Several classes of exact solutions to the 1 + 1 Born–Infeld equation

We obtain closed-form exact solutions to the 1 + 1 Born–Infeld equation arising in nonlinear electrodynamics. In particular, we obtain general traveling wave solutions of one wave variable, solutions of two wave variables, similarity solutions, multiplicatively separable solutions, and additively separable solutions. Then, putting the Born–Infeld model into correspondence with the minimal surface equation using a Wick rotation, we are able to construct complex helicoid solutions, transformed catenoid solutions, and complex analogues of Scherk’s first and second surfaces. Some of the obtained solutions are new, whereas others are generalizations of solutions in the literature. These exact solutions demonstrate the fact that solutions to the Born–Infeld model can exhibit a variety of behaviors. Exploiting the integrability of the Born–Infeld equation, the solutions are constructed elegantly, without the need for complicated analytical algorithms.     

Stochastic Evolution of Augmented Born–Infeld Equations

This paper compares the results of applying a recently developed method of stochastic uncertainty quantification designed for fluid dynamics to the Born–Infeld model of nonlinear electromagnetism. The similarities in the results are striking. Namely, the introduction of Stratonovich cylindrical noise into each of their Hamiltonian formulations introduces stochastic Lie transport into their dynamics in the same form for both theories. Moreover, the resulting stochastic partial differential equations retain their unperturbed form, except for an additional term representing induced Lie transport by the set of divergence-free vector fields associated with the spatial correlations of the cylindrical noise. The explanation for this remarkable similarity lies in the method of construction of the Hamiltonian for the Stratonovich stochastic contribution to the motion in both cases, which is done via pairing spatial correlation eigenvectors for cylindrical noise with the momentum map for the deterministic motion. This momentum map is responsible for the well-known analogy between hydrodynamics and electromagnetism. The momentum map for the Maxwell and Born–Infeld theories of electromagnetism treated here is the 1-form density known as the Poynting vector. Two appendices treat the Hamiltonian structures underlying these results.

     

Multiple Vortices in the Aharony–Bergman–Jafferis–Maldacena Model

Vortices in non-Abelian gauge field theory play important roles in confinement mechanism and are governed by systems of nonlinear elliptic equations of complicated structures. In this paper, we present a series of existence and uniqueness theorems for multiple vortex solutions of the BPS vortex equations, arising in the dual-layered Chern–Simons field theory developed by Aharony, Bergman, Jafferis, and Maldacena, over ${\mathbb{R}^2}$ and on a doubly periodic domain. In the full-plane setting, we show that the solution realizing a prescribed distribution of vortices exists and is unique. In the compact setting, we show that a solution realizing n prescribed vortices exists over a doubly periodic domain ${\Omega}$ if and only if the condition $$n < \frac{\lambda |\Omega|}{2 \pi}$$ holds, where ${\lambda >0 }$ is the Higgs coupling constant. In this case, if a solution exists, it must be unique. Our methods are based on calculus of variations.     

Born–Infeld solitons,maximal surfaces,and Ramanujan’s identities

We show that a Born–Infeld soliton can be realised either as a spacelike minimal graph or timelike minimal graph over a timelike plane or a combination of both away from singular points. We also obtain some exact solutions of the Born–Infeld equation from already known solutions to the maximal surface equation. Further we present a method to construct a one parameter family of complex solitons from a given one parameter family of maximal surfaces. Finally, using Ramanujan’s identities and the Weierstrass–Enneper representation of maximal surfaces, we derive further non-trivial identities.     

Existence and non‐existence of steady states to a cross‐diffusion system arising in a Leslie predator–prey model

This paper is concerned with a cross‐diffusion system arising in a Leslie predator–prey population model in a bounded domain with no flux boundary condition. We investigate sufficient condition for the existence and the non‐existence of non‐constant positive solution. We obtain that if natural diffusion coefficient of predator is large enough and cross‐diffusion coefficients are fixed, then under some conditions there exists non‐constant positive solution. Furthermore, we show that if natural diffusion coefficients of predator and prey are both large enough, and cross‐diffusion coefficients are small enough, then there exists no non‐constant positive solution. Copyright © 2012 John Wiley & Sons, Ltd.     

Global solvability for a large flux of a three‐dimensional time‐dependent Navier–Stokes problem in a straight cylinder

The unique global existence of a solution to nonstationary Navier–Stokes system with prescribed nonzero flux F(t) in an infinite three‐dimensional pipe is proved. The obtained solution remains close to the corresponding nonstationary Poiseuille flow. Moreover, it converges to the Poiseuille flow as |x₃|→∞. Copyright © 2008 John Wiley & Sons, Ltd.     

An impulsive diffusion predator-prey system in three-species with Beddington-DeAngelis response

This paper is purported to investigate an impulsive diffusion predator-prey system in three-species with Beddington-DeAngelis response in a bounded domain with no flux boundary condition. Employing the upper and lower solution method and compare theory, we investigate the sufficient conditions of the upper ultimate boundedness and permanence of this system which imply that impulse always changes the situation of survival for species. The conditions for the existence of unique globally stable positive periodic solution are also obtained.     

On the shape operator of surfaces in space forms

Necessary and sufficient conditions are studied for the existence of surfaces in space forms with prescribed shape operator S. We consider three cases: that of the non-existence of a solution, the case of a unique solution and that of more than one solution. Examples are given for local and global S-deformations and S-rigidity.     

Γ-Convergence of 2D Ginzburg-Landau functionals with vortex concentration along curves

We study the variational convergence of a family of twodimensional Ginzburg-Landau functionals arising in the study of superfluidity or thin-film superconductivity as the Ginzburg-Landau parameter ε tends to 0. In this regime and for large enough applied rotations (for superfluids) or magnetic fields (for superconductors), the minimizers acquire quantized point singularities (vortices). We focus on situations in which an unbounded number of vortices accumulate along a prescribed Jordan curve or a simple arc in the domain. This is known to occur in a circular annulus under uniform rotation, or in a simply connected domain with an appropriately chosen rotational vector field. We prove that if suitably normalized, the energy functionals Γ-converge to a classical energy from potential theory. Applied to global minimizers, our results describe the limiting distribution of vortices along the curve in terms of Green equilibrium measures.     

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.     

On boundary and initial values of solutions of a second-order parabolic equation that degenerate on the domain boundary

Properties of the solutions of a parabolic equation in the case of a Keldysh-type degeneracy on the boundary of the domain are investigated. The unique solvability of the first mixed problem for this equation is studied. Necessary and sufficient conditions for the existence of limits of the solution on the lateral surface of a cylindrical domain and on its lower base in L₂-type spaces are found.     

时标上具Allee效应的动力学方程的概周期解

本文研究了时标上具Allee效应的可再生资源动力学方程的概周期解的存在性与稳定性.利用线性系统指数二分性与压缩映射不动点定理,得到了方程存在唯一概周期解的充分条件.此外,通过构建适当的Laypunov函数,得到了概周期解是全局指数稳定的充分条件.     

Global strong solution to the three-dimensional density-dependent incompressible magnetohydrodynamic flows

An initial-boundary value problem is considered for the density-dependent incompressible viscous magnetohydrodynamic flow in a three-dimensional bounded domain. The homogeneous Dirichlet boundary condition is prescribed on the velocity, and the perfectly conducting wall condition is prescribed on the magnetic field. For the initial density away from vacuum, the existence and uniqueness are established for the local strong solution with large initial data as well as for the global strong solution with small initial data. Furthermore, the weak-strong uniqueness of solutions is also proved, which shows that the weak solution is equal to the strong solution with certain initial data.     

Modeling and analysis of reactive solute transport in deformable channels with wall adsorption–desorption

We show well posedness for a model of nonlinear reactive transport of chemical in a deformable channel. The channel walls deform due to fluid–structure interaction between an unsteady flow of an incompressible, viscous fluid inside the channel and elastic channel walls. Chemical solutes, which are dissolved in the viscous, incompressible fluid, satisfy a convection–diffusion equation in the bulk fluid, while on the deforming walls, the solutes undergo nonlinear adsorption–desorption physico‐chemical reactions. The problem addresses scenarios that arise, for example, in studies of drug transport in blood vessels. We show the existence of a unique weak solution with solute concentrations that are non‐negative for all times. The analysis of the problem is carried out in the context of semi‐linear parabolic PDEs on moving domains. The arbitrary Lagrangian–Eulerian approach is used to address the domain movement, and the Galerkin method with the Picard–Lindelöf theorem is used to prove existence and uniqueness of approximate solutions. Energy estimates combined with the compactness arguments based on the Aubin–Lions lemma are used to prove convergence of the approximating sequences to the unique weak solution of the problem. It is shown that the solution satisfies the positivity property, that is, that the density of the solute remains non‐negative at all times, as long as the prescribed fluid domain motion is ‘reasonable’. This is the first well‐posedness result for reactive transport problems defined on moving domains of this type. Copyright © 2015 John Wiley & Sons, Ltd.     

One-parameter family of solitons from minimal surfaces

In this paper, we discuss a one parameter family of complex Born–Infeld solitons arising from a one parameter family of minimal surfaces. The process enables us to generate a new solution of the B–I equation from a given complex solution of a special type (which are abundant). We illustrate this with many examples. We find that the action or the energy of this family of solitons remains invariant in this family and find that the well-known Lorentz symmetry of the B–I equations is responsible for it.     

Analysis of fluid flow and heat transfer over an unsteady stretching surface

This article considers two situations involving unsteady laminar boundary layer flow due to a stretching surface in a quiescent viscous incompressible fluid. In one configuration, the surface is impermeable with prescribed heat flux, in the other, the surface is permeable with prescribed temperature. The boundary value problems governing a similarity reduction for each of these situations are investigated and the existence of a solution is proved for all relevant values of physical parameters. The uniqueness of the solution is also proved for some (but not all) values of the parameters. Finally, a priori bounds are obtained for the skin friction coefficient and local Nusselt number.     

Blow-up Surfaces for Nonlinear Wave Equations,I

We introduce a systematic procedure for reducing nonlinear wave equations to characteristic problems of Fuchsian type. This reduction is combined with an existence theorem to produce solutions blowing up on a prescribed hypersurface. This first part develops the procedure on the example □u = exp(u); we find necessary and sufficient conditions for the existence of a solution of the form ln(2/?²) + v, where {? = 0} is the blow-up surface, and v is analytic. This gives a natural way of continuing solutions after blow-up.     

Dynamics of Extended Objects and a Stationary Axially Symmetric Cosmological Model

We find a new solution describing a homogeneous stationary axially symmetric model, which, in contrast to the Gödel model, does not contain closed timelike lines. We find exact solutions corresponding to the motion of a null string in the rotating Universe and show that these solutions crucially depend on the initial data. To obtain more detailed information on the cosmic string dynamics, we performed numerical simulations indicating essential differences in the behavior of strings and null strings in the presence of global rotation of the Universe. These numerical solutions show that the string manifests involved oscillations, varies its shape with the appearance of loops and cusps, and twists into a spiral.     

Global strong solutions to the Shliomis system for ferrofluids in a bounded domain

We consider the partial differential equations proposed by Shliomis to model the dynamics of an incompressible viscous ferrofluid submitted to an external magnetic field. The Shliomis system consists of the incompressible Navier‐Stokes equations, the magnetization equations, and the magnetostatic equations. The magnetization equations is of Bloch type, and no regularizing term is added. We prove the global existence of unique strong solution to the initial boundary value problem for the system in a bounded domain, with the small initial data and external magnetic field but without any restrictions on the physical parameters. The novelty of the analysis is to introduce a linear combination of magnetic fields.