In this paper, we construct the bilinear identities for the wave functions of an extended B-type Kadomtsev-Petviashvili (BKP) hierarchy, which contains two types of (2+1)-dimensional Sawada-Kotera equation with a self-consistent source (2d-SKwS-I and 2d-SKwS-II). By introducing an auxiliary variable corresponding to the extended flow for the BKP hierarchy, we find the tau-function and the bilinear identities for this extended BKP hierarchy. The bilinear identities can generate all the Hirota's bilinear equations for the zero-curvature forms of this extended BKP hierarchy. As examples, the Hirota's bilinear equations for the two types of 2d-SKwS (both 2d-SKwS-I and 2d-SKwS-II) will be given explicitly.

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Two nonlinear diffusion equations for thin film epitaxy, with or without slope selection, are studied in this work. The nonlinearity models the Ehrlich–Schwoebel effect – the kinetic asymmetry in attachment and detachment of adatoms to and from terrace boundaries. Both perturbation analysis and numerical simulation are presented to show that such an atomistic effect is the origin of a nonlinear morphological instability, in a rough-smooth-rough pattern, that has been experimentally observed as transient in an early stage of epitaxial growth on rough surfaces. Initial-boundary-value problems for both equations are proven to be well-posed, and the solution regularity is also obtained. Galerkin spectral approximations are studied to provide both a priori bounds for proving the well-posedness and numerical schemes for simulation. Numerical results are presented to confirm part of the analysis and to explore the difference between the two models on coarsening dynamics.

We use a small positive parameter to change the totalvariation function for unconstrainedMRimage reconstruction to a strictly convex perturbed function. Bregman iteration is applied to solve the modified total-variation MR image (TVMRI) reconstruction problem. A lagged diffusivity fixed-point algorithm is applied to solve the minimization problem in the Bregman iteration. We use the periodic boundary condition and a Fourier transform to accelerate TVMRI reconstruction.RealMR images are used to test the approach in numerical experiments. The experimental results demonstrate that the proposed method is very efficient for TVMRI reconstruction.

In 1986, R. D. Cook proposed differential geometry to assess local influence of minor perturbations of statistical models. We construct a conformally invariant curvature, the conformal normal curvature, for the same purpose. This curvature provides a measure of local influence ranging from 0 to 1, with objective benchmarks to judge largeness. We study various approaches to using the conformal normal curvature and the relationships between these approaches.