Symmetry-Breaking Bifurcation for Free Elastic Shell of Biological Cluster,Part 2.

We will be concerned with a two-dimensional mathematical model for a free elastic shell of biological cluster. The cluster boundary is connected with its kernel by elastic links. The inside part is filled with compressed gas or fluid. Equilibrium forms of the shell of biological cluster may be found as solutions of a certain nonlinear functional-differential equation with several physical parameters. For each multiparameter this equation has a radially symmetric solution. Our goal is to study the bifurcation which breaks symmetry. In order to establish critical values of bifurcation parameter and buckling modes we will investigate an appropriate linear problem. Our main result on the existence of symmetrybreaking bifurcation will be proved by the use of a variational version of the Crandall-Rabinowitz theorem.     

Exact solutions of a two-dimensional nonlinear Schrödinger equation

In the present study, we converted the resulting nonlinear equation for the evolution of weakly nonlinear hydrodynamic disturbances on a static cosmological background with self-focusing in a two-dimensional nonlinear Schrödinger (NLS) equation. Applying the function transformation method, the NLS equation was transformed to an ordinary differential equation, which depended only on one function ξ and can be solved. The general solution of the latter equation in ζ leads to a general solution of NLS equation. A new set of exact solutions for the two-dimensional NLS equation is obtained.     

Attractors of Navier-Stokes systems and of parabolic equations,and estimates for their dimensions

One investigates the problem of the existence of an attractor α of the semi-group S_t, generated by the solutions of the nonlinear nonstationary equations $$\frac{{\partial u}}{{\partial t}} = A(u), u|_{t = 0} = u_0 (x); S_t u_0 \equiv u(t)$$ . One proves a very general theorem on the existence of an attractor α of the semigroup S_t for t→∞. One gives examples of differential equations having attractors: a second-order quasilinear parabolic equation, a two-dimensional Navier—Stokes system, a monotone parabolic equation of any order. One proves a theorem on the finiteness of the Hausdorff dimension of the attractor α. One gives an estimate for the Hausdorff dimension of the attractor α for a two-dimensional Navier—Stokes system.     

The dynamics of the motion of a flat punch on the boundary of an elastic half-plane

The dynamic contact problem of the motion of a flat punch on the boundary of an elastic half-plane is considered. During motion, the punch deforms the elastic half-plane, penetrating it in such a manner that its base remains parallel to the boundary of the half-plane at each instant of time. In movable coordinates connected to the moving punch, the contact problem reduces to solving a two-dimensional integral equation, whose two-dimensional kernel depends on the difference between the arguments for each of the variables. An approximate solution of the integral equation of the problem is constructed in the form of a Neumann series, whose zeroth term is represented in the form of the superposition of the solutions of two-dimensional integral equations on the coordinate semiaxis minus the solution of the integral equation on the entire axis. This approach provides a way to construct the solution of the two-dimensional integral equation of the problem in four velocity ranges of motion of the punch, which cover the entire spectrum of its velocities, as well as to perform a detailed analysis of the special features of the contact stresses and vertical displacements of the free surface on the boundary of the contract area. An approximate method for solving the integral equation, which is based on a special approximation of the integrand of the kernel of the integral equation in the complex plane, is proposed for obtaining effective solutions of the problem that do not contain singular quadratures.     

Construction of an Inertial Manifold of a Parabolic Equation with a Monotonic Nonlinear Part

Sufficient conditions are obtained for the existence of an inertial manifold of a parabolic equation with a monotonic nonlinear part in a bounded region of not more than three dimensions. The existence of an inertial manifold is proved for a reaction diffusion equation viewed on a two-dimensional torus.     

A finite volume simulation model for saturated–unsaturated flow and application to Gooburrum,Bundaberg, Queensland,Australia

In this paper, a two-dimensional control volume finite-element computational model is developed for simulating saltwater intrusion in a heterogeneous coastal alluvial aquifer system at Gooburrum located near Bundaberg in Queensland, Australia. The model consists of a coupled system of two non-linear partial differential equations. The first equation describes the flow of a variable-density fluid, and the second equation describes the transport of dissolved salt via a form of the Fokker–Planck equation. The outcomes of the work demonstrate that transport simulation techniques provide excellent tools for hydraulic investigations even when complex transition zones are involved.     

A Fast, Bandlimited Solver for Scattering Problems in Inhomogeneous Media

The numerical treatment of two-dimensional scattering in inhomogeneous media is considered. A novel approach in treating convolution operators with low regularity is used to construct an iterative solver for the Lippmann-Schwinger integral equation. In this way, accurate approximations within a choice of bandwidth can be obtained in a rapid manner. The performance of the method is tested on a discontinuous scattering object for which the exact solution is known.     

Asymptotic behavior of the solutions of the Helmholtz equation in the caustic shadow domain. I

One makes use of the complex ray method in order to construct the uniform asymptotics of the wave field in the shadow zone beyond the caustic for the Helmholtz equation with an analytic refraction index. The complex eikonal is obtained as a result of the analytic continuation of the eikonal equation into the two-dimensional complex coordinate space. One considers a special example.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 128, pp. 172–185, 1983.The author expresses his gratitude to V. S. Buldyrev for the discussion of the results.     

Non-local symmetries of first-order equations

It is shown how one can transform scalar first-order ordinarydifferential equations which admit non-local symmetries of theexponential type to integrable equations admitting canonicalexponential non-local symmetries. As examples we invoke theAbel equation of the second kind, the Riccati equation and naturalgeneralizations of these. Moreover, our method describes howa double reduction of order for a second-order ordinary differentialequation which admits a two-dimensional Lie algebra of generatorsof point symmetries can be affected if the second-order equationis first reduced in order once by a symmetry which does notspan an ideal of the two-dimensional Lie algebra.     

Shallow water waves over an uneven bottom and an inhomogeneous KP equation

Three-dimensional shallow water waves over an uneven bottom are considered. The depth is assumed to be slow in variation. As a model, an inhomogeneous Kadomtsev-Petviashvili equation is presented. Some reductions of this equation are used to describe deformation of a line soliton due to the depth change. The model equation is valid for a wide class of two-dimensional nonlinear waves in inhomogeneous systems.     

On the numerical solution of a complete two-dimensional hypersingular integral equation by the method of discrete singularities

We study the solvability of a complete two-dimensional linear integral equation with a hypersingular integral understood in the sense of the Hadamard principal value. We justify the convergence of a quadrature-type numerical method for the case in which the equation in question is uniquely solvable. We present an application of the results to the numerical solution of the Neumann boundary value problem on a plane screen for the Helmholtz equation by the surface potential method.     

On the existence and regularity of solutions of a quasilinear mixed equation of Leray-Lions type

Our aim in this article is to study the existence and regularity of solutions of a quasilinear elliptic-hyperbolic equation. This equation appears in the design of blade cascade profiles. This leads to an inverse problem for designing two-dimensional channels with prescribed velocity distributions along channel walls. The governing equation is obtained by transformation of the physical domain to the plane defined by the streamlines and the potential lines of fluid. We establish an existence and regularity result of solutions for a more general framework which includes our physical problem as a specific example.     

Matrix Bernoulli equations. II

In this paper, we find sufficient conditions for the solvability by quadratures of J. Bernoulli’s equation defined over the set M ₂ of square matrices of order 2. We consider the cases when such equations are stated in terms of bases of a two-dimensional abelian algebra and a three-dimensional solvable Lie algebra over M ₂. We adduce an example of the third degree J. Bernoulli’s equation over a commutative algebra.     

A numerical scheme based on a solution of nonlinear advection–diffusion equations

A mathematical formulation of the two-dimensional Cole–Hopf transformation is investigated in detail. By making use of the Cole–Hopf transformation, a nonlinear two-dimensional unsteady advection–diffusion equation is transformed into a linear equation, and the transformed equation is solved by the spectral method previously proposed by one of the authors. Thus a solution to initial value problems of nonlinear two-dimensional unsteady advection–diffusion equations is derived. On the base of the solution, a numerical scheme explicit with respect to time is presented for nonlinear advection–diffusion equations. Numerical experiments show that the present scheme possesses the total variation diminishing properties and gives solutions with good quality.     

Galerkin and inertial manifold methods for finite-dimensional approximation of nonlocal equations of nonlinear optics

The article examines the class of models of nonlinear optical systems with two-dimensional feedback described by a nonlocal diffusion equation and a Schrödinger equation. Rate of convergence bounds are obtained for the Galerkin scheme with an eigenelement basis. The existence of a finite-dimensional inertial manifold with the global attraction property is established.     

相同模式内弧立波的强斜相互作用*

水文讨论分层流体中相同模式向孤立波的强斜相互作用,包括浅流体情形和深流体情形．采用Lazrange描述方法,发现在浅流体情形相互作用由KP方程描述;在深流体情形相互作用由二维的中等长波方程描述;在无限深情形相互作用由二维的BO方程描述．     

Boundary Control of PDEs via Curvature Flows: the View from the Boundary, II

We describe some results on the exact boundary controllability of the wave equation on an orientable two-dimensional Riemannian manifold with nonempty boundary. If the boundary has positive geodesic curvature, we show that the problem is controllable in finite time if (and only if) there are no closed geodesics in the interior of the manifold. This is done by solving a parabolic problem to construct a convex function. We exhibit an example for which control from a subset of the boundary is possible, but cannot be proved by means of convex functions. We also describe a numerical implementation of this method.     

Dynamic crack propagation in a 2D elastic body. The out-of plane case

We discuss the propagation of a running crack under shear waves in a rigorous mathematical way for a simplified model. This model is described by two coupled equations in the actual configuration: a two-dimensional scalar wave equation in a cracked bounded domain and an ordinary differential equation derived from an energy balance law. The unknowns are the displacement fields u = u (y, t) and the one-dimensional crack tip trajectory h = h (t). We handle both equations separately, assuming at first that the crack position is known. Existence and uniqueness of strong solutions of the wave equation are studied and the crack-tip singularities are derived under the assumption that the crack is straight and moves tangentially. Using an energy balance law and the crack tip behaviour of the displacement fields we finally arrive at an ordinary differential equation for h (t), called equation of motion for the crack tip. We demonstrate the crack-tip motion with corresponding nonuniformly crack speed by numerical simulations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation

Summary We describe a quadrature method for the numerical solution of the logarithmic integral equation of the first kind arising from the single-layer approach to the Dirichlet problem for the two-dimensional Helmholtz equation in smooth domains. We develop an error analysis in a Sobolev space setting and prove fast convergence rates for smooth boundary data.     

Two-dimensional numerical modelling of shallow water flows over multilayer movable beds

The two-dimensional modelling of shallow water flows over multi-sediment erodible beds is presented. A novel approach is developed for the treatment of multiple sediment types in morphodynamics. The governing equations include the two-dimensional shallow water equations for hydrodynamics, an Exner-type equation for morphodynamics, a two-dimensional transport equation for the suspended sediments, and a set of empirical equations for entrainment and deposition. Multilayer sedimentary beds are formed of different erodible soils with sediment properties and new exchange conditions between the bed layers are developed for the model. The coupled equations yield a hyperbolic system of balance laws with source terms. As a numerical solver for the system, we implement a fast finite volume characteristics method. The numerical fluxes are reconstructed using the method of characteristics which employs projection techniques. The proposed finite volume solver is simple to implement, satisfies the conservation property and can be used for two-dimensional sediment transport problems in non-homogeneous isotropic beds without need of complicated three-dimensional equations. To assess the performance of the proposed models, we present numerical results for a wide variety of shallow water flows over sedimentary layers. Comparisons to experimental data for dam-break problems over movable beds are also included in this study.