Oblique amplitude modulation of dust-acoustic plasma waves

Theoretical and numerical studies are presented of the nonlinear amplitude modulation of dust-acoustic (DA) waves propagating in an unmagnetized three component, weakly-coupled, fully ionized plasma consisting of electrons, positive ions and charged dust particles, considering perturbations oblique to the carrier wave propagation direction. The stability analysis, based on a nonlinear Schroedinger-type equation (NLSE), shows that the wave may become unstable; the stability criteria depend on the angle $\theta$ between the modulation and propagation directions. Explicit expressions for the instability rate and threshold have been obtained in terms of the dispersion laws of the system. The possibility and conditions for the existence of different types of localized excitations have also been discussed.Comment: 21 pages, 6 figures, to appear in Physica Script

Electron-acoustic plasma waves: oblique modulation and envelope solitons

Theoretical and numerical studies are presented of the amplitude modulation of electron-acoustic waves (EAWs) propagating in space plasmas whose constituents are inertial cold electrons, Boltzmann distributed hot electrons and stationary ions. Perturbations oblique to the carrier EAW propagation direction have been considered. The stability analysis, based on a nonlinear Schroedinger equation (NLSE), reveals that the EAW may become unstable; the stability criteria depend on the angle $\theta$ between the modulation and propagation directions. Different types of localized EA excitations are shown to exist.Comment: 10 pages, 5 figures; to appear in Phys. Rev.

An integrable discretization of KdV at large times

An "exact discretization" of the Schroedinger operator is considered and its direct and inverse scattering problems are solved. It is shown that a differential-difference nonlinear evolution equation depending on two arbitrary constants can be solved by using this spectral transform and that for a special choice of the constants it can be considered an integrable discretization of the KdV equation at large times. An integrable difference-difference equation is also obtained.Comment: 12 page

A discrete Schrodinger spectral problem and associated evolution equations

A recently proposed discrete version of the Schrodinger spectral problem is considered. The whole hierarchy of differential-difference nonlinear evolution equations associated to this spectral problem is derived. It is shown that a discrete version of the KdV, sine-Gordon and Liouville equations are included and that the so called `inverse' class in the hierarchy is local. The whole class of related Darboux and Backlund transformations is also exhibited.Comment: 14 pages, LaTeX2

Scattering theory for Klein-Gordon equations with non-positive energy

We study the scattering theory for charged Klein-Gordon equations: \{{array}{l} (\p_{t}- \i v(x))^{2}\phi(t,x) \epsilon^{2}(x, D_{x})\phi(t,x)=0,[2mm] \phi(0, x)= f_{0}, [2mm] \i^{-1} \p_{t}\phi(0, x)= f_{1}, {array}. where: \epsilon^{2}(x, D_{x})= \sum_{1\leq j, k\leq n}(\p_{x_{j}} \i b_{j}(x))A^{jk}(x)(\p_{x_{k}} \i b_{k}(x))+ m^{2}(x), describing a Klein-Gordon field minimally coupled to an external electromagnetic field described by the electric potential $v(x)$ and magnetic potential $\vec{b}(x)$. The flow of the Klein-Gordon equation preserves the energy: h[f, f]:= \int_{\rr^{n}}\bar{f}_{1}(x) f_{1}(x)+ \bar{f}_{0}(x)\epsilon^{2}(x, D_{x})f_{0}(x) - \bar{f}_{0}(x) v^{2}(x) f_{0}(x) \d x. We consider the situation when the energy is not positive. In this case the flow cannot be written as a unitary group on a Hilbert space, and the Klein-Gordon equation may have complex eigenfrequencies. Using the theory of definitizable operators on Krein spaces and time-dependent methods, we prove the existence and completeness of wave operators, both in the short- and long-range cases. The range of the wave operators are characterized in terms of the spectral theory of the generator, as in the usual Hilbert space case