Hydromagnetic Stability of a Streaming Cylindrical Incompressible Plasma

A dispersion relation is derived and analyzed for the case where the equilibrium velocity of an incompressible, nonresistive, cylindrical plasma has a spiral motion along magnetic field lines. The symmetric hydromagnetic equations are used to derive the plasma hydromagnetic pressure. The dispersion relation is found by matching plasma and outer-region hydromagnetic pressures across a sharp-moving interface. The zeros of the dispersion relation are obtained by a sequence of mappings between three complex planes. The presence of flow introduces overstable modes. For m = 0 the time-divergences are removed by flow. For m = 1 the divergences are enhanced by flow such that the growth rates and oscillation frequencies increase linearly with the flow velocity. The smaller is the wavelength of the disturbance in the z direction, the larger are the overstable eigenvalues

Numerical Solitons of Generalized Korteweg-de Vries Equations

We propose a numerical method for finding solitary wave solutions of generalized Korteweg-de Vries equations by solving the nonlinear eigenvalue problem on an unbounded domain. The artificial boundary conditions are obtained to make the domain finite. We specially discuss the soliton solutions of the K(m, n) equation and KdV-K(m,n) equation. Furthermore for the mixed models of linear and nonlinear dispersion, the collision behaviors of soliton-soliton and soliton-antisoliton are observed.Comment: 9 pages, 4 figure

Hidden solitons in the Zabusky-Kruskal experiment: Analysis using the periodic, inverse scattering transform

Recent numerical work on the Zabusky--Kruskal experiment has revealed, amongst other things, the existence of hidden solitons in the wave profile. Here, using Osborne's nonlinear Fourier analysis, which is based on the periodic, inverse scattering transform, the hidden soliton hypothesis is corroborated, and the \emph{exact} number of solitons, their amplitudes and their reference level is computed. Other "less nonlinear" oscillation modes, which are not solitons, are also found to have nontrivial energy contributions over certain ranges of the dispersion parameter. In addition, the reference level is found to be a non-monotone function of the dispersion parameter. Finally, in the case of large dispersion, we show that the one-term nonlinear Fourier series yields a very accurate approximate solution in terms of Jacobian elliptic functions.Comment: 10 pages, 4 figures (9 images); v2: minor revision, version accepted for publication in Math. Comput. Simula

Dynamics of Vesicles in shear and rotational flows: Modal Dynamics and Phase Diagram

Despite the recent upsurge of theoretical reduced models for vesicle shape dynamics, comparisons with experiments have not been accomplished. We review the implications of some of the recently proposed models for vesicle dynamics, especially the Tumbling-Trembling domain regions of the phase plane and show that they all fail to capture the essential behavior of real vesicles for excess areas, \Delta, greater than 0.4. We emphasize new observations of shape harmonics and the role of thermal fluctuations.Comment: (removed forgotten leftover figure files

Nonlinear wave propagation through cold plasma

Electromagnetic wave propagation through cold collision free plasma is studied using the nonlinear perturbation method. It is found that the equations can be reduced to the modified Kortweg-de Vries equation

Stability of Periodic Soliton Equations under Short Range Perturbations

We consider the stability of (quasi-)periodic solutions of soliton equations under short range perturbations and give a complete description of the long time asymptotics in this situation. We show that, apart from the phenomenon of the solitons travelling on the quasi-periodic background, the perturbed solution asymptotically approaches a modulated solution. We use the Toda lattice as a model but the same methods and ideas are applicable to all soliton equations in one space dimension. More precisely, let $g$ be the genus of the hyperelliptic Riemann surface associated with the unperturbed solution. We show that the $n/t$-pane contains $g+2$ areas where the perturbed solution is close to a quasi-periodic solution in the same isospectral torus. In between there are $g+1$ regions where the perturbed solution is asymptotically close to a modulated lattice which undergoes a continuous phase transition (in the Jacobian variety) and which interpolates between these isospectral solutions. In the special case of the free solution ($g=0$) the isospectral torus consists of just one point and we recover the classical result. Both the solutions in the isospectral torus and the phase transition are explicitly characterized in terms of Abelian integrals on the underlying hyperelliptic Riemann surface.Comment: 4 pages, 2 figure

Shock interactions with heavy gaseous elliptic cylinders: Two leeward-side shock competition modes and a heuristic model for interfacial circulation deposition at early times

We identify two different modes, types I and II, of the interaction for planar shocks accelerating heavy prolate gaseous ellipses. These modes arise from different interactions of the incident shock (IS) and transmitted shock (TS) on the leeward side of the ellipse. A time ratio t_T/t_I(M,η,λ,γ_0,γ_b), which characterizes the mode of interaction, is derived heuristically. Here, the principal parameters governing the interaction are the Mach number of the shock (M), the ratio of the density of the ellipse to the ambient gas density, (η>1), γ_0, γ_b (the ratios of specific heats of the two gases), λ (the aspect ratio). Salient events in shock–ellipse interactions are identified and correlated with their signatures in circulation budgets and on-axis space–time pressure diagrams. The two modes yield different mechanisms of the baroclinic vorticity generation. We present a heuristic model for the net baroclinic circulation generated on the interface at the end of the early-time phase by both the IS and TS and validate the model via numerical simulations of the Euler equations. In the range 1.2⩽M⩽3.5, 1.54⩽η⩽5.04, and λ=1.5 and 3.0, our model predicts the baroclinic circulation on the interface within a band of ±10% in comparison to converged numerical simulations

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

We discuss a new non-linear PDE, u_t + (2 u_xx/u) u_x = epsilon u_xxx, 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.Comment: 15 pages, 4 figures, corrected sign typo in KdV Lagrangian above equation 3