Exact solutions for fractional DDEs via auxiliary equation method coupled with the fractional complex transform

Dynamical behavior of many nonlinear systems can be described by fractional‐order equations. This study is devoted to fractional differential–difference equations of rational type. Our focus is on the construction of exact solutions by means of the (G'/G)‐expansion method coupled with the so‐called fractional complex transform. The solution procedure is elucidated through two generalized time‐fractional differential–difference equations of rational type. As a result, three types of discrete solutions emerged: hyperbolic, trigonometric, and rational. Copyright © 2016 John Wiley & Sons, Ltd.     

Symmetrisers and generalised solutions for strictly hyperbolic systems with singular coefficients

This paper is devoted to strictly hyperbolic systems and equations with non‐smooth coefficients. Below a certain level of smoothness, distributional solutions may fail to exist. We construct generalised solutions in the Colombeau algebra of generalised functions. Extending earlier results on symmetric hyperbolic systems, we introduce generalised strict hyperbolicity, construct symmetrisers, prove an appropriate Gårding inequality and establish existence, uniqueness and regularity of generalised solutions. Under additional regularity assumptions on the coefficients, when a classical solution of the Cauchy problem (or of a transmission problem in the piecewise regular case) exists, the generalised solution is shown to be associated with the classical solution (or the piecewise classical solution satisfying the appropriate transmission conditions).     

Perturbation index of linear partial differential-algebraic equations with a hyperbolic part

This paper deals with linear partial differential-algebraic equations (PDAEs) which have a hyperbolic part. If the spatial differential operator satisfies a Gårding-type inequality in a suitable function space setting, a perturbation index can be defined. Theoretical and practical examples are considered.     

Introductory Statement

This work continues the account given in Part I of the paper¹ by presenting a short summary of some of the mathematical techniques employed in the wave front analysis of quasi‐linear hyperbolic partial differential equations. Starting from a number of important physical examples, the classification of quasi‐linear first‐order systems is discussed and followed by a simple account of the theory of characteristics for systems involving n dependent and two independent variables. A special example is discussed showing how discontinuities arise in solutions, and the paper is concluded by an account of wave front analysis as applied to the piston problem of gas dynamics.     

Complete minimal graphs with prescribed asymptotic boundary on rotationally symmetric Hadamard surfaces

It is proved that if M is a rotationally symmetric Hadamard surface which is conformally equivalent to the hyperbolic disk then the asymptotic Dirichlet problem for the minimal surface equation is uniquely solvable for any continuous asymptotic boundary data. This result gives a partial answer of a question in Gálvez and Rosenberg (Am J Math 132:1249?C1273, 2010) about the existence of entire minimal graphs on Hadamard surfaces with sectional curvature possibly degenerating at infinity.     

非线性耦合Klein-Gordon方程组和耦合Schr\"{o}dinger-Boussinesq方程组的双行波解

本文提出了一种全新复合$(\frac{G''}{G})$展开方法,运用这种新方法并借助符号计算软件构造了非线性耦合Klein-Gordon方程组和耦合Schr\"{o}dinger-Boussinesq方程组的多种双行波解,包括双双曲正切函数解,双正切函数解,双有理函数解以及它们的混合解. 复合$(\frac{G''}{G})$展开方法不但直接有效地求出了两类非线性偏微分方程的双行波解,而且扩大了解的范围.这种新方法对于研究非线性偏微分方程具有广泛的应用意义.     

The Stability of Large‐Amplitude Shallow Interfacial Non‐Boussinesq Flows

The system of equations describing the shallow‐water limit dynamics of the interface between two layers of immiscible fluids of different densities is formulated. The flow is bounded by horizontal top and bottom walls. The resulting equations are of mixed type: hyperbolic when the shear is weak and the behavior of the system is internal‐wave like, and elliptic for strong shear. This ellipticity, or ill‐posedness is shown to be a manifestation of large‐scale shear instability. This paper gives sharp nonlinear stability conditions for this nonlinear system of equations. For initial data that are initially hyperbolic, two different types of evolution may occur: the system may remain hyperbolic up to internal wave breaking, or it may become elliptic prior to wave breaking. Using simple waves that give a priori bounds on the solutions, we are able to characterize the condition preventing the second behavior, thus providing a long‐time well‐posedness, or nonlinear stability result. Our formulation also provides a systematic way to pass to the Boussinesq limit, whereby the density differences affect buoyancy but not momentum, and to recover the result that shear instability cannot occur from hyperbolic initial data in that case.     

(G'/G²)-expansion Solutions to MBBM and OBBM Equations

In this paper,with the aid of symbolic computation, themodified Benjamin- Bona-Mahony and Ostrovsky-Benjamin-Bona-Mahony equations are investigated by extended (G'/G2)-expansion method. As a consequence, some trigonometric, hyperbolic and rational function solutions with multiple arbitrary parameters for the two equations are revealed, which helps to illustrate the effectiveness of this method.     

On the existence of special solutions of the generalized Davey–Stewartson system

In this note we show that for certain choice of parameters the hyperbolic–elliptic–elliptic generalized Davey–Stewartson system admits time-dependent travelling wave solutions of the kind given in [V.A. Arkadiev, A.K. Pogrebkov, M.C. Polivanov, Inverse scattering transform method and soliton solutions for Davey–Stewartson II equation, Physica D 36 (1989) 189–197] for the hyperbolic Davey–Stewartson system. These solutions lead to radial solutions as well. We also find the sufficient conditions for non-existence of travelling wave solutions for the hyperbolic–elliptic–elliptic generalized Davey–Stewartson system by using the point of view developed in [A. Eden, T.B. Gürel, E. Kuz, Focusing and defocusing cases of the purely elliptic generalized Davey–Stewartson system, IMA J. Appl. Math. (in press)].     

On Nonlinear Stabilization of Linearly Unstable Maps

We examine the phenomenon of nonlinear stabilization, exhibiting a variety of related examples and counterexamples. For Gâteaux differentiable maps, we discuss a mechanism of nonlinear stabilization, in finite and infinite dimensions, which applies in particular to hyperbolic partial differential equations, and for Fréchet differentiable maps with linearized operators that are normal, we give a sharp criterion for nonlinear exponential instability at the linear rate. These results highlight the fundamental open question whether Fréchet differentiability is sufficient for linear exponential instability to imply nonlinear exponential instability, at possibly slower rate.     

Modelling traffic flow with a time‐dependent fundamental diagram

We model traffic flow with a time‐dependent fundamental diagram. A time‐dependent fundamental diagram arises naturally from various factors such as weather conditions, traffic jam or modern traffic congestion managements, etc. The model is derived from a car‐following model which takes into account the situation changes over the time elapsed time. It is a system of non‐concave hyperbolic conservation laws with time‐dependent flux and the sources. The global existence and uniqueness of the solution to the Cauchy problem is established under the condition that the variation in time of the fundamental diagram is bounded. The zero relaxation limit of the solutions is found to be the unique entropy solution of the equilibrium equation. Copyright © 2004 John Wiley & Sons, Ltd.     

Exact travelling wave solutions for nonlinear Schr\"{o}dinger equation with variable coefficients

In this paper, two nonlinear Schr\"{o}dinger equations with variable coefficients in nonlinear optics are investigated. Based on travelling wave transformation and the extended $(\frac{G''}{G})$-expansion method, exact travelling wave solutions to nonlinear Schr\"{o}dinger equation with time-dependent coefficients are derived successfully, which include bright and dark soliton solutions, triangular function periodic solutions, hyperbolic function solutions and rational function solutions.     

Semiclassical resonances associated with a periodic orbit

We consider resonances for a h-pseudo-differential operator H(x, hD _x; h) induced by a periodic orbit of hyperbolic type. We generalize the framework of Gérard and Sjöstrand, in the sense that we allow hyperbolic and elliptic eigenvalues of the Poincarémap, and look for so-called semi-excited resonances with imaginary part of magnitude ?h log h, or h ^δ, with 0 < δ < 1.     

Attractor for a conserved phase‐field system with hyperbolic heat conduction

We consider a conserved phase‐field system of Caginalp type, characterized by the assumption that both the internal energy and the heat flux depend on the past history of the temperature and its gradient, respectively. The latter dependence is a law of Gurtin–Pipkin type, so that the equation ruling the temperature evolution is hyperbolic. Thus, the system consists of a hyperbolic integrodifferential equation coupled with a fourth‐order evolution equation for the phase‐field. This model, endowed with suitable boundary conditions, has already been analysed within the theory of dissipative dynamical systems, and the existence of an absorbing set has been obtained. Here we prove the existence of the universal attractor. Copyright © 2004 John Wiley & Sons, Ltd.     

Convergence analysis for parallel‐in‐time solution of hyperbolic systems

Parallel‐in‐time algorithms have been successfully employed for reducing time‐to‐solution of a variety of partial differential equations, especially for diffusive (parabolic‐type) equations. A major failing of parallel‐in‐time approaches to date, however, is that most methods show instabilities or poor convergence for hyperbolic problems. This paper focuses on the analysis of the convergence behavior of multigrid methods for the parallel‐in‐time solution of hyperbolic problems. Three analysis tools are considered that differ, in particular, in the treatment of the time dimension: (a) space–time local Fourier analysis, using a Fourier ansatz in space and time; (b) semi‐algebraic mode analysis, coupling standard local Fourier analysis approaches in space with algebraic computation in time; and (c) a two‐level reduction analysis, considering error propagation only on the coarse time grid. In this paper, we show how insights from reduction analysis can be used to improve feasibility of the semi‐algebraic mode analysis, resulting in a tool that offers the best features of both analysis techniques. Following validating numerical results, we investigate what insights the combined analysis framework can offer for two model hyperbolic problems, the linear advection equation in one space dimension and linear elasticity in two space dimensions.     

Abundant explicit exact solutions to the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law nonlinearities

This paper is concerned with the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law. Abundant explicit and exact solutions of the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law are derived uniformly by using the first integral method. These exact solutions are include that of extended hyperbolic function solutions, periodic wave solutions of triangle functions type, exponential form solution, and complex hyperbolic trigonometric function solutions and so on. The results obtained confirm that the first integral method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial DEs. Copyright © 2014 John Wiley & Sons, Ltd.     

The Riccati Equation Method Combined with the Generalized Extended $(G'/G)$-Expansion Method for Solving the Nonlinear KPP Equation

An analytic study of the nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation is presented in this paper. The Riccati equation method combined with the generalized extended $(G''/G)$-expansion method is an interesting approach to find more general exact solutions of the nonlinear evolution equations in mathematical physics. We obtain the traveling wave solutions involving parameters, which are expressed by the hyperbolic and trigonometric function solutions. When the parameters are taken as special values, the solitary and periodic wave solutions are given. Comparison of our new results in this paper with the well-known results are given.     

A parabolic variant of H-measures

H-measures, as originally introduced by Luc Tartar and Patrick Gérard, are suited to hyperbolic problems. However, they turned out not to be well adjusted to the study of parabolic equations. A variant of H-measures is proposed, which is much better adapted to such kind of problems. We present the new parabolic scaling and the main ingredients for the proof of existence of the new variant. Some applications to the Schrödinger equation and vibrating plate equation are shown, together with an outlook to possible applications in other problems.     

A brief report on a number of recently discovered sets of notes on Riemann's lectures and on the transmission of the Riemann Nachlass

The collection of Riemann's mathematical papers preserved in Göttingen University Library since 1895 includes none of Riemann's scientific correspondence nor any of his more personal papers. The present report gives an account of the documents (correspondence, lecture notes, etc.) discovered outside Göttingen in the course of a larger research project on Riemann, and briefly describes the history of the Riemann Nachlass. At the same time, readers are kindly requested to inform the author of the whereabouts of any further material relating to Riemann, so that it can be included in the collection of texts and sources currently in preparation.     

An integral equation method for the electromagnetic scattering from cavities

Consider a time‐harmonic electromagnetic plane wave incident on a cavity in a ground plane. The physical process is modelled by Maxwell's equations. In this paper, integral representations of the solutions to the model problem in both fundamental polarizations are derived and studied. Existence and uniqueness of the solutions for the integral equations are established. The integral equations approach forms a basis for numerical solution of the model problem. In particular, for each fundamental polarization, an integral formulation with Gårding‐type estimates is derived. These formulations provide a basis for variational boundary element methods for solving the cavity problem. The Gårding‐type estimates imply convergence results for conforming boundary element methods. Copyright © 2000 John Wiley & Sons, Ltd