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The magnetic effects of the quiet‐time proton belt have been studied by means of data obtained from a proton detector aboard Explorer 12. From the measured energy spectrums, intensities, and pitch‐angle distributions of the protons with energies above 100 kev, the current density on a magnetic meridian plane was calculated using the theory of Akasofu and Chapman, modified to eliminate several simplifying assumptions. The algebraic sum of all the currents was 0.59 million amperes, and the magnetic moment of the current loop was 0.029 ME. The magnetic perturbations on the meridian plane were obtained from the electric current distribution. At the magnetic equator on the earth's surface the proton belt produced a 9 γ decrease in the field, which is considerably less than the 38 γ decrease reported by Akasofu, Cain, and Chapman. Their calculations were based on very preliminary data from the same detector which yielded a total kinetic energy of the trapped protons of 2.38×1022 ergs, whereas the analysis performed here yields an energy of only 5.7×1021 ergs. The maximum perturbation appeared at L=3.6 on the magnetic equator, where the field attained a value of −23 γ.

The method of symmetry reduction is systematically applied to derive several classes of invariant solutions for the generalized Weierstrass system inducing constant mean curvature surfaces and to the associated two-dimensional nonlinear sigma model. A classification of subgroups with generic orbits of codimension one of the Lie point symmetry group for these systems provides a tool for introducing symmetry variables and reduces the initial systems to different nonequivalent systems of ordinary differential equations. We perform a singularity analysis for them in order to establish whether these ordinary differential equations have the Painlevé property. These ordinary differential equations can then be transformed to standard forms and next solved in terms of elementary and Jacobi elliptic functions. This results in a large number of new solutions and in some cases new interesting constant mean curvature surfaces are found. Furthermore, this symmetry analysis is extended to include conditional symmetries by subjecting the original systems to certain differential constraints. In this case, several types of nonsplitting algebraic, trigonometric and hyperbolic multi-soliton solutions have been obtained in explicit form. A new procedure for constructing solutions of the overdetermined system which is composed of the generalized Weierstrass system and the complex eikonal equations is studied. Finally, an approach to the classical configurations of strings in three-dimensional Euclidean space based on the obtained solutions of the generalized Weierstrass system is presented. RésuméLa méthode de réduction par symétries est appliquée systématiquement pour dériver plusieurs classes de solutions invariantes du système de Weierstrass généralisé induisant des surfaces de courbure moyenne constante et du modèle sigma euclidien bidimensionnel associéà ce système. Une classification des sous-groupes avec orbites génériques de codimension un des groupes de Lie de symétrie ponctuels pour ces systèmes fournit un moyen d'introduire des variables de symétrie et réduit le système initialà différents systèmes inéquivalents d'équations différentielles ordinaires. Nous effectuons une analyse des singularités afin d'établir si ceséquations différentielles ordinaires possèdent la propriété de Painlevé. Ceséquations différentielles ordinaires peuvent alorsêtre transformées dans des formes standards et ensuite résolues en termes de fonctionsélémentaires et de fonctions elliptiques de Jacobi. Cela résulte en un grand nombre de nouvelles solutions et, dans certains cas, de nouvelles surfacesà courbure moyenne constante sont trouvées. De plus, cette analyse des symétries estétendue au cas des symétries conditionnelles en soumettant les systèmes initiauxà certaines contraintes différentielles. Dans ce cas, plusieurs solutions multi-solitoniques non-séparantes de type algébrique, trigonomé-trique et elliptique sont obtenues sous une forme explicite. Une nouvelle procédure pour construire des solutions du système surdéterminé composé du ...

Pade approximants are able to sum effectively the Rayleigh-Schrodinger perturbation series for the ground state energy of the quartic anharmonic oscillator, as well as the corresponding renormalized perturbation expansion [E.J. Weniger, J. Cizek, and F. Vinette, J. Math. Phys. 34, 571 (1993)]. In the sextic case, Pade approximants are still able to sum these perturbation series, but convergence is so slow that they are computationally useless. In the octic case, Pade approximants are not powerful enough and fail. On the other hand, the inclusion of only a few additional data from the strong coupling domain [E.J. Weniger, Ann. Phys. (N.Y.) (to be published)] greatly enhances the power of summation methods. The summation techniques that we consider are two-point Pade approximants and effective characteristic polynomials. It is shown that these summation methods give good results for the quartic and sextic anharmonic oscillators, and even in the case of the octic anharmonic oscillator, which represents an extremely challenging summation problem, two-point Pade approximants give relatively good results. PACS number(s): 02.70.c, 03.65w, 02.30.Lt I. THE SUMMATION OF DIVERGENT PERTURBATION EXPANSIONS Perturbation theory is one of the few principal methods of approximating solutions to eigenvalue problems in theoretical physics [1-4]. Accordingly, there is an extensive literature. Mathematical aspects of perturbation theory are treated in monographs by Friedrichs [5], Kato

The integrability of a system which describes constant mean curvature surfaces by means of the adapted Weierstrass-Enneper inducing formula is studied. This is carried out by using a specific transformation which reduces the initial system to the completely integrable 2-dimensional Euclidean nonlinear sigma model. Through the use of the apparatus of differential forms and Cartan theory of systems in involution, it is demonstrated that the general analytic solutions of both systems possess the same degree of freedom. Furthermore, a new linear spectral problem equivalent to the initial Weierstrass-Enneper system is derived via the method of differential constraints. A new procedure for constucting solutions to this system is proposed and illustrated by several elementary examples, including a multi-soliton solution.

The symmetry group method is applied to a generalized Korteweg-de Vries equation and several classes of group invariant solutions for it are obtained by means of this technique. Polynomial, trigonometric, and elliptic function solutions can be calculated. It is shown that this generalized equation can be reduced to a first-order equation under a particular second-order differential constraint which resembles a Schrödinger equation. For a particular instance in which the constraint is satisfied, the generalized equation is reduced to a quadrature. A condition which ensures that the reciprocal of a solution is also a solution is given, and a first integral to this constraint is found

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