Asymptotic Decay for a Generalized Boussinesq System

In this paper we study the large time behavior of solutions to a generalization of the Boussinesq system of equations in n 2 spatial dimensions. We establish the existence and algebraic decay of the L ₂-norm of the solution.     

Chaos and integrability in a nonlinear wave equation

We consider the parametrized family of equations _tt ,u- _xx u-au+u ₂ ² u=O,x(0,L), with Dirichlet boundary conditions. This equation has finite-dimensional invariant manifolds of solutions. Studying the reduced equation to a four-dimensional manifold, we prove the existence of transversal homoclinic orbits to periodic solutions and of invariant sets with chaotic dynamics, provided that =2, 3, 4,.... For =1 we prove the existence of infinitely many first integrals pairwise in involution.     

Center Manifolds for Homoclinic Solutions

In this article, center-manifold theory is developed for homoclinic solutions of ordinary differential equations or semilinear parabolic equations. A center manifold along a homoclinic solution is a locally invariant manifold containing all solutions which stay close to the homoclinic orbit in phase space for all times. Therefore, as usual, the low-dimensional center manifold contains the interesting recurrent dynamics near the homoclinic orbit, and a considerable reduction of dimension is achieved. The manifold is of class C ^1, for some >0. As an application, results of Shilnikov about the occurrence of complicated dynamics near homoclinic solutions approaching saddle-foci equilibria are generalized to semilinear parabolic equations.     

Nonlinear Stability of a One-Dimensional Boussinesq Equation

We study nonlinear orbital stability and instability of the set of ground state solitary wave solutions of a one-dimensional Boussinesq equation or one-dimensional Benney–Luke equation. It is shown that a solitary wave (traveling wave with finite energy) may be orbitally stable or unstable depending on the range of the wave's speed of propagation.     

Existence and Integral Representations of Weak Solutions for Elastic Plates with Cracks

The existence and continuous dependence on the data are investigated in Sobolev spaces for the problem of bending of a Reissner-Mindlin-type plate weakened by a crack when the displacements or the moments and force are prescribed along the two sides of the crack. The cases of both an infinite and a finite plate are considered, and representations are sought for the solutions in terms of single layer and double layer potentials with distributional densities. This revised version was published online in August 2006 with corrections to the Cover Date.     

Instability of solitary waves for generalized Boussinesq equations

An equation of Boussinesq-type of the formu _tt -u _xx +(f(u)+u_xx)_xx=0 is considered. It is shown that a traveling wave may be stable or unstable, depending on the range of the wave's speed of propagation and on the nonlinearity. Sharp conditions to that effect are given.This research is supported in part by NSF Grant DMS 90-23864.     

BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS IN VARIANT BOUSSINESQ EQUATIONS

The bifurcations of solitary waves and kink waves for variant Boussinesq equations are studied by using the bifurcation theory of planar dynamical systems. The bifurcation sets and the numbers of solitary waves and kink waves for the variant Boussinesq equations are presented. Several types explicit formulas of solitary waves solutions and kink waves solutions are obtained. In the end, several formulas of periodic wave solutions are presented.     

Resonance Capture in a Three Degree-of-Freedom Mechanical System

We study the phenomena of resonance capture in a three degree-of-freedom dynamical system modeling the dynamics of an unbalanced rotor, subject to a small constant torque, supported by orthogonal, linearly elastic supports, which is constrained to move in the plane. In the physical system the resonance exists between translational motions of the frame and the angular velocity of the unbalanced rotor. These equations, valid in the neighborhood of the resonance, possess a small parameter which is related to the imbalance. In the limit 0, the unperturbed system possesses a homoclinic orbit which separates bounded periodic motion corresponding to resonant solutions from unbounded motion which corresponds to solutions passing through the resonance. Using a generalized Melnikov integral, we characterize the splitting distance between the invariant manifolds which govern capture and escape from resonance for 0. It is shown that as certain slowly varying parameters evolve, the separation distance alternates sign, indicating that both capture into, and escape from resonance occur. We find that although a measurable set of initial conditions enter into a sustained resonance, as the system further evolves the orientation of the manifolds reverses and many of these captured solutions will subsequently escape.     

Steady Groundwater Flow to Drains on a Sloping Bed: Comparison of Solutions Based on Boussinesq Equation and Richards Equation

Analytical expressions for the scaling factor (A) in the Wooding and Chapman (J Geophys Res 71:2895–2902, 1966) solution for steady-state flow to drains on a sloping bed are presented. Otherwise A needs to be obtained by matching numerical and solutions. Corrections to various errors in other analytical solutions are given. The HYDRUS2D numerical model was used to generate results for steady-state flow to drains on a sloping bed which were compared to published Hele-Shaw cell results. The numerical results were used to compute both the pressure head on the bottom and the height of the phreatic surface. The numerical results for maximum water-table height are almost exactly the same as the published Hele-Shaw cell results and are greater than the numerical values for the maximum pressure heads on the sloping base. These HYDRUS2D model results were then compared with various analytical solutions, and it was found that Towner’s (Water Resour Res 11:144–147, 1975) solution gave the best results for both estimation of the maximum height of the phreatic surface and the position on the slope where this occurs.     

On an Iterative Method for Approximate Solutions of a Generalized Boussinesq Model

An iterative method is proposed for finding approximate solutions of an initial and boundary value problem for a nonstationary generalized Boussinesq model for thermally driven convection of fluids with temperature dependent viscosity and thermal conductivity. Under certain conditions, it is proved that such approximate solutions converge to a solution of the original problem; moreover, convergence-rate bounds for the constructed approximate solutions are also obtained.     

Homoclinic Saddle-Node Bifurcations in Singularly Perturbed Systems

In this paper we study the creation of homoclinic orbits by saddle-node bifurcations. Inspired on similar phenomena appearing in the analysis of so-called localized structures in modulation or amplitude equations, we consider a family of nearly integrable, singularly perturbed three dimensional vector fields with two bifurcation parameters a and b. The O() perturbation destroys a manifold consisting of a family of integrable homoclinic orbits: it breaks open into two manifolds, W ^s() and W ^u(), the stable and unstable manifolds of a slow manifold . Homoclinic orbits to correspond to intersections W ^s()W ^u(); W ^s()W ^u()= for a<a*, a pair of 1-pulse homoclinic orbits emerges as first intersection of W ^s() and W ^u() as a>a*. The bifurcation at a=a* is followed by a sequence of nearby, O( ²(log)²) close, homoclinic saddle-node bifurcations at which pairs of N-pulse homoclinic orbits are created (these orbits make N circuits through the fast field). The second parameter b distinguishes between two significantly different cases: in the cooperating (respectively counteracting) case the averaged effect of the fast field is in the same (respectively opposite) direction as the slow flow on . The structure of W ^s()W ^u() becomes highly complicated in the counteracting case: we show the existence of many new types of sometimes exponentially close homoclinic saddle-node bifurcations. The analysis in this paper is mainly of a geometrical nature.     

Boussinesq方程的发展形式与Airy波理论差异性研究

应用势流理论，采用递推函数方法推导出一个新形式的Ｂｏｕｓｉｎｅｓｑ方程。通过对新方程的参数设置，可以讨论出Ｂｏｕｓｓｉｎｅｓｑ方程发展趋势和不同的发展形式。对浅水波动的描述方程，Ｂｏｕｓｓｉｎｅｓｑ方程的发展趋势为适用水深范围的拓展。拓展应用范围的大小则由其方程频散特征向Ａｉｒｙ波频散解逼近程度来决定。而Ｂｏｕｓｉｎｅｑ方程又不同于Ａｉｒｙ波，主要原因是Ｂｏｕｓｓｉｎｅｓｑ方程中含有线性频散项，Ａｉｒｙ波则只是长波首项近似，无线性频散项。其频散特征为精确的线性频散解。对实际水波传播而言，Ａｉｒｙ波理论的局限性是不言而喻的。     

Generalized potential flow theory and direct calculation of velocities applied to the numerical solution of the Navier-Stokes and the Boussinesq equations

A formulation based on three scalar functions or potentials is applied to analyse the Navier-Stokes and Boussinesq equations in three dimensions. In this formulation an explicit expression for the pressure exists, the so-called generalized Bernoulli equation. Therefore the scalar functions formulation may be considered as a generalization of the well-known potential flow and Bernoulli theory for irrotational fluid motion. The many advantages of this formulation applied to three-dimensional Navier-Stokes and Boussinesq flow will be discussed, and a numerical example is given as an illustration.     

Weak discontinuities in a chemically reacting atmosphere

Summary The growth and decay of a weak discontinuity headed by a singular surface of arbitrary shape in three dimensions is investigated in a chemically reacting atmosphere, in the absence of dissipative mechanisms such as viscosity, diffusion and heat conduction. The combined effects of the disequilibrium due to the chemical reaction and a wave front curvature on the propagation of discontinuities have been examined and discussed. It has been observed that the chemical disequilibrium, with its Arrhenius rate dependence, causes the compression wave to steepen more swiftly that it does in an inert atmosphere. The critical values of the initial discontinuity, and time for shock formation, in cases of diverging and converging waves, have been determined.     

Numerical analysis of time-dependent Boussinesq models

In this paper we analyse numerical models for time-dependent Boussinesq equations. These equations arise when so-called Boussinesq terms are introduced into the shallow water equations. We use the Boussinesq terms proposed by Katapodes and Dingemans. These terms generalize the constant depth terms given by Broer. The shallow water equations are discretized by using fourth-order finite difference formulae for the space derivatives and a fourth-order explicit time integrator. The effect on the stability and accuracy of various discrete Boussinesq terms is investigated. Numerical experiments are presented in the case of a fourth-order Runge-Kutta time integrator.     

Global Weak Solutions to Equations of Motion for Magnetic Fluids

We study the differential system governing the flow of an incompressible ferrofluid under the action of a magnetic field. The system consists of the Navier–Stokes equations, the angular momentum equation, the magnetization equation, and the magnetostatic equations. We prove, by using the Galerkin method, a global in time existence of weak solutions with finite energy of an initial boundary-value problem and establish the long-time behavior of such solutions. The main difficulty is due to the singularity of the gradient magnetic force.      

Solving a fully nonlinear highly dispersive Boussinesq model with mesh‐less least square‐based finite difference method

Combining mesh‐less finite difference method and least square approximation, a new numerical model is developed for water wave propagation model in two horizontal dimensions. In the numerical formulation of the method, the approximation of the unknown functions and their derivatives are constructed on a set of nodes in a local circular‐shaped region. The Boussinesq equations studied in this paper is a fully nonlinear and highly dispersive model, which is composed of the exact boundary conditions and the truncated series expansion solution of the Laplace equation. The resultant system involves a sparse, unsymmetrical matrix to be solved at each time step of the simulation. Matrix solutions are studied to reduce the computing resource requirements and improve the efficiency and accuracy. The convergence properties of the present numerical method are investigated. Preliminary verifications are given for nonlinear wave shoaling problems; the numerical results agree well with experimental data available in the literature. Copyright © 2006 John Wiley & Sons, Ltd.     

Partial Regularity of the Suitable Weak Solutions to the Multi-dimensional Incompressible Boussinesq Equations

This work is concerned with the partial regularity of the suitable weak solutions to the Boussinesq equations in \(\mathbb {R}^{n}\) where \(n=3,\,4\). By means of the De Giorgi iteration method developed in Vasseur (Nonlinear Differ Equ Appl 14(5–6):753–785, 2007), Wang, Wu (J Differ Equ 256(3):1224–1249, 2014), we obtain that \(n-2\) dimensional parabolic Hausdorff measure of the possible singular points set of the suitable weak solutions to this system is zero. Particularly, we obtain some interior regularity criteria only in terms of the scaled mixed norm of velocity for the suitable weak solutions to the Boussinesq equations, which implies that the potential singular points may only stem from the velocity field.