We study the Gross-Pitaevskii equation with a slowly varying smooth potential, V (x) = W (hx). We show that up to time log(1/h)/h and errors of size h in H, the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian, (ξ + sech ∗ V (x))/2. This provides an improvement (h→ h) compared to previous works, and is strikingly confirmed by numerical simula...

In the paper we establish some new results depending on the comparative growth properties of composition of entire functions using relative $L^{ast }$-order (relative $L^{ast }$-lower order) as compared to their corresponding left and right factors where $Lequiv Lleft( rright) $ is a slowly changing function.

We derive reciprocal integral relations between phases and amplitude moduli for a class of wave functions that are cyclically varying in time. The relations imply that changes of a certain kind (e.g. not arising from the dynamic phase) obligate changes in the other. Numerical results indicate the approximate validity of the relationships for arbitrarily (non-cyclically) varying states in the ad...

Functional differential equations with deviating arguments are studied for the first time in the framework of Karamata regularly varying functions. A sharp condition is established for the existence of slowly varying solutions for a class of second order linear equations of the form x′′ = q(t)x(g(t)), both in the retarded and in the advanced case.

An analytic study of the reaction probability integrals corresponding to the various forms of the slowly varying cross-section factor S(E) is attempted. Exact expressions for reaction probability integrals are expressed in terms of the extended gamma functions.

A new algorithm for learning invariance manifolds is introduced that allows a neuron to learn a non-linear transfer function to extract invariant or rather slowly varying features from a vectorial input sequence. This is generalized to a group of neurons, referred to as a Gibson-clique, to learn slowly varying features that are uncorrelated. Since the transfer functions are non-linear, this tec...