Finite volume schemes for dispersive wave propagation and runup

Finite volume schemes are commonly used to construct approximate solutions to conservation laws. In this study we extend the framework of the finite volume methods to dispersive water wave models, in particular to Boussinesq type systems. We focus mainly on the application of the method to bidirectional nonlinear, dispersive wave propagation in one space dimension. Special emphasis is given to important nonlinear phenomena such as solitary waves interactions, dispersive shock wave formation and the runup of breaking and non-breaking long waves.     

Interactions of Soliton Waves for a Generalized Discrete KdV Equation

It is well known that soliton interactions in discrete integrable systems often possess new properties which are different from the continuous integrable systems, e.g., we found that there are such discrete solitons in a semidiscrete integrable system(the time variable is continuous and the space one is discrete) that the shorter solitary waves travel faster than the taller ones. Very recently, this kind of soliton was also observed in a full discrete generalized Kd V system(the both of time and space variables are discrete) introduced by Kanki et al. In this paper, for the generalized discrete Kd V(gd Kd V) equation, we describe its richer structures of one-soliton solutions. The interactions of two-soliton waves to the gd Kd V equation are studied. Some new features of the soliton interactions are proposed by rigorous theoretical analysis.     

Nonlinear wave propagation in discrete and continuous systems

In this review we try to capture some of the recent excitement induced by a large volume of theoretical and computational studies addressing nonlinear Schrödinger models (discrete and continuous) and the localized structures that they support. We focus on some prototypical structures, namely the breather solutions and solitary waves. In particular, we investigate the bifurcation of travelling wave solution in Discrete NLS system applying dynamical systems methods. Next, we examine the combined effects of cubic and quintic terms of the long range type in the dynamics of a double well potential. The relevant bifurcations, the stability of the branches and their dynamical implications are examined both in the reduced (ODE) and in the full (PDE) setting. We also offer an outlook on interesting possibilities for future work on this theme.     

Discrete exterior geometry approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems

This paper addresses the issue of structure-preserving discretization of open distributed-parameter systems with Hamiltonian dynamics. Employing the formalism of discrete exterior calculus, we introduce a simplicial Dirac structure as a discrete analogue of the Stokes–Dirac structure and demonstrate that it provides a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. The spatial domain, in the continuous theory represented by a finite-dimensional smooth manifold with boundary, is replaced by a homological manifold-like simplicial complex and its augmented circumcentric dual. The smooth differential forms, in discrete setting, are mirrored by cochains on the primal and dual complexes, while the discrete exterior derivative is defined to be the coboundary operator. This approach of discrete differential geometry, rather than discretizing the partial differential equations, allows to first discretize the underlying Stokes–Dirac structure and then to impose the corresponding finite-dimensional port-Hamiltonian dynamics. In this manner, a number of important intrinsically topological and geometrical properties of the system are preserved.     

Nonlinear waves over highly variable topography

This paper reviews the authors' recent work on water waves in heterogeneous media, namely nonlinear reduced models for water waves propagating over a region of highly variable depth. Both surface and internal waves are considered. Through different strategies for the mathematical modeling, at the level of equations, we show how diferent types of water wave models (namely systems of partial differential equations) can arise: weakly or fully dispersive, weakly or strongly nonlinear. The applications for these types of long waves are usually from Geophysics: in particular oceanography and meteorology. Here the emphasis is on coastal waves related to Physical Oceanography. An overview of the mathematical formulation is presented together with different asymptotic simplifications at the level of the partial differential equations (PDEs). References for recent work on the asymptotic analysis of solutions is also provided. These describe the regime where long pulse shaped waves propagate over rapidly varying topographic heterogeneities, which are modeled through rapidly varying coefficients in the PDEs.     

一类非线性色散方程中的新型奇异孤立波

研究一类非线性色散广义DGH方程的新型奇异孤立波及其Painlevé可积性.利用Painlevé分析发现当对流项强度m=2时广义DGH方程是可积的,这是一个新的可积方程.通过构造新的变量代换以及auto-Backlund变换获得该方程丰富的奇异孤立波解,如紧孤立波（compacton）、尖峰孤立波（peakon）、新型带尖点的双孤立波和带爆破点的双孤立波等. 关键词： 非线性色散方程 可积性 奇异孤立波     

A new approach for seeking coefficient function solutions of conformable fractional partial differential equations based on the Jacobi elliptic equation

In this paper, we present an approach for seeking exact solutions with coefficient function forms of conformable fractional partial differential equations. By a combination of an under-determined fractional transformation and the Jacobi elliptic equation, exact solutions with coefficient function forms can be obtained for fractional partial differential equations. The innovation point of the present approach lies in two aspects. One is the fractional transformation, which involve the traveling wave transformations used by many articles as special cases. The other is that more general exact solutions with coefficient function forms can be found, and traveling wave solutions with constants coefficients are only special cases of our results. As of applications, we apply this method to the space-time fractional (2+1)-dimensional dispersive long wave equations and the time fractional Bogoyavlenskii equations. As a result, some exact solutions with coefficient function forms for the two equations are successfully found.     

High-speed photonically assisted analog-to-digital conversion using a continuous wave multiwavelength source and phase modulation

A more simple photonically assisted analog-to-digital conversion system utilizing a cw multiwavelength source and phase modulation instead of a mode-locked laser is presented. The output of the cw multiwavelength source is launched into a dispersive device (such as a single-mode fiber). This fiber creates a pulse train, where the central wavelength of each pulse corresponds to a spectral line of the optical source. The pulses can then be either dispersed again to perform discrete wavelength time stretching or demultiplexed for continuous time analog-to-digital conversion. We experimentally demonstrate the operation of both time stretched and interleaved systems at 38 GHz. The potential of integrating this type of system on a monolithic chip is discussed.     

Soliton interactions,Bäcklund transformations,Lax pair for a variable-coefficient generalized dispersive water-wave system

Under investigation in this paper is a variable-coefficient generalized dispersive water-wave system, which can simulate the propagation of the long weakly non-linear and weakly dispersive surface waves of variable depth in the shallow water. Under certain variable-coefficient constraints, by virtue of the Bell polynomials, Hirota method and symbolic computation, the bilinear forms, one- and two-soliton solutions are obtained. Bäcklund transformations and new Lax pair are also obtained. Our Lax pair is different from that previously reported. Based on the asymptotic and graphic analysis, with different forms of the variable coefficients, we find that there exist the elastic interactions for u, while either the elastic or inelastic interactions for v, with u and v as the horizontal velocity field and deviation height from the equilibrium position of the water, respectively. When the interactions are inelastic, we see the fission and fusion phenomena.     

Doubly periodic waves of a discrete nonlinear Schrödinger system with saturable nonlinearity

Abstract

A system of two discrete nonlinear Schrödinger equations of the Ablowitz-Ladik type with a saturable nonlinearity is shown to admit a doubly periodic wave, whose long wave limit is also derived. As a by-product, several new solutions of the elliptic type are provided for NLS-type discrete and continuous systems.     

Dynamic study and applications of metamaterial systems

We investigate the dynamic characteristics of metamaterial systems, such as the temporal coherence gain of the superlens, the causality limitation on the ideal cloaking systems, the relaxation process and essential elements in the dispersive cloaking systems, and the extending of the working frequency range of cloaking systems. The key point of our study is the physical dispersive properties of metamaterials, which are well-known to be intrinsically strongly dispersive. With physical dispersion, new physical pictures can be obtained for the waves propagating inside metamaterial, such as the “group retarded time” for waves inside the superlens and cloak, the causality limitation on real metamaterial systems, and the essential elements for design optimization. Therefore, we believe the dynamic study of metamaterials will be an important direction for further research. All theoretical derivations and conclusions are demonstrated by powerful finite-difference time-domain simulations.     

Wave propagation phenomena from a spatiotemporal viewpoint: Resonances and bifurcations

By using biorthogonal decompositions, we show how uniformly propagating waves, togehter with their velocity, shape, and amplitude, can be extracted from a spatiotemporal signal consisting of the superposition of various traveling waves. The interaction between the different waves manifests itself in space-time resonances in case of a discrete biorthogonal spectrum and in resonant wavepackets in case of a continuous biorthogonal spectrum. Resonances appear as invariant subspaces under the biorthogonal operator, which leads to closed sets of algebraic equations. The analysis is then extended to superpositions of dispersive waves for which the (Fourier) dispersion relation is no longer linear. We then show how a space-time bifurcation, namely a qualitative change in the spatiotemporal nature of the solution, occurs when the biorthogonal operator is a nonholomorphic function of a parameter. This takes place when two eigenvalues are degenerate in the biorthogonal spectrum and when the spatial and temporal eigenvectors rotate within each eigenspace. Such a scenario applied to the superposition of traveling waves leads to the generation of additional waves propagating at new velocities, which can be computed from the spatial and temporal eigenmodes involved in the process (namely the shape of the propagating waves slightly before the bifurcation). An eigenvalue degeneracy, however, does not necessarily lead to a bifurcation, a situation we refer to as being self-avoiding. We illustrate our theoretical predictions by giving examples of bifurcating and self-avoiding events in propagating phenomena.     

Noether's theory of Lagrange systems in discrete case

In this paper, Noether theory of Lagrange systems in discrete case are studied. First, we briefly overview the well-known Noether theory of Lagrange system in the continuous case. Then, we introduce some definitions and notations, such as the operators of discrete translation to the right and the left and the operators of discrete differentiation to the right and the left, and give the conditions for the invariance of the difference functional on the uniform lattice and the non-uniform one, respectively. We also deduce the discrete analog of the Noether-type identity. Finally, the discrete analog of Noether's theorem is presented. An example was discussed to illustrate these results.     

Soliton solutions for a generalized fifth-order KdV equation with t-dependent coefficients

We consider a generalized fifth-order KdV equation with time-dependent coefficients exhibiting higher-degree nonlinear terms. This nonlinear evolution equation describes the interaction between a water wave and a floating ice cover and gravity-capillary waves. By means of the subsidiary ordinary differential equation method, some new exact soliton solutions are derived. Among these solutions, we can find the well known bright and dark solitons with sech and tanh function shapes, and other soliton-like solutions. These solutions may be useful to explain the nonlinear dynamics of waves in an inhomogeneous KdV system supporting high-order dispersive and nonlinear effects.     

Parity and time inversion symmetries of electromagnetic systems

By forming the intersections of the parity and time reversal equivalence classes of physical entities that are represented by differential forms and differential form densities, a number of subsets of discrete symmetry classes for electromagnetic systems can be generated. Only one of these subsets is consistent with elementary thermodynamic arguments for dissipative systems and at the same time yields the notion that both charge and mass are spacetime scalars. This subset is not in correspondence with the two self-consistent presentations that now are implied in the literature.     

General Higher-Order Breather and Hybrid Solutions of the Fokas System

Fokas system is the simplest (2+1)-dimensional extension of the nonlinear Schr?dinger (NLS) equation (Eq.(2), Inverse Problems 10 (1994) L19-L22).By appropriately limiting on soliton solutions generated by the Hirota bilinear method, the explicit forms of $n$-th breathers and semi-rational solutions for the Fokas system are derived. The obtained first-order breather exhibits arange of interesting dynamics. For high-order breather, it has more rich dynamical behaviors.The first-order and the second-order breather solutions are given graphically. Using the long wave limit in soliton solutions, rational solutions are obtained, which are used to analyze the mechanism of the rogue wave and lump respectively.By taking a long waves limit of a part of exponential functions in $f$ and $g$ appeared in the bilinear form of the Fokas system, many interesting hybrid solutions are constructed. The hybrid solutions illustrate various superposed wave structures involving rogue waves, lumps, solitons, and periodic line waves. Their rather complicated dynamics are revealed.     

The mechanical and the wave-theoretical aspects of momentum considering discrete action

The mechanical aspect of momentum, basically its role as a tangent vector of the trajectory of the particle, is related to properties of the momentum found in the contexts of Hamilton's optico-mechanical analogy, de Broglie's matter waves, and quantum mechanics. These properties are treated in a systematic way by considering an approximation of the particle mechanical action of the particle by a step function. A special method of discretizing partial differential equations is shown to be required. Using this method, a discrete dynamics is developed. It is shown that particle dynamics can be regarded as the limit case of the discrete dynamics as the step functions tend to the continuous ones. The equation of motion of a free particle in an arbitrary reference system is deduced in two ways: (i) in continuous dynamics by making use of the invariance of action within changes of reference systems, and (ii) by taking the mentioned limit in discrete dynamics of an equation which expresses that the mechanical and wave-theoretical aspects of the momentum are interrelated in specific way.     

光纤中的电磁对偶变换与导波的模式分析

讨论了麦克斯韦方程在各向同性均匀电介质中的电磁对偶变换不变性,定义了电磁混合比,求出了对偶场的充要条件.把上述不变性和充要条件应用于阶跃光纤,分析其导波模式,证明在非轴对称情况下,阶跃光纤中不存在具有色散的受导简正模.存在的解只有临界折射模(n=n₂时的特解).求出了它的分立频谱.受导简正模只存在于轴对称情况中.以上结论与实验事实不矛盾,并由此解释了基模的特性 关键词： 光纤光学 光波导 电磁对偶变换 受导波     

Wave Breaking in a Class of Nonlocal Dispersive Wave Equations

Abstract

The Korteweg de Vries (KdV) equation is well known as an approximation model for small amplitude and long waves in di!erent physical contexts, but wave breaking phenomena related to short wavelengths are not captured in. In this work we consider a class of nonlocal dispersive wave equations which also incorporate physics of short wavelength scales. The model is identified by a renormalization of an infinite dispersive di!erential operator, followed by further specifications in terms of conservation laws associated with the underlying equation. Several well-known models are thus rediscovered. Wave breaking criteria are obtained for several models including the Burgers-Poisson system, the Camassa-Holm type equation and an Euler-Poisson system. The wave breaking criteria for these models are shown to depend only on the negativity of the initial velocity slope relative to other global quantities.