Michael A. Allen

A solitary-wave solution to a perturbed KdV equation

We derive the approximate form and speed of a solitary-wave solution to a perturbed KdV equation. Using a conventional perturbation expansion, one can derive a first-order correction to the solitary-wave speed, but at the next order, algebraically secular terms appear, which produce divergences that render the solution unphysical. These terms must be treated by a regrouping procedure developed by us previously. In this way, higher-order corrections to the speed are obtained, along with a form of solution that is bounded in space. For this particular perturbed KdV equation, it is found that there is only one possible solitary wave that has a form similar to the unperturbed soliton solution.

Antibody-Conjugated Paramagnetic Nanobeads: Kinetics of Bead-Cell Binding

Specific labelling of target cell surfaces using antibody-conjugated paramagnetic nanobeads is essential for efficient magnetic cell separation. However, studies examining parameters determining the kinetics of bead-cell binding are scarce. The present study determines the binding rates for specific and unspecific binding of 150 nm paramagnetic nanobeads to highly purified target and non-target cells. Beads bound to cells were enumerated spectrophotometrically. Results show that the initial bead-cell binding rate and saturation levels depend on initial bead concentration and fit curves of the form A(1 − exp(−kt)). Unspecific binding within conventional experimental time-spans (up to 60 min) was not detectable photometrically. For CD3-positive cells, the probability of specific binding was found to be around 80 times larger than that of unspecific binding.

Time evolution of perturbed solitons of modified Kadomtsev-Petviashvili equations

We solve the (2+1)-dimensional Schamel-Kadomtsev- Petviashvili equations with negative and positive dispersion numerically with one or two perturbed plane solitons as initial conditions. In the negative dispersion case, the plane soliton is stable and retains its identity. For the equation with positive dispersion, the plane solitons decay into two- dimensional lump solitons. We show that in contrast to one- dimensional solitons, collisions between two lump solitons are far from elastic. We also demonstrate that the solitons emerging from the collision can be very sensitive to the alignment of the solitons prior to collision.

Numerical Solutions of the Nonlinear Schrödinger Equation with a Square Root Nonlinearity

The time evolution of bright solitons in an ion-acoustic plasma with non-isothermal electrons are determined numerically by using the Crank-Nicolson scheme. Head-on collisions between two solitons with the same and different amplitudes are investigated. Both types of collision are found to be inelastic -- the amplitudes and speeds of the outgoing solitons are a little different from those of the incoming solitons.

Weakly nonlinear waves in magnetized plasma with a slightly non-Maxwellian electron distribution. Part 1, Stability of solitary waves

We determine the growth rate of linear instabilities resulting from long-wavelength transverse perturbations applied to periodic nonlinear wave solutions to the Schamel - Korteweg - de Vries - Zakharov - Kuznetsov (SKdVZK) equation which governs weakly nonlinear waves in a strongly magnetized cold-ion plasma whose electron distribution is given by two Maxwellians at slightly different temperatures. To obtain the growth rate it is necessary to evaluate non-trivial integrals whose number is kept to a minimum by using recursion relations. It is shown that a key instance of one such relation cannot be used for classes of solution whose minimum value is zero. and an additional integral must be evaluated explicitly instead. The SKdVZK equation contains two nonlinear terms whose ratio b increases as the electron distribution becomes increasingly flat-topped. As b and hence the deviation from electron isothermality increases, it is found that for cnoidal wave solutions that travel faster than long-wavelength linear waves, there is a more pronounced variation of the growth rate with the angle theta at which the perturbation is applied. Solutions whose minimum values are zero and which travel slower than long-wavelength linear waves are found, at first order, to be stable to perpendicular perturbations and have a relatively narrow range of theta for which the first-order growth rate is not zero.

Viewing tropical forest succession as a three-dimensional dynamical system

As tropical forests are complex systems, they tend to be modelled either roughly via scaling relationships or in a detailed manner as high-dimensional systems with many variables. We propose an approach which lies between the two whereby succession in a tropical forest is viewed as a trajectory in the configuration space of a dynamical system with just three dependent variables, namely, the mean leaf-area index (LAI) and its standard deviation (SD) or coefficient of variation along a transect, and the mean diameter at breast height (DBH) of trees above the 90th percentile of the distribution of tree DBHs near the transect. Four stages in this forest succession are identified: (I) naturally afforesting grassland: the initial stage with scattered trees in grassland; (II) very young forest: mostly covered by trees with a few remaining gaps; (III) young smooth forest: almost complete cover by trees of mostly similar age resulting in a low SD; and (IV) old growth or mature forest: the attracting region in configuration space characterized by fluctuating SD from tree deaths and regrowth. High-resolution LAI measurements and other field data from Khao Yai National Park, Thailand show how the system passes through these stages in configuration space, as do simple considerations and a crude cellular automaton model.

Nonlinear magnetoacoustic waves in a cold plasma

The equations describing planar magnetoacoustic waves of permanent form in a cold plasma are rewritten so as to highlight the presence of a naturally small parameter equal to the ratio of the electron and ion masses. If the magnetic field is not nearly perpendicular to the direction of wave propagation, this allows us to use a multiple-scale expansion to demonstrate the existence and nature of nonlinear wave solutions. Such solutions are found to have a rapid oscillation of constant amplitude superimposed on the underlying large-scale variation. The approximate equations for the large-scale variation are obtained by making an adiabatic approximation and in one limit, new explicit solitary pulse solutions are found. In the case of a perpendicular magnetic field, conditions for the existence of solitary pulses are derived. Our results are consistent with earlier studies which were restricted to waves having a velocity close to that of long-wavelength linear magnetoacoustic waves.

Strongly restricted permutations and tiling with fences

We identify bijections between strongly restricted permutations of { 1 , 2 , ? , n } of the form π ( i ) - i ? W , where W is any finite set of integers which is independent of i and n , and tilings of an n -board (a linear array of n square cells of unit width) using square tiles and ( 1 2 , g ) -fence tiles where g ? Z + . A ( 1 2 , g ) -fence is composed of two pieces of width 1 2 separated by a gap of width g . The tiling approach allows us to obtain the recurrence relation for the number of permutations when W = { - 1 , d 1 , ? , d r } where d r 0 and the remaining d l are non-negative integers which are independent of i and n . This is a generalization of a previous result. Terms in this recurrence relation, along with terms in other recurrences we obtain for more complicated cases, can be identified with certain groupings of interlocking tiles. The ease of counting tilings gives rise to a straightforward way of obtaining identities concerning the number of occurrences of patterns such as fixed points or excedances in restricted permutations. We also use the tilings to obtain the possible permutation cycles.

Evaluation of Some Improper Integrals Involving Hyperbolic Functions

leaving card Ck j in its place, but changing its suit from hearts to spades, then swapping the remaining k − j cards of rows k − 1 and k, as in Figure 4. Why is it legal to change the suit of card Ck j from hearts to spades? Since Ck j was the first small card, then the spade card Ck−1, j−1 is not small and therefore has a value strictly greater than x j−2. Thus all spade cards can take on values less than or equal to x j−1. Since Ck j is small, its value is at most x j−1, so changing it from hearts to spades is allowable. As before, Ck j remains the first small card of C ′, so (C ′)′ = C and C ′ has permutation π ′, which has opposite parity of π since they differ by a transposition. Thus there is a one-to-one correspondence between the positively counted Vandermonde tables with small cards and the negatively counted Vandermonde tables with small cards. Therefore the determinant of Vn is the number of Vandermonde tables with no small cards, namely, ∏ 0≤i< j≤n(x j − xi ), as desired.