2,614 research outputs found
Collisions of acoustic solitons and their electric fields in plasmas at critical compositions
Acoustic solitons obtained through a reductive perturbation scheme are
normally governed by a Korteweg-de Vries (KdV) equation. In multispecies
plasmas at critical compositions the coefficient of the quadratic nonlinearity
vanishes. Extending the analytic treatment then leads to a modified KdV (mKdV)
equation, which is characterized by a cubic nonlinearity and is even in the
electrostatic potential. The mKdV equation admits solitons having opposite
electrostatic polarities, in contrast to KdV solitons which can only be of one
polarity at a time. A Hirota formalism has been used to derive the two-soliton
solution. That solution covers not only the interaction of same-polarity
solitons but also the collision of compressive and rarefactive solitons. For
the visualisation of the solutions, the focus is on the details of the
interaction region. A novel and detailed discussion is included of typical
electric field signatures that are often observed in ionospheric and
magnetospheric plasmas. It is argued that these signatures can be attributed to
solitons and their interactions. As such, they have received little attention.Comment: 15 pages, 15 figure
Traveling Wave Solutions to Fifth- and Seventh-order Korteweg-de Vries Equations: Sech and Cn Solutions
In this paper we review the physical relevance of a Korteweg-de Vries (KdV)
equation with higher-order dispersion terms which is used in the applied
sciences and engineering. We also present exact traveling wave solutions to
this generalized KdV equation using an elliptic function method which can be
readily applied to any scalar evolution or wave equation with polynomial terms
involving only odd derivatives. We show that the generalized KdV equation still
supports hump-shaped solitary waves as well as cnoidal wave solutions provided
that the coefficients satisfy specific algebraic constraints.
Analytical solutions in closed form serve as benchmarks for numerical solvers
or comparison with experimental data. They often correspond to homoclinic
orbits in the phase space and serve as separatrices of stable and unstable
regions. Some of the solutions presented in this paper correct, complement, and
illustrate results previously reported in the literature, while others are
novel.Comment: 12 pages, 4 figures, minor text modifications, updated bibliograph
Head-on collisions of electrostatic solitons in nonthermal plasmas
In contrast to overtaking interactions, head-on collisions between two electrostatic solitons can only be dealt with by an approximate method, which limits the range of validity but offers valuable insights. Treatments in the plasma physics literature all use assumptions in the stretching of space and time and in the expansion of the dependent variables that are seldom if ever discussed. All models force a separability to lowest order, corresponding to two linear waves with opposite but equally large velocities. A systematic exposition of the underlying hypotheses is illustrated by considering a plasma composed of cold ions and nonthermal electrons. This is general enough to yield critical compositions that lead to modified rather than standard Korteweg-de Vries equations, an aspect not discussed so far. The nonlinear evolution equations for both solitons and their phase shifts due to the collision are established. A Korteweg-de Vries description is the generic conclusion, except when the plasma composition is critical, rendering the nonlinearity in the evolution equations cubic, with concomitant repercussions on the phase shifts. In the latter case, the solitons can have either polarity, so that combinations of negative and positive solitons can occur, contrary to the generic case, where both solitons necessarily have the same polarity
Dark Radiation Emerging After Big Bang Nucleosynthesis?
We show how recent data from observations of the cosmic microwave background
may suggest the presence of additional radiation density which appeared after
big bang nucleosynthesis. We propose a general scheme by which this radiation
could be produced from the decay of non-relativistic matter, we place
constraints on the properties of such matter, and we give specific examples of
scenarios in which this general scheme may be realized.Comment: v3: 5 pages, 1 figure. References added, typos corrected, notation
changed throughout. v2: 5 pages, 1 figure. Reformatted, references added,
acknowledgments updated, effect of radiation on CMB clarified. v1: 11 pages,
1 figur
Numerical periodic normalization for codim 2 bifurcations of limit cycles : computational formulas, numerical implementation, and examples
Explicit computational formulas for the coefficients of the periodic normal forms for codimension 2 (codim 2) bifurcations of limit cycles in generic autonomous ODEs are derived. All cases (except the weak resonances) with no more than three Floquet multipliers on the unit circle are covered. The resulting formulas are independent of the dimension of the phase space and involve solutions of certain boundary-value problems on the interval [0, T], where T is the period of the critical cycle, as well as multilinear functions from the Taylor expansion of the ODE right-hand side near the cycle. The formulas allow one to distinguish between various bifurcation scenarios near codim 2 bifurcations of limit cycles. Our formulation makes it possible to use robust numerical boundary-value algorithms based on orthogonal collocation, rather than shooting techniques, which greatly expands its applicability. The implementation is described in detail with numerical examples, where numerous codim 2 bifurcations of limit cycles are analyzed for the first time
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