Instability of near-extreme solutions to the Whitham equation

The Whitham equation is a model for the evolution of small-amplitude, unidirectional waves of all wavelengths in shallow water. It has been shown to accurately model the evolution of waves in laboratory experiments. We compute -periodic traveling wave solutions of the Whitham equation and numerically study their stability with a focus on solutions with large steepness. We show that the Hamiltonian oscillates at least twice as a function of wave steepness when the solutions are sufficiently steep. We show that a superharmonic instability is created at each extremum of the Hamiltonian and that between each extremum the stability spectra undergo similar bifurcations. Finally, we compare these results with those from the Euler equations.     

Some remarks on the Gronwall integral inequality

In this paper, a generalized Volterra-type integral inequality is developed. On the basis of this inequality, the effect of fractional-order $\omega$ on the application of the integer-order Gronwall integral inequality (IOGII) is discussed. Specially speaking, the IOGII cannot be directly used to reckon the solution of integral inequality with the order . It seems that both the IOGII and the generalized Volterra-type integral inequality can be applied to estimate the solution of integral inequality with the order $\omega \ge 1$, and results are consistent, but this is just a coincidence.     

Algebraic techniques for least squares problems in commutative quaternionic theory

Due to the rise of commutative quaternion in Hopfield neural networks, digital signal, and image processing, one encounters the approximate solution problems of the commutative quaternion linear equations $AX\approx B$ and $AXC\approx B$. This paper, by means of real representation and complex representation of commutative quaternion matrices, introduces concepts of norms of commutative quaternion matrices and derives two algebraic techniques for finding solutions of least squares problems in commutative quaternionic theory.     

One-dimensional average on spheres and approximation

In this paper, we consider the approximation of the degeneration of classic average on sphere on Hardy-Sobolev spaces I_λ(H^p) . We prove that the degenerate expression of average on sphere of an L^p function is convergent almost everywhere, and the speed of the approximation depends on λ. And when λ ≥ 2, the approximation of the average operator will be saturated. On the other hand, we also study the generalization of the average on the product of Hardy-Sobolev spaces.