We have been interested in studying a nonlinear variational wave equation whose wave speed is a sinusoidal function of the wave amplitude, arising naturally from liquid crystals. High-frequency waves of small amplitudes, the so-called weakly nonlinear waves, near a constant state a are governed by two asymptotic equations: the first-order asymptotic equation if a is not a critical point of the sinusoidal function, or the second-order asymptotic equation if a is either a maximal or a minimal point of the sinusoidal function. Our earlier work on the first-order asymptotic equation has greatly helped the study of the nonlinear variational wave equation with monotone wave speed functions. It is apparent in our research that investigation of the second-order asymptotic equation is both crucial and equally illuminating for the study of the nonlinear variational wave equation with sinusoidal wave speed functions. We succeed in this paper in handling what may be appropriately called the ‘concentration-annihilation’ phenomena in the historical spirit of compensated-compactness (Tartar et al.), concentration-compactness (Lions), and concentration-cancellation or concentration-evanesces (DiPerna and Majda). More precisely, the second-order asymptotic equation has a product term uv2 for which v2 may have concentration on a set where u vanishes in a sequence of approximate solutions, while the product retains no concentration. Although absent in the first-order asymptotic equation, this concentration-annihilation phenomenon has been demonstrated through an explicit example for the nonlinear variational wave equation with sinusoidal wave speed functions in an earlier work. We use this concentration-annihilation to establish the global existence of weak solutions to the second-order asymptotic equation with initial data of bounded total variations.

In this paper, we study n-dimensional complete submanifolds with constant scalar curvature in the Euclidean space En+p and n-dimensional compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1). We prove that the totally umbilical sphere Sn(r), totally geodesic Euclidean space En and generalized cylinder Sn−1(c) × E1 are the only n-dimensional (n > 2) complete submanifolds Mn with constant scalar curvature n(n − 1)r in the Euclidean space En+p, which satisfy the following condition: where S denotes the squared norm of the second fundamental form of Mn. For compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1), we also obtain a corresponding result (see theorem 1.3).

We use singularity theory to classify forced symmetry-breaking bifurcation problems where f1 is 𝕆(2)-equivariant and f2 is 𝔻n-equivariant with the orthogonal group actions on z ∈ ℝ2. Forced symmetry breaking occurs when the symmetry of the equation changes when parameters are varied. We explicitly apply our results to the branching of subharmonic solutions in a model periodic perturbation of an autonomous equation and sketch further applications.

The connection between the discrete and the continuous coagulation–fragmentation models is investigated. A weak stability principle relying on a priori estimates and weak compactness in L1 is developed for the continuous model. We approximate the continuous model by a sequence of discrete models and, writing the discrete models as modified continuous ones, we prove the convergence of the latter towards the former with the help of the above-mentioned stability principle. Another application of this stability principle is the convergence of an explicit time and size discretization of the continuous coagulation-fragmentation model.

By means of a certain variational principle, where rank-one convexity enters in a fundamental way as part of feasibility, we construct a whole class of rank-one convex functions. We show the existence of optimal paths and motivate our analysis by the issue of the equivalence of quasiconvexity and rank-one convexity for 2 × 2 matrices.

This work is concerned with basic structural properties of first-order hyperbolic systems with source terms divided by a small parameter ε. We identify a relaxation criterion necessary for the solution sequences indexed with ε to have reasonable limits as ε goes to zero. This relaxation criterion is shown to imply hyperbolicity of the reduced systems governing the limits. Moreover, we introduce a so-called GC-stability theory and strengthen the hyperbolicity result. The latter shows that there are no linearly stable hyperbolic relaxation approximations for non-hyperbolic conservation laws.