Conservation laws for Camassa–Holm equation, Dullin–Gottwald–Holm equation and generalized Dullin–Gottwald–Holm equation

In this paper we construct the conservation laws for the Camassa–Holm equation, the Dullin–Gottwald–Holm equation (DGH) and the generalized Dullin–Gottwald–Holm equation (generalized DGH). The variational derivative approach is used to derive the conservation laws. Only first order multipliers are considered. Two multipliers are obtained for the Camassa–Holm equation. For the DGH and generalized DGH equations the variational derivative approach yields two multipliers; thus two conserved vectors are obtained.     

Global weak solutions for a periodic two‐component μ‐Camassa–Holm system

In this paper, we consider the global existence of weak solutions for a two‐component μ‐Camassa–Holm system in the periodic setting. Global existence for strong solutions to the system with smooth approximate initial value is derived. Then, we show that the limit of approximate solutions is a global‐in‐time weak solution of the two‐component μ‐Camassa–Holm system. Copyright © 2013 John Wiley & Sons, Ltd.     

Development of a numerical phase optimized upwinding combined compact difference scheme for solving the Camassa–Holm equation with different initial solitary waves

In this article, the solution of Camassa–Holm (CH) equation is solved by the proposed two‐step method. In the first step, the sixth‐order spatially accurate upwinding combined compact difference scheme with minimized phase error is developed in a stencil of four points to approximate the first‐order derivative term. For the purpose of retaining both of the long‐term accurate Hamiltonian property and the geometric structure inherited in the CH equation, the time integrator used in this study should be able to conserve symplecticity. In the second step, the Helmholtz equation governing the pressure‐like variable is approximated by the sixth‐order accurate three‐point centered compact difference scheme. Through the fundamental and numerical verification studies, the integrity of the proposed high‐order scheme is demonstrated. Another aim of this study is to reveal the wave propagation nature for the investigated shallow water equation subject to different initial wave profiles, whose peaks take the smooth, peakon, and cuspon forms. The transport phenomena for the cases with/without inclusion of the linear first‐order advection term κu_x in the CH equation will be addressed. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1645–1664, 2015     

Low regularity solutions,blowup, and global existence for a generalization of Camassa–Holm‐type equation

We consider a generalization of Camassa–Holm‐type equation including the Camassa–Holm equation and the Novikov equation. We mainly establish the existence of solutions in lower order Sobolev space with . Then, we present a precise blowup scenario and give a global existence result of strong solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

The compact and noncompact structures for two types of generalized Camassa–Holm–KP equations

The objective of this paper is to investigate two types of generalized nonlinear Camassa–Holm–KP equations in (2+1) dimensional space. Compactons, solitons, solitary patterns, periodic solutions and algebraic travelling wave solutions are expressed analytically under various circumstances. The conditions that cause the qualitative change in the physical structures of the solutions are emphasized.     

Application of variational iteration method for modified Camassa‐Holm and Degasperis‐Procesi equations

In this article, the variational iteration method (VIM) is used to obtain approximate analytical solutions of the modified Camassa‐Holm and Degasperis‐Procesi equations. The method is capable of reducing the size of calculation and easily overcomes the difficulty of the perturbation technique or Adomian polynomials. The results reveal that the VIM is very effective. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

On the Cauchy problem for the periodic b-family of equations and of the non-uniform continuity of Degasperis–Procesi equation

We consider the problem of the existence of the global solutions and formation of singularities for a b-family of equations which includes the Camassa–Holm and Degasperis–Procesi equation. We also consider the problem of the uniformly continuity of Degasperis–Procesi equation.     

A three‐level linearized finite difference scheme for the camassa–holm equation

The Camassa–Holm (CH) system is a strong nonlinear third‐order evolution equation. So far, the numerical methods for solving this problem are only a few. This article deals with the finite difference solution to the CH equation. A three‐level linearized finite difference scheme is derived. The scheme is proved to be conservative, uniquely solvable, and conditionally second‐order convergent in both time and space in the discrete L_∞ norm. Several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 451–471, 2014     

Wave‐breaking phenomenon for a generalized spatially periodic Camassa–Holm system

Considered herein is a generalized two‐component Camassa–Holm system in spatially periodic setting. We first prove two conservation laws; then under proper assumptions on the initial data, we show the precise blow‐up scenarios and sufficient conditions guaranteeing the formation of singularities to the solutions of the generalized Camassa–Holm system. Copyright © 2014 John Wiley & Sons, Ltd.     

Global existence of weak solutions to a weakly dissipative modified two‐component Dullin–Gottwald–Holm system

We consider a weakly dissipative modified two‐component Dullin–Gottwald–Holm system. The existence of global weak solutions to the system is established. We first give the well‐posedness result of viscous approximate problem and obtain the basic energy estimates. Then, we show that the limit of the viscous approximation solutions is a global weak solution to the system. Copyright © 2014 John Wiley & Sons, Ltd.     

Conservative compact finite difference scheme for the N‐coupled nonlinear Klein–Gordon equations

In this article, a compact finite difference method is developed for the periodic initial value problem of the N‐coupled nonlinear Klein–Gordon equations. The present scheme is proved to preserve the total energy in the discrete sense. Due to the difficulty in obtaining the priori estimate from the discrete energy conservation law, the cut‐off function technique is employed to prove the convergence, which shows the new scheme possesses second order accuracy in time and fourth order accuracy in space, respectively. Additionally, several numerical results are reported to confirm our theoretical analysis. Lastly, we apply the reliable method to simulate and study the collisions of solitary waves numerically.     

On the global weak solutions for a modified two‐component Camassa‐Holm equation

In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})$ and some a priori estimates on the first‐order derivatives of approximation solutions.     

A new kind of nonlocal symmetry for the μ‐Camassa‐Holm equation with linear dispersion

The μ‐Camassa‐Holm equation with linear dispersion is a completely integrable model. In this paper, it is shown that this equation has quadratic pseudo‐potentials that allow us to construct pseudo‐potential–type nonlocal symmetries. As an application, we obtain its recursion operator by using this kind of nonlocal symmetry, and we construct a Darboux transformation for the μ‐Camassa‐Holm equation.     

Convergence analysis of the Navier–Stokes alpha model

We consider the Navier–Stokes‐alpha (NS‐α) model as an approximation of turbulent flows under nonperiodic boundary conditions. We prove global existence and uniqueness of weak solutions of the particular model. Further, we give full discretization of the model using the finite element approximations. Finally, we prove convergence of the method to the continuous NS‐α solution as h → 0 for a constant α. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Stability and convergence of the mark and cell finite difference scheme for Darcy‐Stokes‐Brinkman equations on non‐uniform grids

In this paper, we consider the mark and cell (MAC) method for Darcy‐Stokes‐Brinkman equations and analyze the stability and convergence of the method on nonuniform grids. Firstly, to obtain the stability for both velocity and pressure, we establish the discrete inf‐sup condition. Then we introduce an auxiliary function depending on the velocity and discretizing parameters to analyze the super‐convergence. Finally, we obtain the second‐order convergence in L2 norm for both velocity and pressure for the MAC scheme, when the perturbation parameter ? is not approaching 0. We also obtain the second‐order convergence for some terms of ∥·∥_? norm of the velocity, and the other terms of ∥·∥_? norm are second‐order convergence on uniform grid. Numerical experiments are carried out to verify the theoretical results.     

Navier–Stokes equations in aperture domains: Global existence with bounded flux and time‐periodic solutions

We consider the Navier–Stokes equations in an aperture domain of the three‐dimensional Euclidean space. We are interested in proving the existence of regular solutions corresponding to small initial data and flux through the aperture. The flux is assumed to be smooth and bounded on (0, +∞). As a consequence, we prove the existence of a time‐periodic solution corresponding to a time‐periodic flux through the aperture. Finally, we compare our solution with a solution belonging to a wider class, showing that, if such a solution does exist, then the two solutions coincide. Copyright © 2007 John Wiley & Sons, Ltd.     

Stationary solutions of Euler–Poisson equations for non‐isentropic gaseous stars

The motion of the self‐gravitational gaseous stars can be described by the Euler–Poisson equations. For some velocity fields and entropy functions that solve the conservation of mass and energy, we consider the existence of stationary solutions of Euler–Poisson equations. Under various restriction to the strength of velocity field, different assumptions on the isentropic function and adiabatic exponent, we get the existence, multiplicity and uniqueness of the stationary solutions to the Euler–Poisson system, respectively. Copyright © 2012 John Wiley & Sons, Ltd.     

Non‐homogeneous Navier–Stokes systems with order‐parameter‐dependent stresses

We consider the Navier–Stokes system with variable density and variable viscosity coupled to a transport equation for an order‐parameter c. Moreover, an extra stress depending on c and ?c, which describes surface tension like effects, is included in the Navier–Stokes system. Such a system arises, e.g. for certain models of granular flows and as a diffuse interface model for a two‐phase flow of viscous incompressible fluids. The so‐called density‐dependent Navier–Stokes system is also a special case of our system. We prove short‐time existence of strong solution in L^q‐Sobolev spaces with q>d. We consider the case of a bounded domain and an asymptotically flat layer with a combination of a Dirichlet boundary condition and a free surface boundary condition. The result is based on a maximal regularity result for the linearized system. Copyright © 2010 John Wiley & Sons, Ltd.