An observation on the periodic solutions to nonlinear physical models by means of the auxiliary equation with a sixth-degree nonlinear term

It is a fact that in auxiliary equation methods, the exact solutions of different types of auxiliary equations may produce new types of exact travelling wave solutions to nonlinear equations. In this manner, various auxiliary equations of first-order nonlinear ordinary differential equation with distinct-degree nonlinear terms are examined and, by means of symbolic computation, the new solutions of original auxiliary equation of first-order nonlinear ordinary differential equation with sixth-degree nonlinear term are presented. Consequently, the novel exact solutions of the generalized Klein–Gordon equation and the active-dissipative dispersive media equation are found out for illustration purposes. They are also applicable, where conventional perturbation method fails to provide any solution of the nonlinear problems under study.     

Electromagnetic scattering from an anisotropic impedance half-plane at oblique incidence: the exact solution

Scattering of a plane electromagnetic wave from an anisotropicimpedance half-plane at skew incidence is considered. The twomatrix surface impedances involved are assumed to be complexand different. The problem is solved in closed form. The boundary-valueproblem reduces to a system of two first-order difference equationswith periodic coefficients subject to a symmetry condition.The main idea of the method developed is to convert the systemof difference equations into a scalar Riemann–Hilbertproblem on a finite contour of a hyperelliptic surface of genus3. A constructive procedure for its solution and the solutionof the associated Jacobi inversion problem is proposed and describedin detail. Numerical results for the edge diffraction coefficientsare reported.     

Asymptotic behavior of waves

The asymptotic conjugation relation is established for all ƒL²(Rⁿ) under mild assumptions on and g, where denotes Fourier multiplication. The asymptotic estimate for finite energy solutions u of the wave equation is deduced from (*), along with generalizations to a class of first-order symmetric hyperbolic systems of partial differential equations that are homogeneous and constant coefficient, and a weakened version for the Klein-Gordon equation. Also deduced from (*) is the fact that for a free Schrödinger particle the probability of being in the set tA at time t tends to the probability that the velocity is in A as t → ±∞.     

Conservative finite difference schemes for the generalized Zakharov–Kuznetsov equations

This paper is concerned with the construction of conservative finite difference schemes by means of discrete variational method for the generalized Zakharov–Kuznetsov equations and the numerical solvability of the two-dimensional nonlinear wave equations. A finite difference scheme is proposed such that mass and energy conservation laws associated with the generalized Zakharov–Kuznetsov equations hold. Our arguments are based on the procedure that D. Furihata has recently developed for real-valued nonlinear partial differential equations. Numerical results are given to confirm the accuracy as well as validity of the numerical solutions and then exhibit remarkable nonlinear phenomena of the interaction and behavior of pulse wave solutions.     

Perturbations for linear difference equations

In this paper we study boundary value problems for perturbed second-order linear difference equations with a small parameter. The reduced problem obtained when the parameter is equal to zero is a first-order linear difference equation. The solution is represented as a convergent series in the small parameter, whose coefficients are given by means of solutions of the reduced problem.     

A fully coupled method for simulation of wave-current-seabed systems

Coastal flow involves surface wave propagation, current circulation, and seabed evolution, and its prediction remains challenging when they strongly interact with each other, especially during extreme events such as tsunami and storm surge. We propose a fully coupled method to simulate motion of wave-current-seabed systems and associated multiphysics phenomena. The wave action equation, the shallow water equations, and the Exner equation are respectively used for wave, current, and seabed morphology, and the discretization is based on a second-order, flux-limiter, finite difference scheme previously developed for current-seabed systems. The proposed method is tested with analytical solutions, laboratory measurements, and numerical solutions obtained with other schemes. Its advantages are demonstrated in capturing interplay among wave, current, and seabed; it has the capability of first-order upwind schemes to suppress artificial oscillations as well as the accuracy of second-order schemes in resolving flow structures.     

Factorization method for linear and quasilinear singularly perturbed boundary value problems for ordinary differential equations

For linear singularly perturbed boundary value problems, we come up with a method that reduces solving a differential problem to a discrete (difference) problem. Difference equations, which are an exact analog of differential equations, are constructed by the factorization method. Coefficients of difference equations are calculated by solving Cauchy problems for first-order differential equations. In this case nonlinear Ricatti equations with a small parameter are solved by asymptotic methods, and solving linear equations reduces to computing quadratures. A solution for quasilinear singularly perturbed equations is obtained by means of an implicit relaxation method. A solution to a linearized problem is calculated by analogy with a linear problem at each iterative step. The method is tested against solutions to the known Lagerstrom-Cole problem.     

Exact boundary observability for nonautonomous quasilinear wave equations

By means of a direct and constructive method based on the theory of semiglobal C² solution, the local exact boundary observability is shown for nonautonomous 1-D quasilinear wave equations. The essential difference between nonautonomous wave equations and autonomous ones is also revealed.     

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α? (0,1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.     

Open boundary conditions for hyperbolic equations

A previously developed general procedure for deriving accurate difference equations to describe conditions at open boundaries for hyperbolic equations is extended and further illustrated by means of several examples of practical importance. Problems include those with both incoming and outgoing waves at the boundary, the use of locally cylindrical and spherical wave approximations at each point of the boundary, and nonlinear wave propagation. Reflected waves in all cases are minimal and less than 10^?2 of the incident wave.     

Numerical modeling of seismic and acoustic-gravity waves propagation in an “Earth-Atmosphere” model in the presence of wind in the air

In this paper, an effective numerical algorithm for 2.5D seismic and acoustic-gravitational wave propagation is applied to a combined “Earth-Atmosphere” model in the presence of wind in the air. Seismic wave propagation in an elastic half-space is described by a system of first-order dynamic equations of elasticity theory. The propagation of acoustic-gravitational waves in the atmosphere in the presence of wind is described by the linearized Navier-Stokes equations. The algorithm is based on the integral Laguerre transform with respect to time, the finite integral Fourier transform with respect to a spatial coordinate combined with a finite difference method for the reduced problem.     

Cayley–Dickson Procedure, Relativistic Wave Equations and Supersymmetric Oscillators

The relativistic first-order wave equations for massive particles with spin 0,1,1/2 are formulated in terms of a factorization of the Klein–Fock equation by means of the algebra of octonions. An analogous method applied to Hamiltonian of the quantum isotropic oscillator leads to the natural generalization of the model. The class of supersymmetric oscillators with dimension N7 associated with te algebras of the Cayley–Dickson series is introduced.     

Exact travelling wave solutions of the discrete sine–Gordon equation obtained via the exp-function method

In this paper, we generalize the exp-function method, which was used to find new exact travelling wave solutions of nonlinear partial differential equations (NPDEs) or coupled nonlinear partial differential equations, to nonlinear differential–difference equations (NDDEs). As an illustration, two series of exact travelling wave solutions of the discrete sine–Gordon equation are obtained by means of the exp-function method. As some special examples, these new exact travelling wave solutions can degenerate into the kink-type solitary wave solutions reported in the open literature.     

A comparative study of approximate symmetry and approximate homotopy symmetry to a class of perturbed nonlinear wave equations

A comparative study of approximate symmetry and approximate homotopy symmetry to a class of perturbed nonlinear wave equations is performed. First, complete infinite-order approximate symmetry classification of the equation is obtained by means of the method originated by Fushchich and Shtelen. An optimal system of one-dimensional subalgebras is derived and used to construct general formulas of approximate symmetry reductions and similarity solutions. Second, we study approximate homotopy symmetry of the equation and construct connections between the two symmetry methods for the first-order and higher-order cases, respectively. The series solutions derived by the two methods are compared.     

Perturbations of Soliton Solutions to the Unstable Nonlinear Schrödinger and Sine-Gordon Equations

The adiabatic evolution of soliton solutions to the unstable nonlinear Schrödinger (UNS) and sine-Gordon (SG) equations in the presence of small perturbations is reconsidered. The transport equations describing the evolution of the solitary wave parameters are determined by a direct multiple-scale asymptotic expansion and by phase-averaged conservation relations for an arbitrary perturbation. The evolution associated with a dissipative perturbation is explicitly determined and the first-order perturbation fields are also obtained.     

Evolution of Higher-Order Gray Hirota Solitary Waves

The defocusing Hirota equation has dark and gray soliton solutions which are stable on a background of periodic waves of constant amplitude. In this paper, gray solitary wave evolution for a higher-order defocusing Hirota equation is examined. A direct analysis is used to identify families of higher-order gray Hirota solitary waves, which are embedded for certain parameter values. Soliton perturbation theory is used to determine the detailed behavior of an evolving higher-order gray Hirota solitary wave. An integral expression for the first-order correction to the wave is found and analytical expressions for the steady-state and transient components of the solitary wave tail are derived. A subtle and complex picture of the development of solitary wave tails emerges. It is found that solitary wave tails develop for two reasons, one is decay of the solitary wave caused by resonance, the second is corrections at first-order to the background wave. Strong agreement is found between the theoretical predictions of the perturbation theory and numerical solutions of the governing equations.     

Factorization technique and new exact solutions for the modified Camassa-Holm and Degasperis-Procesi equations

In this paper, the recent factorization technique is applied to the modified Camassa-Holm and Degasperis-Procesi equations and two first-order ordinary differential equations are obtained, respectively. Subsequently, some new exact solitary wave solutions for the two equations are proposed. The figures for the bell-type and peakon-type solutions of the modified Camassa-Holm are plotted to describe the properties of the solutions.     

Group properties of generalized quasi-linear wave equations

In this paper, complete group classification of a class of (1+1)-dimensional generalized quasi-linear wave equations is performed by using the Lie-Ovsiannikov method, additional equivalent transformation and furcate split method. Lie reductions of some truly ‘variable coefficient’ wave equations which are singled out from the classification results are investigated. Some classes of exact solutions of these ‘variable coefficient’ wave equations are constructed by means of both the reductions and the additional equivalent transformations. The nonclassical symmetries to the generalized quasi-linear wave equation are also studied. This enabled to obtain some exact solutions of the wave equations which are invariant under certain conditional symmetries.     

Existence of traveling waves in compressible Euler equations with viscosity and capillarity

Given any Lax shock of the compressible Euler dynamics equations, we show that there exists the corresponding traveling wave of the system when viscosity and capillarity are suitably added. For a traveling wave corresponding to a given Lax shock, the governing viscous–capillary system is reduced to a system of two differential equations of first-order, which admits an asymptotically stable equilibrium point and a saddle point. We then develop the method of estimating attraction domain of the asymptotically stable equilibrium point for the compressible Euler equations and show that the saddle point in fact lies on the boundary of this set. Then, we establish a saddle-to-stable connection by pointing out that there is a stable trajectory leaving the saddle point and entering the attraction domain of the asymptotically stable equilibrium point. This gives us a traveling wave of the viscous–capillary compressible Euler equations.     

Multiphase averaging and the inverse spectral solution of the Korteweg—de Vries equation

Inverse spectral theory is used to prescribe and study equations for the slow modulations of N-phase wave trains for the Korteweg-de Vries (KdV) equation. An invariant representation of the modulational equations is deduced. This representation depends upon certain differentials on a Riemann surface. When evaluated near ∞ on the surface, the invariant representation reduces to averaged conservations laws; when evaluated near the branch points, the representation shows that the simple eigenvalues provide Riemann invariants for the modulational equations. Integrals of the invariant representation over certain cycles on the Riemann surface yield “conservation of waves.” Explicit formulas for the characteristic speeds of the modulational equations are derived. These results generalize known results for a single-phase traveling wave, and indicate that complete integrability can induce enough structure into the modulational equations to diagonalize (in the sense of Riemann invariants) their first-order terms.