In this article, a new technique for deriving integrable hierarchy is discussed, i.e., such that are derived by combining the Tu scheme with the vector product. Several classes of spectral problems are introduced by three-dimensional loop algebra and six-dimensional loop algebra whose commutators are vector product, and the six-dimensional loop algebra is derived from the enlargement of the three-dimensional loop algebra. It is important that we make use of the variational method to create a new vector-product trace identity for which the Hamiltonian structure of the isospectral integrable hierarchy is worked out. The derived integrable hierarchies are reduced to the modified Korteweg-de Vries (mKdV) equation, generalized coupled mKdV integrable system and non-isospectral mKdV equation under specific parameter selection. Starting from a 3×3 matrix spectral problem, we subsequently construct an explicit $N$-fold Darboux transformation for integrable system (2.8) with the help of a gauge transformation of the corresponding spectral problem. At the same time, the determining equations of nonclassical symmetries associated with mKdV equation are presented in this paper. It follows that we investigate the coverings and the nonlocal symmetries of the nonisospectral mKdV equation by applying the classical Frobenius theorem and the coordinates of a infinitely-dimensional manifold in the form of Cartesian product.

In this paper, we are concerned with the inverse transmission eigenvalue problem to recover the shape as well as the constant refractive index of a penetrable medium scatterer. The linear sampling method is employed to determine the transmission eigenvalues within a certain wavenumber interval based on far-field measurements. Based on a prior information given by the linear sampling method, the neural network approach is proposed for the reconstruction of the unknown scatterer. We divide the wavenumber intervals into several subintervals, ensuring that each transmission eigenvalue is located in its corresponding subinterval. In each such subinterval, the wavenumber that yields the maximum value of the indicator functional will be included in the input set during the generation of the training data. This technique for data generation effectively ensures the consistent dimensions of model input. The refractive index and shape are taken as the output of the network. Due to the fact that transmission eigenvalues considered in our method are relatively small, certain super-resolution effects can also be generated. Numerical experiments are presented to verify the effectiveness and promising features of the proposed method in two and three dimensions.

The monotonicity of discrete Laplacian, i.e., inverse positivity of stiffness matrix, implies discrete maximum principle, which is in general not true for high order accurate schemes on unstructured meshes. On the other hand, it is possible to construct high order accurate monotone schemes on structured meshes. All previously known high order accurate inverse positive schemes are or can be regarded as fourth order accurate finite difference schemes, which is either an M-matrix or a product of two M-matrices. For the $Q^3$ spectral element method for the two-dimensional Laplacian, we prove its stiffness matrix is a product of four M-matrices thus it is unconditionally monotone. Such a scheme can be regarded as a fifth order accurate finite difference scheme. Numerical tests suggest that the unconditional monotonicity of $Q^k$ spectral element methods will be lost for $k≥9$ in two dimensions, and for $k≥4$ in three dimensions. In other words, for obtaining a high order monotone scheme, only $Q^2$ and $Q^3$ spectral element methods can be unconditionally monotone in three dimensions.

We investigate the numerical approximation for stabilizing the semidiscrete linearized Boussinesq system around an unstable stationary state. Stabilization is attained through internal feedback controls applied to the velocity and temperature equations, localized within an arbitrary open subset. This study follows the framework presented in [14], considering the continuous linearized Boussinesq system. The primary objective is to explore the penalization-based approximation of the free divergence condition in the semidiscrete case and provide a numerical validation of these results in a two-dimensional setting.