• Abstract

In this paper, we investigate the blowup mechanism for the 2D isentropic compressible Euler equations with a damping term. Generally, damped Euler equations have a global classical solution for small smooth initial data. However, we identify a class of large initial data for the damped Euler equations, under the condition of azimuthal symmetry and by choosing a suitable self-similar transformation, resulting in the solution to the Cauchy problem blowing up in a finite time. Additionally, we introduce modulation variables to accurately describe the blowup time and location. Furthermore, the blowup profile exhibits a cusp singularity with Hölder $C^{1/3}$ regularity at the blowup point.

• General Summary

• Abstract

In this paper, we introduce energy-stable schemes based on operator splitting methods for Maxwell’s equations in two-dimensional Lorentz dispersive media with transverse electric polarization, namely the sequential splitting scheme (SS-ML) and the Strang-Marchuk splitting scheme (SM-ML). Each splitting scheme involves two substages per time step, where 1D discrete sub-problems are solved using the Crank-Nicolson method for time discretization. Both schemes ensure energy decay and unconditional stability. The convergence analysis reveals that the SS-ML scheme exhibits first-order accuracy in time and second-order accuracy in space based on the energy technique, while the SM-ML scheme achieves second-order accuracy in both time and space. Additionally, numerical dispersion analysis yields two discrete numerical dispersion relation identities for each scheme. Theoretical results are supported by examples and numerical experiments.

• General Summary

• Abstract

The previous studies have shown that the defocusing NLSE has no the modulational instability, and was not found to admit the rogue wave phenomenon so far. In this paper, we address the question of the higher-order rogue wave solutions of the nonlocal $\mathscr{P}\mathscr{T}$-symmetric NLSE in the defocusing regime. Based on Darboux transformation and iterations, we derive an explicit solution for the higher-order rogue waves by adopting a variable separation and Taylor expansion technique. The higher-order rogue wave solutions are expressed in separation-of-variables form. Furthermore, in order to understand these solutions better, patterns of the rogue waves for lowest three order are explored clearly and conveniently. The reported results may be useful for the design of experiments for observation of rogue waves in the defocusing nonlinear physical systems.

• General Summary

• Abstract

A kind of higher-order Boussinesq equation is studied in this work. Based on the Bell polynomials theories, bilinear representations of the equation are derived, and diverse interaction solutions are constructed through Hirota’s bilinear method. These include interaction solutions characterized by hyperbolic cosine and cosine functions, as well as interactions between lump and soliton solutions. In addition, generalized bilinear operators are used in order to construct a new higher-order Boussinesq-like equation, while lump and breather solutions are also developed utilizing Hirota’s bilinear technique. For the various explicit solutions obtained in this work, several of them are considered to selected particular values for the relevant parameters in order to plot different kind of three-dimensional surfaces with associated two-dimensional density profiles to give a comprehensive understanding of the evolution for various solutions.

• General Summary

• Abstract

Multi-view data analysis has gained increasing popularity, in particular multi-view spectral clustering has attracted much attention for its outstanding performance in mining heterogeneity information in multi-view data. However, most spectral clustering methods exhibit the following disadvantages: firstly, learning consensus representation directly from multi-view data that may contain noise renders a distorted description; secondly, the traditional two-step process may fall into a suboptimal solution. To overcome these disadvantages, a novel multi-view spectral clustering method is proposed by unifying the optimization of consensus Laplacian matrix and the learning and discretization of spectral embedding into one step. We consider that the optimal Laplacian matrix is in the neighborhood of view-specific Laplacian matrix, as the view-specific Laplacian matrix only contains partial information from multi-view data, resulting in certain deviation from the optimal Laplacian matrix. The consensus Laplacian matrix was obtained in a dynamic optimization way with the spectral rotation and embedding information simultaneously determined. Extensive experiments have been conducted to demonstrate the effectiveness and superiority of our proposed method.

• General Summary

• Abstract

In this paper, an $hp$ spectral element approximation for distributed optimal control problem governed by an elliptic equation is investigated, whose objective functional does not include the control variable. And the constraint set on control variable is stated with $L^2$-norm. Optimality condition of the continuous and discretized systems are deduced. In order to solve the equivalent systems with high accuracy, $hp$ spectral element method is employed to discretize the constrained optimal control systems. Based on the property of some interpolation operators, a posteriori error estimates are also established by using some properties of some interpolation operators carefully. Finally, a projection gradient algorithm and a numerical example are provided, which confirm our analytical results. Such estimators guarantee the construction of reliable adaptive methods for optimal control problems.

• General Summary