Barbara Abraham-Shrauner

Integration of second order ordinary differential equations not possessing Lie point symmetries

Abstract A class of integrable second order ordinary differential equations not possessing Lie point symmetries is shown to be rich in nonlocal symmetries which provide one route for integration and to have general solutions which are uniform functions.

Hidden symmetries associated with the projective group of nonlinear first-order ordinary differential equations

Hidden symmetries, those not found by the classical Lie group method for point symmetries, are reported for nonlinear first-order ordinary differential equations (ODEs) which arise frequently in physical problems. These are for the special class of the eight nonAbelian, two-parameter subgroups of the eight-parameter projective group. The first-order ODEs can be transformed by non-local transformations to new separable first-order ODEs which then can be reduced to quadratures. The first-order ODEs include Riccati equations and equations which in particular cases are of the form of Abels equation. The procedure demonstrates the feasibility of integrating nonlinear ODEs that do not show any apparent Lie group point symmetry. Applications to the Vlasov characteristic equation and the reaction-diffusion equation are given.

Provenance of Type II hidden symmetries from nonlinear partial differential equations

Abstract The provenance of Type II hidden point symmetries of differential equations reduced from nonlinear partial differential equations is analyzed. The hidden symmetries are extra symmetries in addition to the inherited symmetries of the differential equations when the number of independent and dependent variables is reduced by a Lie point symmetry. These Type II hidden symmetries do not arise from contact symmetries or nonlocal symmetries as in the case of ordinary differential equations. The Lie point symmetries of a model equation and the two-dimensional Burgers’ equation and their descendants are used to identify the hidden symmetries. The significant new result is the provenance of the Type II Lie point hidden symmetries found for differential equations reduced from partial differential equations. Two methods for determining the source of the hidden symmetries are developed.

Hidden and contact symmetries of ordinary differential equations

The Lie algebra for the maximal contact symmetries of third-order ordinary differential equations (ODEs) is examined for type I and II hidden symmetries where the analysis of hidden symmetries for point symmetries is extended to contact symmetries. Ones invariant under the group associated with the ten-dimensional (maximal) Lie algebra may produce type I hidden symmetries for two-parameter subgroups and type II hidden symmetries for certain solvable non-Abelian three-parameter subgroups in the third-order ODEs when they are reduced in order. A new class of type II hidden symmetries is recognized in which contact symmetries transform to point symmetries in some reduction paths. Two examples of ODEs invariant under subgroups of the ten-parameter group under which y```=0 is invariant demonstrate the new class of type II hidden symmetries.

Hidden symmetries and linearization of the modified Painlevé-ince equation

The linearization and hidden symmetries of the modified Painleve–Ince equation, y‘+σyy’+βy3=0, where σ and β are constants, are presented. The linearization of this equation by a nonlocal transformation yields a damped (stable) or growing (unstable) harmonic oscillator equation for β≳0. Hidden symmetries are analyzed by transforming the modified Painleve–Ince equation to a third‐order ordinary differential equation (ODE) which, in general, is invariant under a three‐parameter group by a Riccati transformation. A type I hidden symmetry is found of a second‐order ODE found from the third‐order ODE where a symmetry is lost in the reduction of order by the non‐normal subgroup invariants. A type II hidden symmetry occurs in the third‐order ODE because the symmetries of a second‐order ODE, reduced from the third‐order ODE by another set of normal subgroup invariants, are increased.

Generalized gouy-chapman potential of charged phospholipid membranes with divalent cations

SummaryAn exact result for the electrostatic potential of a phospholipid membrane modeled by a fixed charge sheet and diffuse double layer of the Gouy-Chapman theory is given. The dependence of the potential with distance is expressed in simple form for positive divalent ions (alkaline earth cations) added to monovalent cations and anions in the bathing solution. The result has applications in the study of the effect of divalent cations on nerve or muscle excitation and on the formation of blood clots.

Nonlinear poisson-Boltzmann potential for a uniformly charged dielectric sphere in an electrolyte

A perturbation expansion for the electrostatic potential of a uniformly charged, dielectric sphere in an electrolyte is given in the large radius limit as a solution of the nonlinear Poisson-Boltzmann equation for qφ/κT > 1. The potential in principle calculable to any order n, is valid for distances smaller than the sphere radius and for λD/a, ratio of Debye length to sphere radius, small compared to one. A two term expansion agrees with numerical results of Wiersema at the sphere for the above conditions and also for all λD/a if qφ/κT ≥ 12.

Stability of Bernstein–Greene–Kruskal equilibria

The recent linear stability analysis of Lewis and Symon for spatially inhomogeneous Vlasov equilibria is illustrated with the case of an unstable periodic wavetrain of strongly inhomogeneous Bernstein–Greene–Kruskal equilibria. The stability formalism involves expanding an auxiliary function related to the perturbation distribution function in terms of the equilibrium Liouville eigenfunctions, and expanding the perturbation potential in terms of the eigenfunctions of an appropriately chosen field operator. The infinite‐dimensional dispersion matrix is truncated to M×M by assuming that the normal mode of interest of the perturbation potential can be adequately represented by M eigenfunctions of the field operator; the eigenfrequencies ω are the zeroes of the determinant of the dispersion matrix. A particular Bernstein–Greene–Kruskal equilibrium was chosen as a numerical example, and the growth rate and normal mode of the instability were determined by numerical simulation. The agreement of the theory with ...

Hidden Symmetries, First Integrals and Reduction of Order of Nonlinear Ordinary Differential Equations

Abstract The reduction of nonlinear ordinary differential equations by a combination of first integrals and Lie group symmetries is investigated. The retention, loss or even gain in symmetries in the integration of a nonlinear ordinary differential equation to a first integral are studied for several examples. The differential equations and first integrals are expressed in terms of the invariants of Lie group symmetries. The first integral is treated as a differential equation where the special case of the first integral equal to zero is examined in addition to the nonzero first integral. The inverse problem for which the first integral is the fundamental quantity enables some predictions of the change in Lie group symmetries when the differential equation is integrated. New types of hidden symmetries are introduced.

Exact, Stationary Wave Solutions of the Nonlinear Vlasov Equation

A procedure for systematically calculating a wide class of exact, nonlinear wave solutions of the Vlasov equation that are stationary in the wave frame is given. The essential feature of the method is the expansion of the current density in an infinite series of the vector potential or the expansion of the charge density in an infinite series in the scalar potential. The potentials obey a differential equation of the form of the equation of motion for a point particle in a conservative potential. This nonlinear differential equation has numerous analytic solutions depending on the choice of physical parameters.