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

Symbolic Computation of Polynomial Conserved Densities, Generalized Symmetries, and Recursion Operators for Nonlinear Differential-Difference Equations

Algorithms for the symbolic computation of polynomial conserved densities, fluxes, generalized symmetries, and recursion operators for systems of nonlinear differential-difference equations are presented. In the algorithms we use discrete versions of the Frechet and variational derivatives and the Euler and homotopy operators. The algorithms are illustrated for prototypical nonlinear polynomial lattices, including the Kac-van Moerbeke (Volterra) and Toda lattices. Results are shown for the modified Volterra and Ablowitz-Ladik lattices

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

Densities and Fluxes of Differential-Difference Equations

inverse, called the down-shift operator, is given by D -1 u k = u k-1 . Obviously, u k =D u 0 . The actions of D and D -1 are extended to functions by acting on their arguments. For example, D g(u p ,u p+1 ,...,u q )=g(Du p , Du p+1 ,...,D u q )=g(u p+1 ,u p+2 ,...u q+1 ). In particular, g(u p ,u p+1 ,...,u q ) #u k+1 g(u p+1 ,u p+2 ,...,u q+1 ). Moreover, for equations of type (1.1), the shift operator commutes with the time derivative; that is, dt un (Dun ) . Thus, with the use of the shift operator, the entire system (1.1) which may be an infinite set of ordinary di#erential equations is generated from a single equation dt u 0 = f(u-#,u ,...,u 0 ,...,u m-1,u m ) (1.2) #u #u m #=0. Next, we define the (forward) di#erence operator, # = D I, by #u k =(D- I) u k = u k+1 u k , where I is the identity operator. The di#erence operator extends to functions by # g =Dg g. This operator takes the role of a spatial derivative on the shifted variables as many exam