Convergence of approximate solutions for waves in a stratified fluid with approximated density distribution

Equations defining in linear approximation the wave motions in an arbitrarily stratified fluid are derived. Investigation of convergence of wave solutions with approximations ρ_n (z) of the mean density profile ρ_o (z) shows that uniform convergence of ρ_n (z) to ρ_o (z) is the sufficient condition of convergence of wave equation solutions. The convergence of solutions is uniform on sets of upper bound wave numbers and lower bound phase velocities of waves. Examples that show that when the continuous function ρ_o (z) is approximated by step-wide functions ρ_n (z) the convergence of solutions for internal waves is not uniform over the whole set of admissible wave numbers and phase velocities of waves.     

Multisymplecticity of the centred box discretization for hamiltonian PDEs with m ≥ 2 space dimensions

In this letter, it is shown that the centred box discretization for Hamiltonian PDEs with m ≥ 2 space dimensions is multisymplectic in the sense of Bridges and Reich in [1–6]. Multisymplectic discretizations for the generalized KP equation and the wave equation with 2 space dimensions, respectively, are given. A multisymplectically numerical scheme of the wave equation is derived.     

Long-term analysis of semilinear wave equations with slowly varying wave speed

A semilinear wave equation with slowly varying wave speed is considered in one to three space dimensions on a bounded interval, a rectangle or a box, respectively. It is shown that the action, which is the harmonic energy divided by the wave speed and multiplied with the diameter of the spatial domain, is an adiabatic invariant: it remains nearly conserved over long times, longer than any fixed power of the time scale of changes in the wave speed in the case of one space dimension, and longer than can be attained by standard perturbation arguments in the two- and three-dimensional cases. The long-time near-conservation of the action yields long-time existence of the solution. The proofs use modulated Fourier expansions in time.     

A Model Equation for Wavepacket Solitary Waves Arising from Capillary-Gravity Flows

A model equation governing the primitive dynamics of wave packets near an extremum of the linear dispersion relation at finite wavenumber is derived. In two spatial dimensions, we include the effects of weak variation of the wave in the direction transverse to the direction of propagation. The resulting equation is contrasted with the Kadomtsev–Petviashvilli and Nonlinear Schrödinger (NLS) equations. The model is derived as an approximation to the equations for deep water gravity-capillary waves, but has wider applications. Both line solitary waves and solitary waves which decay in both the transverse and propagating directions—lump solitary waves—are computed. The stability of these waves is investigated and their dynamics are studied via numerical time evolution of the equation.     

Numerical simulation of blow-up of a 2D generalized Ginzburg-Landau equation

We study the Cauchy problem for the following generalized Ginzburg-Landau equation u_t = (ν+i)Δu − (κ+iβ)|u|^2qu + γu in two spatial dimensions for q > 1 (here , β, γ are real parameters and ν,κ > 0). A blow-up of solutions is found via numerical simulation in several cases for q > 1.     

Tanh-travelling wave solutions, truncated Painlevé expansion and reduction of Bullough–Dodd equation to a quadrature in magnetohydrodynamic equilibrium

The equations of magnetohydrodynamic (MHD) equilibria for a plasma in gravitational field are investigated. For equilibria with one ignorable spatial coordinate, the MHD equations are reduced to a single nonlinear elliptic equation for the magnetic potential , known as the Grad–Shafranov equation. Specifying the arbitrary functions in this equation, the Bullough–Dodd equation can be obtained. The truncated Painlevé expansion and reduction of the partial differential equation to a quadrature problem (RQ method) are described and applied to obtain the travelling wave solutions of the Bullough–Dodd equation for the case of isothermal magnetostatic atmosphere, in which the current density J is proportional to the exponentially of the magnetic flux and moreover falls off exponentially with distance vertical to the base, with an “e-folding” distance equal to the gravitational scale height.     

Linear Waves and Nonlinear Wave Interactions in a Bounded Three‐Layer Fluid System

We investigate possible linear waves and nonlinear wave interactions in a bounded three‐layer fluid system using both analysis and numerical simulations. For sharp interfaces, we obtain analytic solutions for the admissible linear mode‐one parent/signature waves that exist in the system. For diffuse interfaces, we compute the overtaking interaction of nonlinear mode‐two solitary waves. Mathematically, owing to a small loss of energy to dispersive tails during the interaction, the waves are not solitons. However, this energy loss is extremely minute, and because the dispersively coupled waves in the system exhibit the three types of Lax KdV interactions, we conclude that for all intents and purposes the solitary waves exhibit soliton behavior.     

The inverse scattering transform: Tools for the nonlinear fourier analysis and filtering of ocean surface waves

Surface wave data from the Adriatic Sea are analysed in the light of new data analysis techniques which may be viewed as a nonlinear generalization of the ordinary Fourier transform. Nonlinear Fourier analysis as applied herein arises from the exact spectral solution to large classes of nonlinear wave equations which are integrable by the inverse scattering transform (IST). Numerical methods are discussed which allow for implementation of the approach as a tool for the time series analysis of oceanic wave data. The case for unidirectional propagation in shallow water, where integrable nonlinear wave motion is governed by the Korteweg-deVries (KdV) equation with periodic/quasi-periodic boundary conditions, is considered. Numerical procedures given herein can be used to compute a nonlinear Fourier representation for a given measured time series. The nonlinear oscillation modes (the IST ‘basis functions’) of KdV obey a linear superposition law, just as do the sine waves of a linear Fourier series. However, the KdV basis functions themselves are highly nonlinear, undergo nonlinear interactions with each other and are distinctly non-sinusoidal. Numerical IST is used to analyse Adriatic Sea data and the concept of nonlinear filtering is applied to improve understanding of the dominant nonlinear interactions in the measured wavetrains.     

On the resonant reflection of weak,nonlinear sound waves off an entropy wave

We analyze the resonant reflection of very weak, nonlinear sound waves off a weak sawtooth entropy wave for spatially periodic solutions of the one‐dimensional, nonisentropic gas dynamics equations. The case of an entropy wave with a sawtooth profile is of interest because the oscillations of the reflected sound waves are nondispersive with frequency independent of their wavenumber, leading to an unusual type of nonlinear dynamics. On an appropriate long time scale, we show that a complex amplitude function for the spatial profile of the sound waves satisfies a degenerate quasilinear Schrödinger equation. We present some numerical solutions of this equation that illustrate the generation of small spatial scales by a resonant four‐wave cascade and front propagation in compactly supported solutions.     

Analytic solutions of an iterative functional differential equation

This paper is concerned with a functional differential equation x^′(z)=1/x(az+bx(z)), where a, b are two complex numbers. By constructing a convergent power series solution y(z) of a auxiliary equation of the form b²y^′(z)=(y(μ²z)−ay(μz))(μy^′(μz)−ay^′(z)), analytic solutions of the form for the original differential equation are obtained.     

Existence and uniqueness of positive solutions to a linear transport equation in a metric space

The linear functional equation ∂_tz = L(z) − rz is considered. The linear operator L acts on a linear metric space of real functions z depending on t and on a parameter ω belonging to a subset of ^m. The existence and uniqueness to a nonnegative solution of the initial value problem is shown. An application to a kinetic equation is performed.     

A theory of electromagnetic and gravitational fields

Careful calculations using classical field theory show that if a macroscopic ball with uniform surface charge (say, a billiard ball with 1E6 excess electrons) is released near the surface of the earth, it will almost instantaneously accelerate to relativistic speed and blow a hole in the ground. This absurd prediction is just the macroscopic version of the self-force problem for charged particles [1]. Furthermore, if one attempts to develop from electromagnetism a parallel theory for gravitational [2], the result is the same, self-acceleration.

The basis of the new theory is a measure of energy density for any wave equation [3–5]. Given any solution of any four-vector wave equation in spacetime (for example, the potentials (c^-1φA)=(A⁰,A¹,A²,A³) in electromagnetism), one can form the 16th first order partial derivatives of the vector components, with respect to the time and space variables (ct,x) = (x⁰, x¹, x², x³). The sum of the squares of the 16 terms is a natural energy function [6, p. 283] (satisfying a conservation law . Such energy functions are routinely utilized by mathematicians as Lyapunov functions in the theory of stability of waves with boundary conditions. A Lagrangian using this sum leads to a new energy tensor for electromagnetic and gravitational fields, an alternative to that in [7].     

Time-linearized,compact methods for the inviscid GRLW equation subject to initial Gaussian conditions

The objective of this paper is three-fold. First, four time-linearization methods that are second- and fourth-order accurate in time and space, respectively, are presented and used to study the dynamics of the modified and generalized regularized-long wave equations (mRLW and GRLW equations, respectively). Two of the methods use the conservation-law form of the equations and treat the wave amplitude and its second-order spatial derivative and the linear and nonlinear advection fluxes as unknowns, whereas the other two employ the non-conservation-law form of the equations and consider the wave amplitude and its first- and second-order spatial derivatives as unknowns. The methods employ three-point fourth-order accurate Padé discretizations for the first- and second-order derivatives, are second-order accurate in time, and yield linear systems of blocktridiagonal matrices. Second, the accuracy of these methods is assessed by comparing their results with those of the exact solution of the mRLW equation. It is reported that the four methods predict nearly the same values of the three invariants and have the same accuracy, and that an accurate prediction of the invariants may not correspond to small errors in space and time. Third, the dynamics of the inviscid GRLW equation is studied first qualitatively in terms of length and time scales and then numerically as a function of the linear advection speed, the exponent of the nonlinear advection flux, the dispersion coefficient and the amplitude and width of the initial bell-shaped or Gaussian conditions. It is shown that wide initial conditions result in wave steepening and breakup and the formation of solitary waves whose amplitude and speed decrease as the time for their formation increases. For narrow initial conditions, it is shown that only a single solitary wave may form. Behind this wave and depending on the parameters that characterized the inviscid GRLW equation, rarefaction or negative amplitude waves that propagate towards the upstream boundary or a train of localized oscillatory waves that do not emerge from the trailing edge of the leading solitary wave may be formed. These oscillatory waves exhibit the characteristics of, but are not dispersive shock waves and their amplitude and frequency increases as the width of the initial conditions is decreased. The results presented here do not only complement previous work by the authors, they also show that the dynamics of the inviscid GRLW equation undergoes new and interesting phenomena as the width of the initial conditions is decreased.     

Generalized solitary waves in the gravity-capillary Whitham equation

We study the existence of traveling wave solutions to a unidirectional shallow water model, which incorporates the full linear dispersion relation for both gravitational and capillary restoring forces. Using functional analytic techniques, we show that for small surface tension (corresponding to Bond numbers between 0 and 1/3) there exists small amplitude solitary waves that decay to asymptotically small periodic waves at spatial infinity. The size of the oscillations in the far field are shown to be small beyond all algebraic orders in the amplitude of the wave.     

Solitary waves and a stability analysis of an equation of short and long dispersive waves

A universal model for the interaction of long nonlinear waves and packets of short waves with long linear carrier waves is given by a system in which an equation of Korteweg–de Vries (KdV) type is coupled to an equation of nonlinear Schrödinger (NLS) type. The system has solutions of steady form in which one component is like a solitary-wave solution of the KdV equation and the other component is like a ground-state solution of the NLS equation. We study the stability of solitary-wave solutions to an equation of short and long waves by using variational methods based on the use of energy–momentum functionals and the techniques of convexity type. We use the concentration compactness method to prove the existence of solitary waves. We prove that the stability of solitary waves is determined by the convexity or concavity of a function of the wave speed.     

A simple bound for the remainder of the neumann series in the case of a self-adjoint compact operator

We consider the functional equation of the second kind \gf − λK\gf = f with K a compact self-adjoint linear operator on a Hilbert space: a Fredholm integral equation of the second kind, for example. We establish the simple bound where λ is any regular value of K; ø is the solution of the equation corresponding to λ; λ₁ is the characteristic value of K smallest in absolute value; and N = 0, 1, 2, .... For \s|λ\s| < \s|λ₁\s|, this is an estimate for the remainder of the partial sums of the Neumann series.     

Cauchy problem for equations of internal waves

The Cauchy problem is considered for the equation of internal waves to which reduce many problems of the linear theory of waves in a continuously stratified fluid. The theorem of uniqueness is proved, and the formula for explicit representation of solution in terms of integrals whose kernels contain the obtained in /1/ fundamental solution of the internal wave operator and its time derivative are derived. Asymptotic analysis of solution in the “distant zone” is carried out for large values of dimensionless time.     

The fundamental solutions for the fractional diffusion-wave equation

The time fractional diffusion-wave equation is obtained from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order 2β with 0 < β ≤ 1/2 or 1/2 < β ≤ 1, respectively. Using the method of the Laplace transform, it is shown that the fundamental solutions of the basic Cauchy and Signalling problems can be expressed in terms of an auxiliary function M(z;β), where z = |x|/t^β is the similarity variable. Such function is proved to be an entire function of Wright type.     

El Naschie’s ε theory and effects of nanoparticle clustering on the heat transport in nanofluids

Effects of nanoparticle clustering on the heat transfer in nanofluids using the scale relativity theory in the topological dimension D_T = 3 are analyzed. In the one-dimensional differentiable case, the clustering morphogenesis process is achieved by cnoidal oscillation modes of the speed field. In such conjecture, a non-autonomous regime implies a relation between the radius and growth speed of the cluster while, a quasi-autonomous regime requires El Naschie’s ε^(∞) theory through the cluster–cluster coherence (El Naschie global coherence). Moreover, these two regimes are separated by the golden mean. In the one-dimensional non-differentiable case, the fractal kink spontaneously breaks the ‘vacuum symmetry’ of the fluid by tunneling and generates coherent structures. This mechanism is similar to the one of superconductivity. Thus, the fractal potential acts as an energy accumulator while, the fractal soliton, implies El Naschie’s ε^(∞) theory (El Naschie local coherence). Since all the properties of the speed field are transferred to the thermal one, for a certain conditions of an external load (e.g. for a certain value of thermal gradient) the soliton and fractal one breaks down (blows up) and release energy. As result, the thermal conductibility in nanofluids unexpectedly increases. Here, El Naschie’s ε^(∞) theory interferes through El Naschie global and local coherences.     

Stochastic resonance with different periodic forces in overdamped two coupled anharmonic oscillators

We study the stochastic resonance phenomenon in the overdamped two coupled anharmonic oscillators with Gaussian noise and driven by different external periodic forces. We consider (i) sine, (ii) square, (iii) symmetric saw-tooth, (iv) asymmetric saw-tooth, (v) modulus of sine and (vi) rectified sinusoidal forces. The external periodic forces and Gaussian noise term are added to one of the two state variables of the system. The effect of each force is studied separately. In the absence of noise term, when the amplitude f of the applied periodic force is varied cross-well motion is realized above a critical value (f_c) of f. This is found for all the forces except the modulus of sine and rectified sinusoidal forces. For fixed values of angular frequency ω of the periodic forces, f_c is minimum for square wave and maximum for asymmetric saw-tooth wave. f_c is found to scale as Ae^0.75ω + B where A and B are constants. Stochastic resonance is observed in the presence of noise and periodic forces. The effect of different forces is compared. The stochastic resonance behaviour is quantized using power spectrum, signal-to-noise ratio, mean residence time and distribution of normalized residence times. The logarithmic plot of mean residence time τ_MR against 1/(D − D_c) where D is the intensity of the noise and D_c is the value of D at which cross-well motion is initiated shows a sharp knee-like structure for all the forces. Signal-to-noise ratio is found to be maximum at the noise intensity D = D_max at which mean residence time is half of the period of the driving force for the forces such as sine, square, symmetric saw-tooth and asymmetric saw-tooth waves. With modulus of sine wave and rectified sine wave, the SNR peaks at a value of D for which sum of τ_MR in two wells of the potential of the system is half of the period of the driving force. For the chosen values of f and ω, signal-to-noise ratio is found to be maximum for square wave while it is minimum for modulus of sine and rectified sinusoidal waves. The values of D_c at which cross-well behaviour is initiated and D_max are found to depend on the shape of the periodic forces.