Related papers: A self-adaptive mesh method for the Camassa-Holm e…
In this paper, we construct integrable self-adaptive moving mesh schemes for multi-component modified short pulse and short pulse equations under general boundary conditions including periodic boundary conditions by using the consistency…
Although the isogeometric analysis has shown its great potential in achieving highly accurate numerical solutions of partial differential equations, its efficiency is the main factor making the method more competitive in practical…
An adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption. The method is based on the so-called moving mesh partial differential…
In this paper, a novel adaptive finite element method is proposed to solve the Kohn-Sham equation based on the moving mesh (nonnested mesh) adaptive technique and the augmented subspace method. Different from the classical self-consistent…
Moving mesh methods (also called r-adaptive methods) are space-adaptive strategies used for the numerical simulation of time-dependent partial differential equations. These methods keep the total number of mesh points fixed during the…
The nonlinear Schr\"{o}dinger equation (NLSE) is one of the most important equations in quantum mechanics, and appears in a wide range of applications including optical fibre communications, plasma physics and biomolecule dynamics. It is a…
Integrable self-adaptive moving mesh schemes for short pulse type equations (the short pulse equation, the coupled short pulse equation, and the complex short pulse equation) are investigated. Two systematic methods, one is based on…
The soliton solutions of the Camassa-Holm equation are derived by the implementation of the dressing method. The form of the one and two soliton solutions coincides with the form obtained by other methods.
Compared with the two-component Camassa-Holm system, the modified two-component Camassa-Holm system introduces a regularized density which makes possible the existence of solutions of lower regularity, and in particular of multipeakon…
We propose a moving mesh adaptive approach for solving time-dependent partial differential equations. The motion of spatial grid points is governed by a moving mesh PDE (MMPDE) in which a mesh relaxation time \tau is employed as a…
This work surveys an r-adaptive moving mesh finite element method for the numerical solution of premixed laminar flame problems. Since the model of chemically reacting flow involves many different modes with diverse length scales, the…
This research addresses the numerical simulation of the Boltzmann transport equation for semiconductor devices by proposing a multidimensional self-adaptive numerical simulation framework. This framework is applied to two important…
We present a novel shape-approximating anisotropic re-meshing algorithm as a geometric generalization of the adaptive moving mesh method. Conventional moving mesh methods reduce the interpolation error of a mesh that discretizes a given…
The short pulse equation was introduced by Schaefer--Wayne (2004) for modeling the propagation of ultrashort optical pulses. While it can describe a wide range of solutions, its ultrashort pulse solutions with a few cycles, which the…
We develop a systematic procedure for constructing soliton solutions of an integrable two-component Camassa-Holm (CH2) system. The parametric representation of the multisoliton solutions is obtained by using a direct method combined with a…
We study an integrable equation whose solutions define a triad of one-forms describing a surface with Gaussian curvature -1. We identify a local group of diffeomorphisms that preserve these solutions and establish conserved quantities. From…
The paper deals with the construction of the asymptotic soliton-like and the asymptotic peakon-like solutions to the modified Camassa-Holm equation with variable coefficicents and a singular perturbation. This equation is a generalization…
In this paper, we implement the Adaptive Moving Mesh method (AMM) to the solution of initial value problems involving the Schr\"odinger equation, and more specifically the Schr\"odinger-Poisson system of equations. This method is based on…
In this paper, we show that the peakon (peaked soliton) solutions can be recovered from the smooth soliton solutions, in the sense that there exists a sequence of smooth N-soliton solutions of the dispersion Camassa-Holm equation converging…
In this paper, multi-component generalizations to the Camassa-Holm equation, the modified Camassa-Holm equation with cubic nonlinearity are introduced. Geometric formulations to the dual version of the Schr\"odinger equation, the complex…