We study the behavior of solitary-wave solutions of some generalized nonlinear Schrödinger equations with an external potential. The equations have the feature that in the absence of the external potential, they have solutions describing inertial motions of stable solitary waves.We construct solutions of the equations with a non-vanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that, over a large interval of time, they describe a solitary wave whose center of mass motion is a solution of Newton's equations of motion for a point particle in the given external potential, up to small corrections corresponding to radiation damping.

We study the nonlinear equationwhich is known to describe the dynamics of pseudo-relativistic boson stars in the meanfield limit. For positive mass parameters, m > 0, we prove existence of travelling solitary waves, ψ(t, x) = e itµ ϕ v (x − vt), with speed |v| < 1, where c = 1 corresponds to the speed of light in our units. Due to the lack of Lorentz covariance, such travelling solitary waves cannot be obtained by applying a Lorentz boost to a solitary wave at rest (with v = 0). To overcome this difficulty, we introduce and study an appropriate variational problem that yields the functions ϕ v ∈ H 1/2 (R 3 ) as minimizers, which we call boosted ground states. Our existence proof makes extensive use of concentration-compactness-type arguments.In addition to their existence, we prove orbital stability of travelling solitary waves ψ(t, x) = e itµ ϕ v (x − vt) and pointwise exponential decay of ϕ v (x) in x.

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Abstract-General expressions for the quality factor (Q) of antennas are minimized to obtain lower-bound formulas for the Q of electrically small, lossy or lossless, combined electric and magnetic dipole antennas confined to an arbitrarily shaped volume. The lowerbound formulas for Q are derived for dipole antennas with specified electric and magnetic dipole moments excited by both electric and magnetic surface currents as well as by electric surface currents alone. With either excitation, separate formulas are found for the dipole antennas containing only lossless or "nondispersive-conductivity" material and for the dipole antennas containing "highly dispersive lossy" material. The formulas involve the quasi-static electric and magnetic polarizabilities of the associated perfectly conducting volume of the antenna, the ratio of the powers radiated by the specified electric and magnetic dipole moments, and the efficiency of the antenna.

Although the stored energy of an antenna is instrumental in the evaluation of antenna Q and the associated physical bounds, it is difficult to strictly define stored energy. Classically, the stored energy is either determined from the input impedance of the antenna or the electromagnetic fields around the antenna. The new energy expressions proposed by Vandenbosch express the stored energy in the current densities in the antenna structure. These expressions are equal to the stored energy defined from the difference between the energy density and the far field energy for many but not all cases.Here, the different approaches to determine the stored energy are compared for dipole, loop, inverted L-antennas, and bow-tie antennas. We use Brune synthesized circuit models to determine the stored energy from the input impedance. We also compare the results with differentiation of the input impedance and the obtained bandwidth. The results indicate that the stored energy in the fields, currents, and circuit models agree well for small antennas. For higher frequencies, the stored energy expressed in the currents agrees with the stored energy determined from Brune synthesized circuit models whereas the stored energy approximated by differentiation of input impedance gives a lower value for some cases. The corresponding results for the bandwidth suggest that the inverse proportionality between the fractional bandwidth and Q-factor depends on the threshold level of the reflection coefficient.

Electric and magnetic current densities are an essential part of electromagnetic theory. The goal of the present paper is to define and investigate stored energies that are valid for structures that can support both electric and magnetic current densities. Stored energies normalized with the dissipated power give us the Q factor, or antenna Q, for the structure. Lower bounds of the Q factor provide information about the available bandwidth for passive antennas that can be realized in the structure. The definition of the stored energies that we propose is valid beyond the leading order small antenna limit. Our starting point is the energy density with subtracted far-field form which we obtain an explicit and numerically attractive current density representation. This representation gives us the insight to propose a coordinate independent stored energy. Furthermore, we find here that lower bounds on antenna Q for structures with e.g., electric dipole radiation can be formulated as convex optimization problems. We determine lower bounds on both open and closed surfaces that support electric and magnetic current densities.The here derived representation of stored energies has in its electrical small limit an associated Q factor that agrees with known small antenna bounds. These stored energies have similarities to earlier efforts to define stored energies. However, one of the advantages with this method is the above mentioned formulation as convex optimization problems, which makes it easy to predict lower bounds for antennas of arbitrary shapes. The present formulation also gives us insight into the components that contribute to Chu's lower bound for spherical shapes. We utilize scalar and vector potentials to obtain a compact direct derivation of these stored energies. Examples and comparisons end the paper. arXiv:1604.08572v2 [physics.class-ph]

We study solutions close to solitary waves of the pseudo-relativistic Hartree equation describing boson stars under the influence of an external gravitational field. In particular, we analyze the long-time effective dynamics of such solutions. In essence, we establish a (long-time) stability result for solutions describing boson stars that move under the influence of an external gravitational field.

We study the motion of solitary-wave solutions of a family of focusing generalized nonlinear Schrödinger equations with a confining, slowly varying external potential, V (x).A Lyapunov-Schmidt decomposition of the solution combined with energy estimates allows us to control the motion of the solitary wave over a long, but finite, time interval.We show that the center of mass of the solitary wave follows a trajectory close to that of a Newtonian point particle in the external potential V (x) over a long time interval.