Linear Dispersion Equation Research Articles

A KdV-like advection–dispersion equation with some remarkable properties

We discuss a new non-linear PDE, ut+(2uxx/u)ux=ϵuxxx, invariant under scaling of dependent variable and referred to here as SIdV. It is one of the simplest such translation and space–time reflection-symmetric first order advection–dispersion equations. This PDE (with dispersion coefficient unity) was discovered in a genetic programming search for equations sharing the KdV solitary wave solution. It provides a bridge between non-linear advection, diffusion and dispersion. Special cases include the mKdV and linear dispersive equations. We identify two conservation laws, though initial investigations indicate that SIdV does not follow from a polynomial Lagrangian of the KdV sort. Nevertheless, it possesses solitary and periodic travelling waves. Moreover, numerical simulations reveal recurrence properties usually associated with integrable systems. KdV and SIdV are the simplest in an infinite dimensional family of equations sharing the KdV solitary wave. SIdV and its generalizations may serve as a testing ground for numerical and analytical techniques and be a rich source for further explorations.

Low frequency collective modes in dense relativistic-degenerate strongly coupled plasma

It is shown that low frequency electrostatic ion mode couples with electromagnetic shear Alfven mode in a dense plasma containing strongly coupled non-degenerate ion and relativistic degenerate electron fluids. By employing the appropriate fluid equations, a linear dispersion equation is obtained which shows modifications due to ion correlations and electron relativistic degeneracy. The results are discussed in the ultra-relativistic and weak-relativistic limits and implications of the results in dense degenerate plasmas of astrophysical origin (e.g., white dwarf stars) are pointed out with possible consequences.

Non-Homogeneous Boundary Value Problems for Linear Dispersive Equations

While the non-homogeneous boundary value problem for elliptic, hyperbolic and parabolic equations is relatively well understood, there are still few results for general dispersive equations. We define here a convenient class of equations comprising the Schrödinger equation, the Airy equation and linear “Boussinesq type” systems, which is in some sense a generalization of strictly hyperbolic equations, and for which we define a generalized Kreiss–Lopatinskiĭ condition. From the construction of generalized Kreiss symmetrizers (adapted from hyperbolic theory) we deduce a priori estimates and well posedness for the pure boundary value problems (BVP) on a half-space associated to this class of equations. The initial boundary value problem (IBVP) is investigated too for the special case of the Schrödinger equation, and possible generalizations of the proof for other problems are indicated.

Whistler oscillitons and capillary-gravity generalized solitons

Nonlinear stationary waveforms in two completely different systems, namely, electromagnetic-fluid waves in a magnetic plasma and capillary-gravity water waves, are compared and contrasted. These systems display common features and are amenable to a Hamiltonian description. More importantly, however, is the fact that the linear dispersion equations for these systems possess the property that the phase speed has an extremum (a maximum in the case of the former and a minimum for the latter) at a certain wavelength. The wave-number of the associated stationary wave is, as a result, complex for wave speeds in excess (or less than) this extremum value, at which the phase and group velocities are equal. In its asymptotic state this gives rise to nonlinear waveforms with a hump (or depression) resembling a soliton on which oscillations on a finer spatial scale are superimposed. In the case of the former, “whistlers”, these waveforms, called “oscillitons”, have recently been extensively studied and a brief review is given here. In the case of the latter, “generalized solitons”, a weakly nonlinear theory, based on a 5th order Korteweg-de Vries (KdeV) equation, is used. The resulting 4th order equation for stationary waves, and its first integral, show that “generalized solitons” (i.e. solitons with superimposed oscillations on a finer scale) of depression propagate at subcritical speeds for Bond numbers less than 1/3. The parameter space of the Froude number (F) and the Bond number (r) divides into regions exhibiting different waveforms, with lines F = 1, r = 1/3 and the curve Fm providing the boundaries of demarcation.

A resistive instability of lower hybrid-like waves in regions with parallel currents

The resistive instabilities and dispersion of obliquely propagating waves near the lower hybrid (LH) frequency are studied in plasma carrying a current parallel to the magnetic field. Possible applications of these instabilities include magnetic reconnection regions where LH-like waves may accelerate and heat both ions and electrons. A resistive instability is found in one of the four LH-like wave modes for electron drift speeds several times greater than the electron thermal speed for representative parameters. Numerical fully electromagnetic kinetic calculations of the solutions to the linear dispersion equation are compared with more approximate analytic calculations and show good agreement. The analytic results indicate that ion magnetization effects play a critical role in the resistive instability.

Ultrasonic waves in crystals with charged dislocations

A system of equations for charged dislocations, where the quadratic nonlinear terms are taken into account, is derived using the variational principle. This system describes the propagation of ultrasonic (US) waves in crystals with charged dislocations. From the linearized system of equations a linear dispersion equation is derived. Formulas for the phase linear velocity of the wave and the absorption coefficient are obtained, which show essential influence of charged dislocations and electrical properties of media on the mentioned quantities. For a nonlinear US wave an equation for the amplitude of the first harmonic is derived and, as a consequence, expressions are obtained for the nonlinear velocity of the US wave, for the attenuation of the first harmonic's amplitude, and for phase variation.

NUMERICAL SOLUTIONS FOR SPACE FRACTIONAL DISPERSION EQUATIONS WITH NONLINEAR SOURCE TERMS

Numerical solutions for the fractional differential dispersion equations with nonlinear forcing terms are considered. The backward Euler finite difference scheme is applied in order to obtain numerical solutions for the equation. Existence and stability of the approximate solutions are carried out by using the right shifted Grunwald formula for the fractional derivative term in the spatial direction. Error estimate of order <TEX>$O({\Delta}x+{\Delta}t)$</TEX> is obtained in the discrete <TEX>$L_2$</TEX> norm. The method is applied to a linear fractional dispersion equations in order to see the theoretical order of convergence. Numerical results for a nonlinear problem show that the numerical solution approach the solution of classical diffusion equation as fractional order approaches 2.

Damping of long‐wavelength kinetic Alfvén fluctuations: Linear theory

The full electromagnetic linear dispersion equation for kinetic Alfvén fluctuations in a homogeneous, isotropic, collisionless, Maxwellian electron‐proton plasma is solved numerically in the long‐wavelength limit. At propagation sufficiently oblique to the background magnetic field Bo, the wave number dependence of the damping rate of such modes is summarized by an analytic expression which scales as k⊥2k∥ where the subscripts denote directions relative to Bo. This damping progressively (although not monotonically) increases with increasing electron and proton β, corresponding to four distinct damping regimes: nonresonant, electron Landau, proton Landau, and proton transit‐time damping.

Why are solitons stable?

The theory of linear dispersive equations predicts that waves should spread out and disperse over time. However, it is a remarkable phenomenon, observed both in theory and practice, that once nonlinear effects are taken into account, solitary wave or soliton solutions can be created, which can be stable enough to persist indefinitely. The construction of such solutions can be relatively straightforward, but the fact that they are stable requires some significant amounts of analysis to establish, in part due to symmetries in the equation (such as translation invariance) which create degeneracy in the stability analysis. The theory is particularly difficult in the critical case in which the nonlinearity is at exactly the right power to potentially allow for a self-similar blowup. In this article we survey some of the highlights of this theory, from the more classical orbital stability analysis of Weinstein and Grillakis-Shatah-Strauss, to the more recent asymptotic stability and blowup analysis of Martel-Merle and Merle-Raphael, as well as current developments in using this theory to rigorously demonstrate controlled blowup for several key equations.

Initial-boundary-value problems for linear and integrable nonlinear dispersive partial differential equations

It is suggested here that an interesting and important line of inquiry is the elaboration of methods of inverse scattering transform (IST) type in contexts where non-homogeneous boundary conditions intercede. The issue, which has practical relevance we indicate by example, appears ripe for development, thanks to recent new ideas interjected into the panoply of IST methodologies. A sketch of the principal steps envisaged in carrying out analysis of boundary-value problems using inverse scattering ideas is provided.

Generalized Hasimoto Transform of One-Dimensional Dispersive Flows into Compact Riemann Surfaces

We study the structure of differential equations of one-dimensional dispersive flows into compact Riemann surfaces. These equations geometrically generalize two-sphere valued systems modeling the motion of vortex filament. We define a generalized Hasimoto transform by constructing a good moving frame, and reduce the equation with values in the induced bundle to a complex valued equation which is easy to handle. We also discuss the relationship between our reduction and the theory of linear dispersive partial differential equations.

A Depth Estimation System for Laboratory Studies using Video Imagery

A simple video-based system has been developed for depth estimation based on wave propagation characteristics. A numerical simulation of a long-crested monochromatic wave propagating over a beach with straight and parallel contours is used for testing the depth inversion system. An oblique video, simulating field conditions, is recorded, digitized and rectified for its further analysis. Pixel intensity time series from a virtual array in the rectified images are analyzed using the depth estimation technique developed by Stockdon and Holman. The linear dispersion equation is applied for depth estimation at every cross-shore position and the resulting values are compared with the depth values used to feed the numerical model. Error analysis confirms good performance for depth estimation using this video-system for completely controlled conditions in small-scale experiments. The relative depth estimation error for this idealized case is 2.3%. This accuracy is explained by the use of a linear model for the wave propagation simulation. The methodology proposed here allows the testing of a new video-system and separation of errors resulting from the depth inversion algorithm from those inherent to photogrammetry techniques in small areas. The use of this system could easily be extended for physical models and field studies.

Conservative Split-Explicit Time Integration Methods for the Compressible Nonhydrostatic Equations

Abstract Historically, time-split schemes for numerically integrating the nonhydrostatic compressible equations of motion have not formally conserved mass and other first-order flux quantities. In this paper, split-explicit integration techniques are developed that numerically conserve these properties by integrating prognostic equations for conserved quantities represented in flux form. These procedures are presented for both terrain-following height and hydrostatic pressure (mass) vertical coordinates, two potentially attractive frameworks for which the equation sets and integration techniques differ significantly. For each set of equations, the linear dispersion equation for acoustic/gravity waves is derived and analyzed to determine which terms must be solved in the small (acoustic) time steps and how these terms are represented in the time integration to achieve stability. Efficient techniques for including numerical filters for acoustic and external modes are also presented. Simulations for several idealized test cases in both the height and mass coordinates are presented to demonstrate that these integration techniques appear robust over a wide range of scales, from subcloud to synoptic.

A reliable technique for solving third‐order dispersion equations

PurposeInitial value problems for the one‐dimensional third‐order dispersion equations are investigated using the reliable Adomian decomposition method (ADM).Design/methodology/approachThe solutions are obtained in the form of rapidly convergent power series with elegantly computable terms.FindingsIt was found that the technique is reliable, powerful and promising. It is easier to implement than the separation of variables method. Modifications of the ADM and the noise terms phenomenon are successfully applied for speeding up the convergence of non‐homogeneous equations.Research limitations/implicationsThe method is restricted to initial value problems in which the space variable fills the whole real axis. Modifications are required to deal with initial boundary value problems. Further, the input initial condition is required to be an infinitely differentiable function and obviously, the convergence radius of the decomposition series depends on the input data.Practical implicationsThe method was mainly illustrated for linear partial differential equations occuring in water resources research, but the natural extension of the ADM to solving nonlinear problems is extremely useful in nonlinear studies and soliton theory.Originality/valueThe study undertaken in this paper provides a reliable approach for solving both linear and nonlinear dispersion equations and new explicit or recursively‐based exact solutions are found.

Terahertz Radiation from Dual-Grating Driven by Two Electron Beams

An approach to producing terahertz (THz) radiation from a rectangular dual grating driven by two electron beams is presented in this paper. The two beams with velocity separation are admixed traveling between the two oppositely placed gratings. The linear dispersion equations for the even and odd modes are derived using the fluid theory and solved through numerical computation. A considerable enhancement in growth rate induced by two-stream instability is demonstrated.

Sturmian nodal set analysis for higher-order parabolic equations and applications

We describe the local pointwise structure of multiple zeros of solutions of $2m$th-order linear uniformly parabolic equations, \begin{equation} \tag*{(0.1)} u_t = \sum _{|{\beta} | \le 2m} \, a_{\beta}(x,t) D^{\beta}_x u \quad {\rm in} \,\,\, {{\bf R}^N} \times [-1,1] \quad (m \ge 2), \end{equation} with bounded and Lipschitz-continuous (for $|{\beta}|=2m$) coefficients, in the existence-uniqueness class $\{|u(x,t)| \le B {\mathrm e}^{b|x|^{\alpha}}\}$, where $B,b>0$ are constants and ${\alpha} = \frac {2m}{2m-1}$. Assuming that $u(0,0)=0$ and using the Sturmian backward continuation blow-up variable $ y=x/(-t)^{\frac 1{2m}} \quad (t < 0), $ we perform a classification of all possible types of formation as $t \to 0^-$ of multiple spatial zeros of the solutions $u(x,t)$. We show that there exists a countable family of multiple zeros evolving as $t \to 0^-$ according to the nodal sets of polynomial eigenfunctions of a non-self-adjoint operator ${{\mathbf B}}^*$ associated with that in (0.1). Next, we show that other related polynomial solutions occur in the collapse of multiple zeros as $t \to 0^+$, which is described in terms of the forward continuation variable $ y=x/t^{\frac 1{2m}} \quad (t>0). $ For the 1D second-order ($m=1$) parabolic equation with smooth coefficients, $$ u_t= a(x,t) u_{xx} + q(x,t) u \quad (a(x,t) \ge a_0>0), $$ this two-step analysis is known as {Sturm's Second Theorem} on zero sets, established by C.~Sturm in 1836, [32]. His more famous First Theorem (the number of zeros of solutions is non-increasing with time ) was derived as a consequence of the second one. In the last thirty years these PDE ideas of Sturm found new applications, generalizations and extensions in various areas of general parabolic theory, stability and orbital connection problems, unique continuation and Poincaré--Bendixson theorems, mean curvature and curve shortening flows, symplectic geometry, etc. Using such a local classification of multiple zeros, we establish a unique continuation theorem for higher-order parabolic PDEs and inequalities, and estimate the Hausdorff dimension of nodal sets of solutions. It turns out that some common features of multiple zeros formation can be observed for solutions of other PDEs including linear dispersion and wave equations, $$ u_t=u_{xxx} \quad \mbox{and} \quad u_{tt}=u_{xx}, $$ and we present a discussion on this issue.

Excitation of Electromagnetic Flute Modes in the Process of Interaction of Plasma Flow with Inhomogeneous Magnetic Field

Laboratory experiments on the interaction of a plasma flow, produced by laser ablation of a solid target with the inhomogeneous magnetic field from the Zebra pulsed power generator demonstrated the presence of strong wave activity in the region of the flow deceleration. The deceleration of the plasma flow can be interpreted as the appearance of a gravity-like force. The drift due to this force can lead to the excitation of flute modes. In this paper a linear dispersion equation for the excitation of electromagnetic flute-type modes with plasma and magnetic field parameters, corresponding to the ongoing experiments is examined. Results indicate that the wavelength of the excited flute modes strongly depends on the strength of the external magnetic field. For magnetic field strengths ∼0.1 MG the excited wavelengths are larger than the width of the laser ablated plasma plume and cannot be observed during the experiment. But for magnetic field strengths ∼1 MG the excited wavelengths are much smaller and can then be detected.

Exact controllability of the nonlinear third-order dispersion equation

Exact controllability of a nonlinear dispersion system has been studied. This work extends the work of Russell and Zhang [D.L. Russell, B.Y. Zhang, Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain, SIAM J. Control Optim. 31 (1993) 659–676], in which the authors considered a linear dispersion system. We obtain controllability results using two standard types of nonlinearities, namely, Lipschitzian and monotone. We also obtain the exact controllability of the same system through the approach of Integral Contractors which is a weaker condition than Lipschitz condition.

Vector theory of diffraction by gratings made of a uniaxial dielectric-magnetic material exhibiting negative refraction

Diffraction of linearly polarized plane electromagnetic waves at the periodically corrugated boundary of vacuum and a linear, homogeneous, nondissipative, uniaxial dielectric-magnetic material is formulated as a boundary-value problem and solved using the differential method. Attention is paid to two classes of diffracting materials: those with negative definite permittivity and permeability tensors and those with indefinite permittivity and permeability tensors. The dispersion equations turn out to be elliptic for the first class of diffracting materials, whereas for the second class they can be hyperbolic, elliptic, or linear, depending on the orientation of the optic axis. When the dispersion equations are elliptic, the optical response of the grating is qualitatively similar to that for conventional gratings: a finite number of refraction channels are supported. On the other hand, hyperbolic or linear dispersion equations imply the possibility of an infinite number of refraction channels. This possibility seriously incapacitates the differential method as the corrugations deepen.

The initial value problem for a third order dispersive equation on the two-dimensional torus

We present the necessary and sufficient conditions for the L 2 L^2 -well-posedness of the initial problem for a third order linear dispersive equation on the two-dimensional torus. Birkhoff’s method of asymptotic solutions is used to prove necessity. Some properties of a system for quadratic algebraic equations associated to the principal symbol play a crucial role in proving sufficiency.