands in two-dimensional periodic media are determined both analytically and numerically in the context of a two-dimensional nonlinear Schrödinger equation with a periodic potential. Using multiscale perturbation methods, the envelope equations of solitary waves near Bloch bands are analytically derived. These envelope equations reveal that solitary waves can bifurcate from edges of Bloch bands under either focusing or defocusing nonlinearity, depending on the signs of the second-order dispersion coefficients at the edge points. Interestingly, at edge points with two linearly independent Bloch modes, the envelope equations lead to a host of solitary wave structures, including reduced-symmetry solitons, dipole-array solitons, vortex-cell solitons, and so on—many of which have not been reported before to our knowledge. It is also shown analytically that the centers of envelope solutions can be positioned at only four possible locations at or between potential peaks. Numerically, families of these solitary waves are directly computed both near and far away from the band edges. Near the band edges, the numerical solutions spread over many lattice sites, and they fully agree with the analytical solutions obtained from the envelope equations. Far away from the band edges, solitary waves are strongly localized, with intensity and phase profiles characteristic of individual families.