We develop an inverse scattering method for an integrable higher-order nonlinear Schrödinger equation (NLSE) with the zero boundary condition at the infinity. An appropriate Riemann-Hilbert problem is related to two cases of scattering data — viz. for $N$ simple poles and a one higher-order pole. This allows obtaining the exact formulae of $N$-th order position and soliton solutions in the form of determinants. In addition, special choices of free parameters allow determining remarkable characteristics of these solutions and discussing them graphically. The results can be also applied to other types of NLSEs such as the standard NLSE, Hirota equation, and complex modified KdV equation. They can help to further explore and enrich related nonlinear wave phenomena.

A Chebyshev polynomial neural network for solving boundary value problems for one- and two-dimensional partial differential equations is constructed. In particular, the input parameters are expanded by Chebyshev polynomials and fed into the network. A loss function is constructed, and approximate solutions are determined by minimizing the loss function. Elliptic equations are used to test a Chebyshev polynomial neural network solver. The numerical examples illustrate the high accuracy of the method.

This paper aims to explore dispersion-managed lump waves in a spatial symmetric KP model. Negative second-order linear dispersive terms play an important role in creating lump waves with the nonlinearity in the model. The starting point is a Hirota bilinear form with an ansatz on quadratic function solutions to the corresponding Hirota bilinear equation. Symbolic computation with Maple is conducted to determine lump waves, and characteristic behaviors are analyzed for the resulting lump wave solutions.

Two extrapolated dynamic string-averaging cutter methods for finding a common fixed point of a finite family of demiclosed cutters in a Hilbert space are developed. One method converges weakly to a common fixed point of the family. The other converges in norm and is a combination of the method mentioned and the steepest-descent method. The proof of the strong convergence does not employ any additional cutter related conditions such as approximate shrinking and bounded regularity of their fixed point sets often used in literature. Particular cases of the last method and applications to a convex optimization problem over the intersection of the level sets and the LASSO problem with computational experiments are provided as illustrations.

This paper investigates the impact of the curriculum learning process in a multilayer neural network (NN) for solving the Lane-Emden and Emden-Fowler models. Starting from the training of a neural network in a small domain, we gradually expanded the domain. The symplectic NN trial solution is used for solving titled models. Feedforward NN and error back-propagation algorithms are used to minimize the error function and modify the parameters. The consistency of the algorithm is demonstrated by solving several problems. Calculating different types of errors (MSE up to 1E-10), we show an excellent agreement between the current simulations and existing results.

Dai et al. [Multiscale Model. Simul. 18 (2020)] proposed a gradient flow model and a numerical scheme for ground state calculations in Kohn-Sham density functional theory. It is a feature that orthonormality of all wave functions can be preserved automatically during the simulation which makes such a method attractive towards simulations for large scale systems. In this paper, two extensions are proposed for further improving the efficiency of the method. The first one is a linearization of the original nonlinear scheme. It is shown analytically that both the orthonormality of wave functions and the decay of the total energy can be preserved well by this linear scheme, while a significant acceleration can be observed from the numerical experiments due to the removal of an iteration process in the nonlinear scheme. The second one is the introduction of the adaptivity in the algorithm both temporally and spatially — i.e. an $h$-adaptive mesh method is employed to control the total amount of mesh grids, and an adaptive stop criterion in time propagation process is designed based on an observation that total energy always decays much faster at the beginning. Plenty of numerical experiments successfully demonstrate effectiveness of our method.

New high-order finite difference multi-resolution weighted essentially non-oscillatory (WENO) schemes with the Lax-Wendroff (LW) time discretization method for solving fractional differential equations with the Caputo fractional derivative are considered. The fractional derivative of order $α ∈ (1, 2)$ is split into an integral part and a second derivative term. The Gauss-Jacobi quadrature method is employed to solve the integral part, and a new multi-resolution WENO method for discretizing the second derivative term is developed. High-order spatial reconstruction procedures use any positive numbers (with a minor restriction) as linear weights. The new multi-resolution WENO-LW methods are one-step explicit high-order finite difference schemes. They are more compact than multi-resolution WENO-RK schemes of the same order. The LW time discretization is more cost efficient than the Runge-Kutta time discretization method. The construction of new multi-resolution WENO-LW schemes is simple and easy to generalize to arbitrary high-order accuracy in multi-dimensions. One- and two-dimensional examples with strong discontinuities verify the good performance of new fourth-, sixth-, and eighth-order multi-resolution WENO-LW schemes.

Numerical solutions of the multi-dimensional spatial fractional Allen-Cahn equations are studied. After the semi-discretization of the spatial fractional Riesz derivative, a system of nonlinear ordinary differential equations with the Toeplitz structure is obtained. In order to reduce the computational complexity, a two-level Strang splitting method is proposed, where the Toeplitz matrix in the system is represented as the sum of circulant and skew-circulant matrices. Therefore, the method can be quickly implemented by the fast Fourier transform, avoiding expensive Toeplitz matrix exponential calculations. It is shown that the discrete maximum principle of the method is unconditionally preserved. Moreover, the analysis of errors in the infinite norm with second-order accuracy is carried out in both time and space. Numerical tests support the theoretical findings and show the efficiency of the method.

Three finite volume iterative schemes for steady incompressible magnetohydrodynamic problems are studied. The theoretical analysis of finite volume methods is more challenging than that of finite element methods because of the presence of a trilinear form and the difficulties with the treatment of nonlinear terms. Nevertheless, we prove the uniform stability of the methods and establish error estimates. It is worth noting that the Newton iterative scheme converges exponentially under viscosity related requirements, while the Oseen iterative method is unconditionally stable and convergent under the uniqueness condition. Some numerical examples confirm the theoretical findings and demonstrate a good performance of the methods under consideration.

An adaptive non-intrusive multi-fidelity reduced basis method for parameterized partial differential equations is developed. Based on snapshots with different fidelity, the method reduces the number of high-fidelity snapshots in the regression model training and improves the accuracy of reduced-order model. One can employ the reduced-order model built on the low-fidelity data to adaptively identify the important parameter values for the high-fidelity evaluations under a given tolerance. The multi-fidelity reduced basis is constructed based on the high-fidelity snapshot matrix and the singular value decomposition of the low-fidelity snapshot matrix. Coefficients of such multi-fidelity reduced basis are determined by projecting low-fidelity snapshots on the low-fidelity reduced basis and using the Gaussian process regression. The projection method is more accurate than the regression method, but it requires low-fidelity snapshots. The regression method trains the Gaussian process regression only once but with slightly lower accuracy. Numerical tests show that the proposed multi-fidelity method can improve the accuracy and efficiency of reduced-order models.

This paper is concerned with a hybrid method for three-dimensional semilinear elliptic equations, constructed by combining the ideas presented in [Huang et al., J. Comput. Phys. 419 (2020)] and [Zhang et al., Comput. Math. Appl. 80 (2020)]. The convergence rate analysis indicates that the method converges rapidly. Numerical examples support the theoretical results and show that the method proposed outperforms the purely deep learning-based and traditional iterative methods.